Boolean Systems: Topics in Asynchronicity 0323954227, 9780323954228

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Boolean Systems

Boolean Systems Topics in Asynchronicity Serban E. Vlad Oradea City Hall and The Society of Mathematical Sciences of Romania Oradea, Romania

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-95422-8 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara E. Conner Acquisitions Editor: Chris Katsaropoulos Editorial Project Manager: Tom Mearns Production Project Manager: Kiruthika Govindaraju Cover Designer: Christian Bilbow Typeset by VTeX

Bye bye Ciupi

Contents Preface 1. Boolean functions

xix 1

1.1. The binary Boole algebra

1

1.2. Affine spaces defined by two points

4

1.3. Boolean functions

6

1.4. Duality

7

1.5. Iterates

8

1.6. Cartesian product of functions

10

1.7. Successors and predecessors

11

1.8. Functions that are compatible with the affine structure of Bn

15

1.9. The Hamming distance. Lipschitz functions

18

2. Morphisms of generator functions

21

2.1. Definition

21

2.2. Examples of morphisms

22

2.3. Composition

23

2.4. Isomorphisms

23

2.5. Synonymous functions

24

2.6. Symmetry relative to translations

26

2.7. Morphisms vs. duality

26

2.8. Morphisms vs. iterates

28

2.9. Morphisms vs. Cartesian product of functions

28

2.10. Morphisms vs. successors and predecessors

29

2.11. Morphisms vs. fixed points

30 vii

viii

Contents

3. State portraits

31

3.1. Preliminaries

31

3.2. State portraits

32

3.3. State portraits vs. generator functions

33

3.4. Examples

35

3.5. State subportrait

36

3.6. Isomorphisms. Duality

37

3.7. Indegree, outdegree, balanced state portraits

38

3.8. Path, path-connectedness

39

3.9. Hamiltonian path, Eulerian path

40

4. Signals

41

4.1. Definition

41

4.2. Initial value and final value, initial time and final time

42

4.3. Duality

42

4.4. Monotonicity

43

4.5. Orbit, orbital equivalence

43

4.6. Omega-limit set, omega-limit equivalence

44

4.7. The forgetful function

47

4.8. The image of a signal via a function

48

4.9. Periodicity

50

5. Computation functions. Progressiveness

53

5.1. Main definitions

53

5.2. Morphisms of progressive computation functions

55

5.3. Special cases of progressive computation functions

57

6. Flows and equations of evolution

59

6.1. Flows

59

6.2. Reachability

62

6.3. Examples

62

Contents

ix

6.4. Consistency, causality and composition

64

6.5. Equations of evolution

67

6.6. Flows with constant generator functions

68

7. Systems

71

7.1. Several equivalent perspectives

71

7.2. Definition

72

7.3. Subsystem

74

7.4. Cartesian product

76

8. Morphisms of flows

79

8.1. Definition

79

8.2. Induced morphisms

81

8.3. Morphisms of generator functions vs. morphisms of flows

82

8.4. Composition

83

8.5. Isomorphisms

84

8.6. Symmetry relative to translations

86

8.7. Morphisms compatible with the subsystems

88

8.8. Morphisms vs. duality

91

8.9. Morphisms vs. orbits and omega-limit sets

91

8.10. Morphisms vs. Cartesian products

92

8.11. Morphisms vs. successors and predecessors

92

8.12. Morphisms vs. limits

93

8.13. Morphisms vs. orbital and omega-limit equivalence

94

8.14. Pseudo-morphisms

95

9. Nullclines

97

9.1. Definition

97

9.2. Examples

97

9.3. Properties 9.4. Special case: NCi

99 = Bn

100

x

Contents

10. Fixed points

103

10.1. Definition

103

10.2. Fixed points vs. final values. Rest position

104

10.3. Morphisms vs. fixed points

105

11. Sources, isolated fixed points, transient points, sinks

107

11.1. Definition

107

11.2. Morphisms

108

11.3. Other properties

109

12. Sets of reachable states

111

12.1. Convergent sequences of sets

111

12.2. Sets of reachable states

112

12.3. Example

115

12.4. Isomorphisms

116

13. Dependence on the initial conditions

119

13.1. Definition

119

13.2. Examples

122

13.3. Subsystem

124

13.4. Cartesian product

124

13.5. Isomorphisms

125

13.6. Versions of dependence on the initial conditions

125

14. Periodicity

127

14.1. Eventual periodicity and double eventual periodicity

127

14.2. Main theorems

128

14.3. Morphisms vs. periodicity

132

14.4. Other definitions of periodicity

133

15. Path-connectedness and topological transitivity 15.1. Path-connectedness

135 135

Contents

xi

15.2. Topological transitivity

136

15.3. Examples

138

15.4. Some properties

140

15.5. Morphisms

142

15.6. Cartesian products

143

15.7. Path-connected components

144

16. Chaos

145

16.1. Definition

145

16.2. Examples

147

16.3. Morphisms

148

17. Nonwandering points and Poisson stability

151

17.1. Nonwandering points

151

17.2. Poisson stability

152

17.3. Properties

153

17.4. Morphisms

154

18. Invariance

155

18.1. Definition

155

18.2. Examples

157

18.3. Invariant subset

160

18.4. Properties

160

18.5. Morphisms

167

18.6. Symmetry relative to translations

168

18.7. Subsystems

169

18.8. Cartesian products

170

18.9. Invariance and path-connectedness vs. topological transitivity

171

18.10. A Lyapunov-Lagrange type invariance theorem

172

18.11. Other possibilities of defining invariance

174

xii

Contents

19. Relatively isolated sets, isolated set

177

19.1. Definition

177

19.2. Examples

178

19.3. Properties

178

19.4. When the orbits included in invariant sets are nullclines

179

19.5. Isomorphisms

181

19.6. Subsystem

181

20. Maximal invariant subset

183

20.1. Definition

183

20.2. Examples

184

20.3. Main properties

186

20.4. Maximality vs. nullclines

190

20.5. Isomorphisms

191

20.6. Subsystems

192

20.7. Cartesian products

193

21. Minimal invariant superset

195

21.1. Definition

195

21.2. Examples

196

21.3. Properties

197

21.4. Minimality vs. nullclines

198

21.5. Isomorphisms

199

21.6. Subsystems

200

21.7. Cartesian products

201

22. Minimal invariant subset

203

22.1. Definition

203

22.2. Examples

204

22.3. Properties

206

22.4. Minimality vs. nullclines

208

Contents

xiii

22.5. Isomorphisms

208

22.6. Cartesian products

209

23. Connectedness and separation

211

23.1. Connectedness

211

23.2. Separation

212

23.3. Examples

213

23.4. Properties

214

23.5. Connectedness vs. topological transitivity

216

23.6. Connectedness vs. path-connectedness

217

23.7. Connected components

218

23.8. Isomorphisms

220

24. Basins of attraction

221

24.1. Definition

221

24.2. Examples

223

24.3. Properties

224

24.4. The basin of attraction of the fixed points

231

24.5. The basin of attraction of the periodic points

233

24.6. Isomorphisms

235

25. Basins of attraction of the states

237

25.1. Definition

237

25.2. Examples

239

25.3. Properties

241

25.4. Isomorphisms

243

26. Local basins of attraction

245

26.1. Definition

245

26.2. Properties

248

26.3. Isomorphisms

251

xiv

Contents

27. Local basins of attraction of the states

253

27.1. Definition

253

27.2. Properties

254

27.3. Isomorphisms

256

28. Attractors

259

28.1. Definition

259

28.2. Examples

260

28.3. Properties

262

28.4. Topological transitivity

265

28.5. Path-connectedness

266

28.6. Isomorphisms

267

28.7. Attractors as omega-limit sets

267

28.8. Cartesian products

271

28.9. Chaos

271

28.10. Repellers

272

28.11. Weak attractors

274

29. Stability

275

29.1. Definition

275

29.2. Examples

278

29.3. Stability vs. the basins of attraction of the fixed points

280

29.4. Morphisms

281

29.5. Subsystems

283

29.6. (In)dependence on the initial conditions

283

30. Time-reversal symmetry

285

30.1. Definition

285

30.2. Examples

288

30.3. The uniqueness of the symmetrical function

289

30.4. Properties

289

Contents

xv

30.5. Morphisms vs. time-reversal symmetry

292

30.6. Cartesian product

293

31. Generator functions with one parameter

295

31.1. Generator functions with one parameter

295

31.2. Iterates

296

31.3. Cartesian product of functions

297

31.4. Successors and predecessors

298

31.5. State portrait families

298

31.6. Bifurcations

300

31.7. Morphisms

302

32. Input flows and equations of evolution

305

32.1. Input flows

305

32.2. Causality and composition

306

32.3. Equations of evolution

309

32.4. Morphisms

310

33. Input systems

313

33.1. Several equivalent perspectives

313

33.2. Definition

314

33.3. Subsystem

315

33.4. State space decomposition

316

33.5. Cartesian product

319

33.6. Autonomy

319

34. The fundamental (operating) mode

321

34.1. An introductory remark

321

34.2. Looking for common sense requests

322

34.3. The fundamental (operating) mode

323

35. Combinational systems with one level

327

xvi

Contents

35.1. Definition

327

35.2. Examples

328

35.3. Stability

329

35.4. Cartesian product

330

35.5. Predecessors and successors

330

35.6. Isomorphisms

332

35.7. Symmetry relative to translations

332

35.8. Invariance

333

35.9. Subsystem

334

36. Combinational systems

335

36.1. Definition

335

36.2. Levels

338

36.3. Example

342

36.4. The input-output function. Stability

343

36.5. Hazards

346

36.6. Cartesian product

346

36.7. Predecessors and successors

349

36.8. Isomorphisms

351

36.9. Symmetry relative to translations

351

36.10. Invariance

353

36.11. Basins of attraction, attractors

353

36.12. Subsystem

354

36.13. The fundamental operating mode

356

37. Wires, gates, and flip flops

361

37.1. Circuits

361

37.2. The wire

361

37.3. The delay element

362

37.4. Gates

368

Contents

xvii

37.5. The SR latch

375

37.6. The gated SR flip flop

380

37.7. The D type flip flop

383

A. Continuous time

387

A.1.

Limits, signals, and computation functions

387

A.2.

Systems, several perspectives

388

B. Theory of Cheng

393

B.1.

Semi-tensor product

393

B.2.

Replacement of B with D

394

B.3.

Structure matrix

395

B.4.

Equations of evolution

400

B.5.

Example

401

C. Symbolic dynamics

405

C.1.

Blocks

405

C.2.

Shift spaces

406

C.3.

Languages

409

C.4.

The timeless model of computation

412

C.5.

The unbounded delay model of computation

417

C.6.

The bounded delay model of computation

419

Notations

425

Bibliography

429

Index

431

Preface Let  : {0, 1}n → {0, 1}n , {0, 1}n  μ → (μ) ∈ {0, 1}n that we call generator function, and the point μ ∈ {0, 1}n . The computation of (μ) may be done (a) asynchronously, when 1 (μ), ..., n (μ) are computed independently on each other, or (b) synchronously, when 1 (μ), ..., n (μ) are computed at the same time. In fact in case (a) we are not sure that (μ) is reached, as far as after the first iteration the coordinates say i (μ), i ∈ {i1 , ..., ip } are computed and the coordinates i (μ), i ∈ {1, ..., n}  {i1 , ..., ip } are not, so that the argument of  has changed. The assumption is that the durations of computation of 1 (μ), ..., n (μ) are unknown. We give the example of the function  : {0, 1}3 → {0, 1}3 , defined by Table 1 and also, equivalently, by the following directed graph G , called state portrait. Time is discrete and the arrows show the increase of time. We have underlined μi the coordinates which, after the computation of i (μ), change their value: i (μ) = μi . These coordinates are called unstable. The initial state is (0, 0, 0). Table 1

Function .

μ1

μ2

μ3

1

2

3

0

0

0

0

1

1

0

0

1

0

1

1

0

1

0

0

1

1

0

1

1

1

1

1

1

0

0

1

0

0

1

0

1

1

0

1

1

1

0

1

1

1

1

1

1

1

0

0

(0, 1, 0)  3 Q  Q  Q  Q  Q  s Q (0, 1, 1) (0, 0, 0) Q 3  Q  Q  Q  Q  s Q  (0, 0, 1)

(1, 0, 1) 6 - (1, 1, 1) 6

- (1, 0, 0)

? (1, 1, 0) xix

xx

Preface

The transfer from (0, 0, 0) to (0, 1, 1) may be done in three different ways, depending on the order that 2 (0, 0, 0), 3 (0, 0, 0) are computed. We are not interested when 1 (0, 0, 0) is computed, since 1 (0, 0, 0) = 0, stable coordinate (no change): – 2 (0, 0, 0) is computed first and 3 (0, 1, 0) is computed afterwards; in this case, the order in which 1 (0, 1, 0), 2 (0, 1, 0) are computed is irrelevant, – 3 (0, 0, 0) is computed first and 2 (0, 0, 1) is computed afterwards; in this case, the order of computation of 1 (0, 0, 1), 3 (0, 0, 1) is irrelevant, – 2 (0, 0, 0), 3 (0, 0, 0) are computed at the same time. We conclude that (0, 1, 1) = (0, 0, 0) is surely reached, in one or two iterations (could be more than two iterations, including some null effect computations). After another iteration, for which the only relevant computation is that of 1 (0, 1, 1) = 1, we get to (1, 1, 1) = (0, 1, 1), and at this moment the work of the dynamical system may bring stability, with either of (1, 0, 1) and (1, 0, 0) as rest position, or instability, with two points (1, 1, 1), (1, 1, 0) between which the system switches. The state function x : N → {0, 1}3 that we analyze depends on the initial value (0, 0, 0), the time instant k ∈ N, and also on a sequence α 0 , α 1 , α 2 , ... ∈ {0, 1}3 , called computation function, that has the meaning: ∀i ∈ {1, 2, 3}, 

αik = 1, at the k-th iteration of , i is computed, αik = 0, at the k-th iteration of , i is not computed.

In other words, x(k) represents the value of a flow x(k) = φ α (μ, k), where μ = (0, 0, 0), and we can show that x(k) represents also the solution of an equation Eq , ∀k ∈ N, ⎧ k k ⎪ ⎨ x1 (k + 1) = 1 (x(k))α1 ∪ x1 (k)α1 , x2 (k + 1) = 2 (x(k))α2k ∪ x2 (k)α2k , ⎪ ⎩ x3 (k + 1) = 3 (x(k))α3k ∪ x3 (k)α3k , (x1 (0), x2 (0), x3 (0)) = (0, 0, 0), called equation of evolution. By system, we understand any of , the state portrait G , the flow φ, and the equation Eq . These are the systems that our monograph is dedicated to, a timeful continuation of the timeless work [48]. As far as, from our point of view, a consistent mathematical theory of the Boolean asynchronous systems does not exist, our purpose is (a) to state some important topics of such a theory, as resulted from the synchronous Boolean systems theory and mostly from the real systems theory, by analogy, (b) to indicate the way that known synchronous deterministic concepts generate new asynchronous nondeterministic concepts. After several introductory chapters about the systems  : {0, 1}n → {0, 1}n , we introduce the dependence on the initial conditions, periodicity, path-connectedness, topological transitivity and chaos. A property of major importance is invariance, which is present in five versions, two of which are equivalent. In relation with it we study the maximal invariant subset, the min-

Preface

xxi

imal invariant superset, the minimal invariant subset, connectedness, separation, as well as the basins of attraction in several versions and the attractors. The stability of the systems and their time-reversal symmetry end the topics that refer to the autonomous systems, i.e. systems without input. The rest of the monograph is concerned with the input systems: the generator functions are of the form  : {0, 1}n × {0, 1}m → {0, 1}n , {0, 1}n × {0, 1}m  (μ, ν) → (μ, ν) ∈ {0, 1}n ; the state portraits become state portrait families, with a state portrait for each value of the parameter ν ∈ {0, 1}m ; the flows depend themselves on the input u : N → {0, 1}m , and so do the equations of evolution Eq . The fundamental operating mode of such a system refers to the request that the input is kept constant until the system stabilizes (until it reaches the rest position). It partially treats the uncertainties related to the computation durations of the coordinates 1 , ..., n , and on the other hand, it represents the connection between the autonomous systems (no input or, equivalently, constant input) and the input systems. Further, the combinational systems are studied (systems without feedback), and the last chapter is of applications. The first appendix addresses the issue of the continuous time, and the second one sketches the important theory of Daizhan Cheng [9], see also [3], which is put in relation with asynchronicity. The third appendix is a bridge between asynchronicity and the symbolic dynamics of [31], and [26] is useful too. From the program stated in [48] two topics have been omitted, the antimorphisms and the technical conditions of proper operation (i.e. race freedom, with its strengthenings and generalizations). This is future work. Two pioneering works in the algebraical theory of switching circuits are [35], and especially [36]. A classical work is obviously [38]. Intuition coming from the engineers is given by the PhD thesis [29], to which we add [1] and [2]. The most significant part of the bibliography consists in literature dedicated to dynamical systems theory and we translate concepts from synchronous real numbers systems to asynchronous Boolean systems by making analogies. We mention in this respect [4], [5], [6], [7], [8], [10], [12], [13], [14], [15], [16], [20], [21], [22], [23], [24], [27], [32], [34], [37], [40], [42], [43], [44], [45], [46], [47], [49], [51]. Volumes [17], [18] present a ‘broad picture of many of the core areas in the mathematical theory of dynamical systems through surveys’ as the editors say. Nice works in systems theory are also [19] and [25]. [33] was used to write a Lyapunov-Lagrange type invariance theorem. An important monograph in Boolean functions is [11], but see also [41]. [28] is an excellent survey in the time-reversal symmetry of the systems. The ‘mathematics for computer science’ from [30] proved to be useful too. In [39] we have found information about the life and the work of George Boole. [50] is a book recommended to everyone.

xxii

Preface

Our work is addressed to mathematicians and computer scientists which are interested in Boolean systems and their use in modeling. The monograph is structured in chapters, sections, sometimes subsections and also paragraphs. The paragraphs are: definitions, notations, theorems, lemmas, corollaries, proofs, remarks, examples. As any text contains (simple or complicated, important or trivial) unsolved things, we have included the paragraphs of type problem. We have tried to shorten the proofs with minimal loss of information. Finally, I wish to thank Elsevier for our good cooperation. Serban E. Vlad Oradea, March 27, 2022

1 Boolean functions George Boole (1815–1864) has the merit of incorporating logic in mathematics as algebra. He introduced the Boolean algebra in his works The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854). The term ‘Boolean algebra’ was probably used for the first time, according to Huntington, by Charles Sanders Peirce (1880) and Henry Maurice Sheffer (1913). The Boolean algebra had no practical applications until the 1930’s, when Akira Nakashima and, a bit later, Claude Elwood Shannon, started applying it in the analysis of the relay contact networks, the grand parents of the present digital electronic circuits. The binary Boole algebra B is introduced in the first section. The points μ, λ ∈ Bn define the affine space [μ, λ] ⊂ Bn of the points situated ‘between’ μ and λ. These affine spaces, from Section 1.2, characterize the monotonic evolutions of a system. The functions  : Bn → Bm are called Boolean and their study is, in a certain sense, the major topic of this work. Duality is a form of symmetry referring to concepts and properties. We mention in Section 1.4 which are the duals of the points μ ∈ Bn and of the Boolean functions  : Bn → Bm . The iterates of  : Bn → Bn from Section 1.5 are two kinds, corresponding to the simultaneous (synchronous) computations of 1 , ..., n , respectively to the independently on each other (asynchronous) computations of 1 , ..., n . In Section 1.6 we address the Cartesian product of functions, in order to fix some notations and properties. By the successors of μ ∈ Bn we mean the set of points where, by asynchronous computations of  : Bn → Bn , we can get to, starting from μ. By the predecessors of μ we mean the set of points wherefrom, by asynchronous computations of , we can arrive in μ. They are defined in Section 1.7. The functions h : Bn → Bn that fulfill h([μ, λ]) = [h(μ), h(λ)] are said to be compatible with the affine structure of Bn and they are the topic of Section 1.8. Many of our examples involve translations, and the translations are compatible with the affine structure of Bn . The Hamming distance and the Lipschitz functions are introduced in Section 1.9.

1.1 The binary Boole algebra Definition 1. The binary Boole (or Boolean) algebra is the set B = {0, 1} endowed with the following laws (see Table 2): ‘ ’ is called (logical) complement, or negation; and ‘·’ is the product, or the intersection, or the conjunction. For a, b ∈ B, a, ab are read ‘not a’, respectively ‘a and b’. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00007-6 Copyright © 2023 Elsevier Inc. All rights reserved.

1

2

Boolean Systems

Table 2 The laws of B. ·

0

1

0

1

0

0

0

1

0

1

0

1

Definition 2. B has the discrete topology, i.e. its subsets ∅, {0}, {1}, {0, 1} are open and closed at the same time. Definition 3. B is ordered by 0 < 1. Definition 4. The set Bn , n ≥ 1 has induced algebraical, topological and order structures: ∀μ ∈ Bn , ∀ν ∈ Bn , μ = (μ1 , ..., μn ), μν = (μ1 ν1 , ..., μn νn ), it has the discrete topology1 and μ ≤ ν means that μ1 ≤ ν1 and ... and μn ≤ νn . Definition 5. Let a, b ∈ B. We define a ∪ b = a b, a ⊕ b = ab ∪ ab, a −→ b = a ∪ b, a ←→ b = a b ∪ ab. ‘∪’ is the sum, or the union, or the disjunction; ‘⊕’ is the exclusive or, or the exclusive disjunction, or the modulo 2 sum, ‘−→’ is the implication, and ‘←→’ is the equivalence. For a, b ∈ B, a ∪ b, a ⊕ b, a −→ b, a ←→ b are read ‘a or b’, ‘a plus b’, ‘a implies b’, and ‘a is equivalent with b’. Remark 1. We have from Definitions 4, 5 that ∪, ⊕, −→, ←→ induce laws on Bn like , · do, i.e. coordinate-wise, for example ∀μ ∈ Bn , ∀ν ∈ Bn , μ ∪ ν = μ ν = (μ1 , ..., μn )(ν1 , ..., νn ) = (μ1 ν1 , ..., μn νn ) = (μ1 ν1 , ..., μn νn ) = (μ1 ∪ ν1 , ..., μn ∪ νn ). Remark 2. We notice that B is a Boole algebra relative to , ·, ∪ indeed. Moreover, B is a field relative to ⊕, · while Bn is a linear space over the field B: for λ ∈ B and μ, ν ∈ Bn , the sum of the vectors and the product of the vectors with scalars are given by: μ ⊕ ν = (μ1 ⊕ ν1 , ..., μn ⊕ νn ), λμ = (λμ1 , ..., λμn ). 1

The subsets of Bn are open and closed at the same time.

Chapter 1 • Boolean functions 3

Notation 1. The vectors of the canonical basis of Bn are ε i = (0, ..., 1, ..., 0) ∈ Bn , i

i ∈ {1, ..., n}. Definition 6. The families ai ∈ B, i ∈ I and x j ∈ Bn , j ∈ J are given, where I, J are arbitrary sets. We define    0, if ∃i ∈ I, ai = 0, , ai = ai = 1, 1, else i∈I

i∈∅

   1, if ∃i ∈ I, ai = 1, , ai = ai = 0, 0, else i∈I

i∈∅

j

and if ∀k ∈ {1, ..., n}, the set {j |j ∈ J, xk = 1} is finite, then we define also  x j ∈ Bn j ∈J

coordinate-wise by  (  x )k = j

j ∈J

 i∈I

ai ,

j

1, if card{j |j ∈ J, xk = 1} is odd, , (  x j )k = 0. j ∈∅ 0, otherwise

 ai ,  x j are read ‘the intersection of ai ∈ B, i ∈ I ’, ‘the union of ai ∈ B, i ∈ I ’, and i∈I

j ∈J

‘the modulo 2 sum of x j ∈ Bn , j ∈ J ’. Definition 7. The support of μ ∈ Bn , denoted supp μ, is defined by supp μ = {i|i ∈ {1, ..., n}, μi = 1}. Notation 2. For μ, λ ∈ Bn , we denote with μ  λ the set μ  λ = {i|i ∈ {1, ..., n}, μi = λi }. Remark 3. We shall use the fact that supp (μλ) = supp μ ∧ supp λ, μ  λ = supp (μ ⊕ λ) = supp μsupp λ. We have denoted with ∧ the set intersection and with  the set symmetrical difference. Theorem 1. We have μ=λ⊕

 εi . i∈μλ

(1.1.1)

4

Boolean Systems

Proof. In order to prove (1.1.1), we see that for any i ∈ {1, ..., n}, two possibilities exist. Case μi = λi Then i ∈ / μ  λ and (1.1.1) is true as μi = λi ⊕ 0. Case μi = λi Then i ∈ μ  λ, and (1.1.1) holds as μi = λi ⊕ 1.

1.2 Affine spaces defined by two points Theorem 2. The set [μ, λ] ⊂ Bn defined for μ, λ ∈ Bn by [μ, λ] = {μ ⊕  ε i |A ⊂ μ  λ} i∈A

is an affine space. Proof. The proof is straight and consists in showing that (a) the set L = {  ε i |A ⊂ μ  λ} i∈A

is a linear space, (b) the function ϕ : [μ, λ] × [μ, λ] −→ L, ∀μ ∈ [μ, λ], ∀μ ∈ [μ, λ], ϕ(μ , μ ) = μ ⊕ μ fulfills the next properties: (b.1)2 ∀μ ∈ [μ, λ], ∀μ ∈ [μ, λ], ∀μ ∈ [μ, λ], ϕ(μ , μ ) ⊕ ϕ(μ , μ ) = ϕ(μ , μ ), (b.2) ∀μ ∈ [μ, λ], ∀δ ∈ L, ∃!μ ∈ [μ, λ], ϕ(μ , μ ) = δ. We have denoted with ∃! the unique existence. Definition 8. The set [μ, λ] is called the affine space defined by the points μ, λ ∈ Bn . Theorem 3. For any μ, λ, λ , τ ∈ Bn , the affine space [μ, λ] fulfills the following properties: (a) ν ∈ [μ, λ] ⇐⇒ μ  ν ⊂ μ  λ, (b) [τ ⊕ μ, τ ⊕ λ] = τ ⊕ [μ, λ], where we have denoted τ ⊕ [μ, λ] = {τ ⊕ ν|ν ∈ [μ, λ]}, (c) [μ, λ] = [λ, μ], (d) [μ, μ] = {μ}, (e) [μ, λ] = [μ, λ ] =⇒ λ = λ , 2

Relation of Clasles.

Chapter 1 • Boolean functions 5

(f) ν ∈ [μ, λ] ⇐⇒ [ν, λ] ⊂ [μ, λ]. (g) Given the numbers 0 = n0 < n1 < ... < np = n, we denote μ1 = (μn0 +1 , ..., μn1 ) ∈ Bn1 −n0 , μ2 = (μn1 +1 , ..., μn2 ) ∈ Bn2 −n1 , ..., μp = (μnp−1 +1 , ..., μnp ) ∈ Bnp −np−1 and similarly for λ. We have [μ, λ] = [μ1 , λ1 ] × [μ2 , λ2 ] × ... × [μp , λp ]. Proof. (a) =⇒ We suppose that ν ∈ [μ, λ]. From the definition of [μ, λ], this means that A ⊂ μ  λ exists such that ν = μ ⊕  ε i . Then μ  ν = {i|i ∈ {1, ..., n}, μi = νi } = A ⊂ μ  λ. i∈A

⇐= The hypothesis states that μ  ν ⊂ μ  λ. Then (1.1.1)

ν = μ ⊕  ε i ∈ {μ ⊕  ε i |A ⊂ μ  λ} = [μ, λ]. i∈A

i∈μν

(e) We suppose that [μ, λ] = [μ, λ ], i.e. μ  λ = μ  λ . We have λ=μ⊕

 εi = μ ⊕

 ε i = λ .

i∈μλ

i∈μλ



μi ε i ⊕  ε i |A ⊂ μ  λ}

(g) We infer [μ, λ] = { ={

i∈{1,...,n}

μi ε ⊕ i



i∈{n0 +1,...,n1 }



i∈{n1 +1,...,n2 }

i∈A

μi ε i ⊕ ... ⊕



i∈{np−1 +1,...,np }

μi ε i

⊕  ε i ⊕  ε i ⊕ ... ⊕  ε i |A1 ⊂ (μ  λ) ∧ {n0 + 1, ..., n1 }, i∈A1

i∈A2

i∈Ap

A2 ⊂ (μ  λ) ∧ {n1 + 1, ..., n2 }, ..., Ap ⊂ (μ  λ) ∧ {np−1 + 1, ..., np }} ={ ×{ ×... × {



μi ε i ⊕  ε i |A1 ⊂ (μ  λ) ∧ {n0 + 1, ..., n1 }}



μi ε ⊕  ε i |A2 ⊂ (μ  λ) ∧ {n1 + 1, ..., n2 }}

i∈{n0 +1,...,n1 }

i∈{n1 +1,...,n2 }



i∈A1

i

i∈A2

μi ε i ⊕  ε i |Ap ⊂ (μ  λ) ∧ {np−1 + 1, ..., np }}

i∈{np−1 +1,...,np }

i∈Ap

= [μ , λ ] × [μ2 , λ2 ] × ... × [μp , λp ]. 1

1

Notation 3. The points μ, λ ∈ Bn are given. The following notations will be useful: [μ, λ) = [μ, λ]  {λ}, (μ, λ] = [μ, λ]  {μ}, (μ, λ) = [μ, λ]  {μ, λ}. Remark 4. [μ, λ] may be interpreted as the set containing the points that are situated ‘between’ μ and λ, some sort of ‘closed interval’ or ‘closed line segment’ with the ends μ and λ.

6

Boolean Systems

Remark 5. We get for μ, λ ∈ Bn : [μ, λ) = {μ ⊕  ε i |A  μ  λ}, i∈A

(μ, λ] = {μ ⊕  ε i |∅  A ⊂ μ  λ}, i∈A

(μ, λ) = {μ ⊕  ε i |∅  A  μ  λ}. i∈A

1.3 Boolean functions Definition 9. The functions  : Bn → Bm are called Boolean functions, n, m ≥ 1. Definition 10. Let τ ∈ Bn . The translation with τ is the function θ τ : Bn → Bn defined by ∀μ ∈ Bn , θ τ (μ) = μ ⊕ τ . Example 1. The identity 1Bn : Bn → Bn is the function defined as ∀μ ∈ Bn , 1Bn (μ) = μ. It coincides with θ (0,...,0) . Remark 6. We can rewrite Theorem 3 (b), page 4 as [θ τ (μ), θ τ (λ)] = θ τ ([μ, λ]), where we have denoted θ τ ([μ, λ]) = {θ τ (ν)|ν ∈ [μ, λ]}. A special form of this statement exists for τ = (1, ..., 1) ∈ Bn , when [μ, λ] = [μ, λ] and we have used the notation [μ, λ] = {ν|ν ∈ [μ, λ]}. Corollary 1. We have for  : Bn → Bn that ∀μ ∈ Bn , (μ) = μ ⊕



εi .

(1.3.1)

i∈μ(μ)

Proof. This follows from Theorem 1, page 3. Theorem 4. The function  : Bn → Bn is given. For any μ ∈ Bn , if [μ, (μ)] = {μ, μ ⊕  ε i , ..., μ ⊕  ε i }, i∈A0

i∈Ak

then (μ) = μ ⊕



i∈A0 ∨...∨Ak

εi .

Proof. We have A0 ⊂ μ  (μ), ..., Ak ⊂ μ  (μ) and A0 ∨ ... ∨ Ak = μ  (μ), thus (μ) = μ ⊕



εi = μ ⊕

i∈μ(μ)



i∈A0 ∨...∨Ak

εi .

Chapter 1 • Boolean functions 7

1.4 Duality Definition 11. The dual of a ∈ B is a ∗ ∈ B, a ∗ = a, and the dual of μ ∈ Bn is μ∗ ∈ Bn , μ∗ = μ. Definition 12. Given  : Bn → Bm , its dual ∗ : Bn → Bm is defined by ∀μ ∈ Bn , ∗ (μ) = (μ). Example 2. We denote with , : B → B, , , ϒ : B2 → B the functions ∀a ∈ B, ∀b ∈ B, (a) = a, (a) = a,

(a, b) = ab, (a, b) = a ∪ b, ϒ(a, b) = a ⊕ b. We have ∗ (a) = (a) = a = a, ∗ (a) = (a) = a = a,

∗ (a, b) = (a, b) = a b = a ∪ b = a ∪ b,  ∗ (a, b) = (a, b) = a ∪ b = a b = ab, ϒ ∗ (a, b) = ϒ(a, b) = ((a ⊕ 1) ⊕ (b ⊕ 1)) ⊕ 1 = a ⊕ b ⊕ 1 = a ⊕ b. Example 3. The translation function is self-dual: ∀τ ∈ Bn , (θ τ )∗ = θ τ , since ∀μ ∈ Bn , (θ τ )∗ (μ) = θ τ (μ) = μ ⊕ (1, ..., 1) ⊕ τ ⊕ (1, ..., 1) = μ ⊕ τ = θ τ (μ). Remark 7. The dual functions , ∗ have many similar properties. Remark 8. By computing the dual of the dual, we get ∀a ∈ B, ∀μ ∈ Bn , (a ∗ )∗ = a, (μ∗ )∗ = μ,

8

Boolean Systems

(∗ )∗ (μ) = ∗ (μ) = (μ) = (μ).

1.5 Iterates Definition 13. For  : Bn → Bn , the function (k) : Bn → Bn , k ∈ N called the k-th iterate of , is defined in the following way: ∀μ ∈ Bn ,  (k)



(μ) =

μ, if k = 0, if k ≥ 1.

((k−1) (μ)),

Definition 14. Given  : Bn → Bn and λ ∈ Bn , the function λ : Bn → Bn that is called the λ−iterate of  is defined by ∀μ ∈ Bn , ∀i ∈ {1, ..., n},  (λ )i (μ) =

μi , if λi = 0, i (μ), if λi = 1.

By definition, (i )λ = (λ )i , thus we shall use the symbol λi for any of (i )λ and (λ )i . Remark 9. At Definitions 13, 14 the functions (k) , λ anticipate the iterations of the generator function  of a dynamical system, that will be discussed later. Note that in (k) all the coordinates of  are iterated, these are the synchronous iterations of , while  in λ ◦ ... ◦ λ only some coordinates are iterated, namely the coordinates i , i ∈ supp λ, ..., i ∈ supp λ , and these are the asynchronous iterations of . Remark 10. The 0-th iterate (0) of , 0 ∈ N, and the (0, ..., 0)-iterate (0,...,0) of , (0, ..., 0) ∈ Bn , equal both the identity 1Bn : Bn → Bn . Theorem 5. For any k ∈ N, τ ∈ Bn and λ ∈ Bn , we get  (θ τ )(k) =

1Bn , if k is even, θ τ , if k is odd,

(θ τ )λ = θ τ λ . Proof. We prove the second statement: ∀μ ∈ Bn , ∀i ∈ {1, ..., n},  (θ τ )λi (μ) =

μi , if λi = 0, = θiτ (μ), if λi = 1



μi , if λi = 0, = μ i ⊕ τ i λi . μi ⊕ τi , if λi = 1

Theorem 6. Let  : Bn → Bn . For any k ∈ N, λ ∈ Bn , we have (∗ )(k) = ((k) )∗ , (∗ )λ = (λ )∗ .

Chapter 1 • Boolean functions 9

Proof. For any μ ∈ Bn , λ ∈ Bn and i ∈ {1, ..., n}, we can write (∗ )λi (μ) =  =



μi , if λi = 0, = ∗i (μ), if λi = 1



μi , if λi = 0, i (μ), if λi = 1

μi , if λi = 0, = λi (μ) = (λ )∗i (μ). i (μ), if λi = 1

Theorem 7. The function  : Bn −→ Bn and μ, λ ∈ Bn , p, i1 , ..., ip , q, j1 , ..., jq ∈ {1, ..., n} are given. (a) If (μ) = μ ⊕ ε i1 ⊕ ... ⊕ ε ip ,

(1.5.1)

λ (μ) = μ ⊕ λi1 ε i1 ⊕ ... ⊕ λip ε ip ;

(1.5.2)

(μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) = μ,

(1.5.3)

λ (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) = μ ⊕ λj1 ε j1 ⊕ ... ⊕ λjq ε jq ;

(1.5.4)

(μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) = μ ⊕ ε i1 ⊕ ... ⊕ ε ip ,

(1.5.5)

then

(b) if

then

(c) if

then λ (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) = μ ⊕ λj1 ε j1 ⊕ ... ⊕ λjq ε jq ⊕ λi1 ε i1 ⊕ ... ⊕ λip ε ip . Proof. (c) We suppose that (1.5.5) is true and let i ∈ {1, ..., n} arbitrary, fixed. Case i ∈ {1, ..., n}  ({i1 , ..., ip } ∨ {j1 , ..., jq })  λi (μ ⊕ ε j1

⊕ ... ⊕ ε ) = jq

 =

μi , if λi = 0, i (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ), if λi = 1

μi , if λi = 0, = μi , μi , if λi = 1

Case i ∈ {i1 , ..., ip }  {j1 , ..., jq }  λi (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) =  =

μi , if λi = 0, i (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ), if λi = 1

μi , if λi = 0, = μi ⊕ λi , μi ⊕ 1, if λi = 1

(1.5.6)

10

Boolean Systems

Case i ∈ {j1 , ..., jq }  {i1 , ..., ip }  λi (μ ⊕ ε j1

⊕ ... ⊕ ε ) = jq



μi ⊕ 1, if λi = 0, i (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ), if λi = 1

μi ⊕ 1, if λi = 0, = μ i ⊕ λi , μi , if λi = 1

= Case i ∈ {i1 , ..., ip } ∧ {j1 , ..., jq }

 λi (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ) =  =

μi ⊕ 1, if λi = 0, i (μ ⊕ ε j1 ⊕ ... ⊕ ε jq ), if λi = 1

μi ⊕ 1, if λi = 0, = μi ⊕ 1 = μi ⊕ λi ⊕ λi . μi ⊕ 1, if λi = 1

1.6 Cartesian product of functions Notation 4. For μ ∈ Bn , ν ∈ Bm , we denote with (μ, ν) the point ξ ∈ Bn+m that fulfills  ξi =

μi , if i ∈ {1, ..., n}, νi−n , if i ∈ {n + 1, ..., n + m}.

Remark 11. The previous notation is the result of the usual identification between Bn+m and Bn × Bm . When we write ξ = (μ, ν), we keep in mind this identification. Definition 15. Let the functions  : Bn+m → Bn+m ,  : Bn → Bn and : Bm → Bm . If ∀μ ∈ Bn , ∀ν ∈ Bm ,  i (μ), if i ∈ {1, ..., n}, i (μ, ν) =

i−n (ν), if i ∈ {n + 1, ..., n + m}, then  is called the Cartesian product of  and and its notation is  =  × . Remark 12. The functions  : Bn → Bn generate systems, as we shall see. If we want that  × generates a system, then we must define  × : Bn+m → Bn+m and identify Bn+m with Bn × Bm . Theorem 8. For any k ∈ N, λ ∈ Bn , and δ ∈ Bm , we have ( × )∗ = ∗ × ∗ , ( × )(k) = (k) × (k) , ( × )(λ,δ) = λ × δ .

Chapter 1 • Boolean functions 11

Proof. For arbitrary μ ∈ Bn , ν ∈ Bm , we have ( × )∗ (μ, ν) = ( × )(μ, ν) = ( × )(μ, ν) = ((μ), (ν)) = ((μ), (ν)) = (∗ (μ), ∗ (ν)) = (∗ × ∗ )(μ, ν).

1.7 Successors and predecessors Definition 16. For the function  : Bn → Bn and μ ∈ Bn , we define the following sets: μ+ = {λ (μ)|λ ∈ Bn },

(1.7.1)

O + (μ) = {(λ ◦ ... ◦ ν )(μ)|λ ∈ Bn , ..., ν ∈ Bn },

(1.7.2)

μ− = {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ},

(1.7.3)

O − (μ) = {δ|δ ∈ Bn , ∃λ ∈ Bn , ..., ∃ν ∈ Bn , (λ ◦ ... ◦ ν )(δ) = μ}

(1.7.4)

called the immediate successors of μ, the successors of μ, the immediate predecessors of μ, and the predecessors of μ. Remark 13. Since μ+ , O + (μ), μ− , O − (μ) are finite, when we have written λ ∈ Bn , ..., ν ∈ Bn we have meant finitely many such choices. Theorem 9. (a) μ ∈ μ+ ∧ O + (μ) ∧ μ− ∧ O − (μ), in particular μ+ , O + (μ), μ− , O − (μ) are nonempty, (b) we have μ+ ⊂ O + (μ) and μ− ⊂ O − (μ), (c) δ ∈ μ+ ⇐⇒ μ ∈ δ − and δ ∈ μ− ⇐⇒ μ ∈ δ + . Proof. (a) This is seen by taking λ = (0, ..., 0) in (1.7.1), ..., λ = ... = ν = (0, ..., 0) in (1.7.4), when (0,...,0) is the identity 1Bn . (c) For example: δ ∈ μ+ ⇐⇒ ∃λ ∈ Bn , δ = λ (μ) ⇐⇒ μ ∈ δ − . Theorem 10. Let  : Bn −→ Bn . For any μ ∈ Bn , (a) the following equivalencies hold: μ+ = {μ} ⇐⇒ (μ) = μ ⇐⇒ O + (μ) = {μ}, μ− = {μ} ⇐⇒ O − (μ) = {μ}; (b) we have μ+ = [μ, (μ)].

(1.7.5)

12

Boolean Systems

Proof. (a) We prove the implication μ+ = {μ} =⇒ O + (μ) = {μ}. If {λ (μ)|λ ∈ Bn } = {μ}, then ∀λ ∈ Bn , λ (μ) = μ and we take an arbitrary δ ∈ O + (μ). We have the existence of λ ∈ Bn , ..., ν ∈ Bn , ξ ∈ Bn with δ = (λ ◦ ... ◦ ν ◦ ξ )(μ), therefore δ = (λ ◦ ... ◦ ν )(ξ (μ)) = (λ ◦ ... ◦ ν )(μ) = ... = λ (μ) = μ. (b) We prove first μ+ ⊂ [μ, (μ)]. Let δ ∈ μ+ arbitrary, thus λ ∈ Bn exists with δ = λ (μ), in other words δ=μ⊕

εi .



(1.7.6)

i∈μλ (μ)

But μ  λ (μ) = {i|i ∈ {1, ..., n}, μi = λi (μ)}  μi , if λi = 0, = {i|i ∈ {1, ..., n}, μi = } i (μ), if λi = 1 = {i|i ∈ {1, ..., n}, λi = 1 and μi = i (μ)} ⊂ {i|i ∈ {1, ..., n}, μi = i (μ)} = μ  (μ), (1.7.6)

wherefrom δ ∈ [μ, (μ)]. We prove now [μ, (μ)] ⊂ μ+ . We take an arbitrary δ ∈ [μ, (μ)], therefore A ⊂ μ  (μ) exists such that δ = μ ⊕  εi .

(1.7.7)

i∈A

We define λ ∈ Bn by ∀i ∈ {1, ..., n},  λi =

0, if i ∈ / A, 1, if i ∈ A

and we infer: ∀i ∈ {1, ..., n},  λi (μ) =  =

μi , if λi = 0, = i (μ), if λi = 1

/ A, μi , if i ∈ = i (μ), if i ∈ A and μi = i (μ)

We infer that δ ∈ μ+ .

 

/ A, μi , if i ∈ i (μ), if i ∈ A / A, μi , if i ∈ μi ⊕ 1, if i ∈ A

(1.7.7)

= δi .

Chapter 1 • Boolean functions 13

Theorem 11. If  is the constant function ∀μ ∈ Bn , (μ) = μ , then μ+ = [μ, μ ], μ− = [μ , μ]. Proof. The first statement results from (1.7.5). We prove μ− ⊂ [μ , μ]. Let ν ∈ μ− arbitrary, thus λ ∈ Bn exists such that λ (ν) = μ. We suppose against all reason that ν ∈ / [μ , μ], i.e. ∃i ∈ {1, ..., n}, νi = μi = μi . We have obtained the contradiction  νi , if λi = 0, λ i (ν) = = μi . μi , if λi = 1 We prove [μ , μ] ⊂ μ− . We take ν ∈ [μ , μ] arbitrary, ν = μ ⊕  εi ,

(1.7.8)

∀i ∈ A, μi = μi .

(1.7.9)

supp λ = A.

(1.7.10)

i∈A

where A ⊂ μ  μ , i.e. We define λ ∈ Bn by We have ∀i ∈ {1, ..., n},  νi , if λi = 0, λi (ν) = μi , if λi = 1

(1.7.10)

=



/ A, νi , if i ∈ μi , if i ∈ A

(1.7.8)



=

/ A, μi , if i ∈ μi , if i ∈ A

(1.7.9)

= μi .

Theorem 12. We can write for μ ∈ Bn , + (μ)+ ∗ = μ  ,

(1.7.11)

+ + O ∗ (μ) = O (μ),

(1.7.12)

− (μ)− ∗ = μ  ,

(1.7.13)

− − O ∗ (μ) = O (μ).

(1.7.14)

Proof. (1.7.11): If we put in Theorem 3 (b), page 4 τ = (1, ..., 1) ∈ Bn (θ τ is the complement) and λ = (μ), we infer (μ)+ ∗

Theorem 10

=

[μ, ∗ (μ)] = [μ, (μ)] = [μ, (μ)]

Theorem 3

=

[μ, (μ)]

Theorem 10

=

μ+ .

14

Boolean Systems

(1.7.13): n n ∗ λ n n λ ∗ (μ)− ∗ = {δ|δ ∈ B , ∃λ ∈ B , ( ) (δ) = μ} = {δ|δ ∈ B , ∃λ ∈ B , ( ) (δ) = μ}

= {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ} = {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ} = {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ} = {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ} = μ− . + Theorem 13. The functions , : Bn −→ Bn are given. We have ∀μ ∈ Bn , μ+  = μ if and only if  = .

Proof. If: Obvious. Only if: We put in Theorem 3 (e), page 4 λ = (μ), λ = (μ). Problem 1. Let , : Bn −→ Bn two functions. An interesting question is which of the statements  = , + ∀μ ∈ Bn , μ+  = μ , − ∀μ ∈ Bn , μ−  = μ , + + ∀μ ∈ Bn , O (μ) = O (μ), − − ∀μ ∈ Bn , O (μ) = O (μ)

are equivalent. Theorem 14. The successors of the successors of μ are successors of μ, and the predecessors of the predecessors of μ are predecessors of μ: ∀μ ∈ O + (μ), O + (μ ) ⊂ O + (μ), ∀μ ∈ O − (μ), O − (μ ) ⊂ O − (μ). Proof. We prove the second inclusion and we take μ ∈ O − (μ), δ ∈ O − (μ ) arbitrary. These mean the existence of λ ∈ Bn , ..., ν ∈ Bn , and also of ρ ∈ Bn , ..., ξ ∈ Bn such that (λ ◦ ... ◦ ν )(μ ) = μ, (ρ ◦ ... ◦ ξ )(δ) = μ . We infer μ = (λ ◦ ... ◦ ν )(μ ) = (λ ◦ ... ◦ ν ◦ ρ ◦ ... ◦ ξ )(δ), therefore δ ∈ O − (μ).

Chapter 1 • Boolean functions 15

Theorem 15. The functions  : Bn −→ Bn , : Bm −→ Bm are given. For any μ ∈ Bn , ν ∈ Bm we have: + + (μ, ν)+ × = μ × ν ,

(1.7.15)

− − (μ, ν)− × = μ × ν ,

(1.7.16)

+ + + (μ, ν) = O (μ) × O (ν), O×

(1.7.17)

− − − O× (μ, ν) = O (μ) × O (ν).

(1.7.18)

Proof. We prove (1.7.15): (λ,ζ ) (μ, ν)+ (μ, ν)|(λ, ζ ) ∈ Bn × Bm } × = {( × )

= {(λ × ζ )(μ, ν)|λ ∈ Bn , ζ ∈ Bm } = {(λ (μ), ζ (ν))|λ ∈ Bn , ζ ∈ Bm } + = {λ (μ)|λ ∈ Bn } × { ζ (ν)|ζ ∈ Bm } = μ+  × ν ,

and the proof of (1.7.16) is similar: n m n m (λ,ζ ) (μ, ν)− (δ, ρ) = (μ, ν)} × = {(δ, ρ)|(δ, ρ) ∈ B × B , ∃(λ, ζ ) ∈ B × B , ( × )

= {(δ, ρ)|(δ, ρ) ∈ Bn × Bm , ∃(λ, ζ ) ∈ Bn × Bm , (λ × ζ )(δ, ρ) = (μ, ν)} = {(δ, ρ)|(δ, ρ) ∈ Bn × Bm , ∃(λ, ζ ) ∈ Bn × Bm , (λ (δ), ζ (ρ)) = (μ, ν)} = {(δ, ρ)|δ ∈ Bn , ρ ∈ Bm , ∃λ ∈ Bn , ∃ζ ∈ Bm , λ (δ) = μ, ζ (ρ) = ν} − = {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = μ} × {ρ|ρ ∈ Bm , ∃ζ ∈ Bm , ζ (ρ) = ν} = μ−  × ν .

1.8 Functions that are compatible with the affine structure of Bn Definition 17. A function h : Bn −→ Bn that satisfies ∀μ ∈ Bn , ∀λ ∈ Bn , h([μ, λ]) = [h(μ), h(λ)]

(1.8.1)

is called compatible with the affine structure of Bn . The set of the functions which are compatible with the affine structure of Bn is denoted by Af (Bn ). Theorem 16. The function h : Bn −→ Bn defined by ∀μ ∈ Bn , h(μ) =



ai μi ε i ⊕ τ,

i∈{1,...,n}

where a, τ ∈ Bn are given, satisfies h ∈ Af (Bn ).

(1.8.2)

16

Boolean Systems

Proof. Let μ, λ ∈ Bn arbitrary, fixed. We see first of all that h(μ) =

ε i ⊕ τ,



i∈supp a∧supp μ

supp h(μ) = supp a ∧ supp μsupp τ.

(1.8.3)

We show that h([μ, λ]) ⊂ [h(μ), h(λ)] and let δ ∈ [μ, λ] arbitrary, thus A ⊂ μ  λ exists such that δ = μ ⊕  εj . j ∈A

On one hand h(δ) = h(μ ⊕  ε j ) = j ∈A

=





ai (



j ∈{1,...,n}

i∈{1,...,n}

μj ε j ⊕  ε j )i ε i ⊕ τ j ∈A

ai μi ε i ⊕ τ ⊕  ai ε i = h(μ) ⊕ i∈A

i∈{1,...,n}



i∈supp a∧A

(1.8.4)

εi ,

and on the other hand supp a ∧ A ⊂ supp a ∧ (μ  λ)

Remark 3, page 3

=

supp a ∧ (supp μsupp λ)

= supp a ∧ supp μsupp a ∧ supp λ = supp h(μ)supp τ supp h(λ)supp τ = supp h(μ)supp h(λ)

Remark 3, page 3

=

h(μ)  h(λ),

thus we have obtained that h(δ) ∈ [h(μ), h(λ)]. In order to prove the inclusion [h(μ), h(λ)] ⊂ h([μ, λ]), we notice first, by making a computation which is similar with the previous one from (1.8.4), that h([μ, λ]) = h({μ ⊕  ε i |H ⊂ μ  λ}) i∈H

= {h(μ ⊕  ε i )|H ⊂ μ  λ} = {h(μ) ⊕ i∈H



i∈supp a∧H

ε i |H ⊂ μ  λ}.

(1.8.5)

Let now δ ∈ [h(μ), h(λ)] arbitrary, i.e. A ⊂ h(μ)  h(λ) = supp a ∧ (μ  λ) exists with δ = h(μ) ⊕  ε i .

(1.8.6)

A ⊂ supp a,

(1.8.7)

i∈A

We have

Chapter 1 • Boolean functions 17

A ⊂ μ  λ,

(1.8.8)

(1.8.7)

A = supp a ∧ A,

(1.8.9)

therefore δ

(1.8.6), (1.8.9)

=

h(μ) ⊕



i∈supp a∧A

εi

(1.8.5), (1.8.8)

∈

h([μ, λ]).

Theorem 17. Let s : {1, ..., n} −→ {1, ..., n} a bijective function and we define h : Bn −→ Bn by ∀μ ∈ Bn , h(μ) =

μs(i) ε i .



i∈{1,...,n}

We have h ∈ Af (Bn ). Proof. Let μ, λ ∈ Bn arbitrary. We notice first that h(μ) =



εi ,

i∈s(supp μ)

supp h(μ) = s(supp μ), and [h(μ), h(λ)] = {h(μ) ⊕  ε i |A ⊂ h(μ)  h(λ)} i∈A

= {h(μ) ⊕  ε i |A ⊂ (supp h(μ)supp h(λ))} i∈A

= {h(μ) ⊕  ε i |A ⊂ (s(supp μ)s(supp λ))} i∈A

= {h(μ) ⊕  ε i |A ⊂ s(supp μsupp λ)}. i∈A

(1.8.10)

We infer that h([μ, λ]) = h({μ ⊕  ε i |A ⊂ μ  λ}) i∈A

= {h(μ ⊕  ε )|A ⊂ supp μsupp λ} = {h(μ) ⊕ h(  ε i )|A ⊂ supp μsupp λ} i

i∈A

i∈A

= {h(μ) ⊕  ε i |A ⊂ supp μsupp λ} i∈s(A)

= {h(μ) ⊕  ε i |s −1 (A) ⊂ supp μsupp λ} i∈A

= {h(μ) ⊕  ε |A ⊂ s(supp μsupp λ)} i

i∈A

(1.8.10)

=

[h(μ), h(λ)].

Example 4. The identity function 1Bn : Bn −→ Bn is compatible with the affine structure of Bn . It results from (1.8.2) for a1 = ... = an = 1 and τ1 = ... = τn = 0.

18

Boolean Systems

Example 5. The translation with τ ∈ Bn , θ τ : Bn −→ Bn satisfies θ τ ∈ Af (Bn ). It results from (1.8.2) when a1 = ... = an = 1. Example 6. We have that the constant function h : Bn −→ Bn equal with τ ∈ Bn satisfies h ∈ Af (Bn ). This is the special case of (1.8.2) when we have a1 = ... = an = 0. Example 7. We give now an example of function that is not compatible with the affine structure of B2 , namely h : B2 −→ B2 , ∀μ ∈ B2 , h(μ1 , μ2 ) = (μ1 μ2 , μ1 μ2 ). We can see that h([(0, 0), (1, 1)]) = h(B2 ) = {(0, 0), (1, 1)} = B2 = [(0, 0), (1, 1)] = [h(0, 0), h(1, 1)]. Remark 14. If h, g ∈ Af (Bn ) then h ◦ g ∈ Af (Bn ), i.e. Af (Bn ) is a semigroup relative to the composition of the functions, whose unit is 1Bn . Theorem 18. We suppose that h ∈ Af (Bn ) is bijective. Then ∀μ ∈ Bn , ∀λ ∈ Bn , h((μ, λ)) = (h(μ), h(λ)). Proof. We get h((μ, λ)) ⊂ h([μ, λ]) = [h(μ), h(λ)]. We suppose against all reason that the statement of the theorem is false, thus ξ ∈ (μ, λ) exists such that h(ξ ) = h(μ) or h(ξ ) = h(λ), where ξ = μ and ξ = λ. Both possibilities are in contradiction with the bijectivity of h. Theorem 19. If h ∈ Af (Bn ) and h is bijective, then h−1 ∈ Af (Bn ). Proof. Let μ, λ ∈ Bn arbitrary and we denote μ = h−1 (μ), λ = h−1 (λ). We infer h−1 ([μ, λ]) = h−1 ([h(μ ), h(λ )]) = h−1 (h([μ , λ ])) = [μ , λ ] = [h−1 (μ), h−1 (λ)].

1.9 The Hamming distance. Lipschitz functions Definition 18. The function d : Bn × Bn −→ {0, 1, ..., n} defined by ∀μ ∈ Bn , ∀λ ∈ Bn , d(μ, λ) = card(μ  λ) is called the Hamming distance (between μ and λ). Theorem 20. For any μ ∈ Bn , λ ∈ Bn , we have card([μ, λ]) = 2d(μ,λ) . Proof. We infer for arbitrary μ, λ: card([μ, λ]) = card({μ ⊕  ε i |A ⊂ μ  λ}) i∈A

= card({A|A ⊂ μ  λ}) = 2card(μλ) = 2d(μ,λ) .

Chapter 1 • Boolean functions 19

Definition 19. The function h : Bn −→ Bn is called Lipschitz if ∀μ ∈ Bn , ∀λ ∈ Bn , d(h(μ), h(λ)) ≤ d(μ, λ)

(1.9.1)

is true, where ≤ is the inequality of the natural numbers. Example 8. Let a, τ ∈ Bn and the function h : Bn −→ Bn defined by ∀μ ∈ Bn , h(μ) =  ai μi ε i ⊕ τ . We have, see the proof of Theorem 16, page 15,

i∈{1,...,n}

d(h(μ), h(λ)) = card(h(μ)  h(λ)) = card(supp h(μ)supp h(λ)) (1.8.3)

= card(supp a ∧ supp μsupp τ supp a ∧ supp λsupp τ ) = card(supp a ∧ (supp μsupp λ)) = card(supp a ∧ (μ  λ)) ≤ card(μ  λ) = d(μ, λ),

i.e. h is Lipschitz. The special cases when h is the identity, the translation with τ and the constant function equal with τ are Lipschitz too. For example, we have d(τ ⊕ μ, τ ⊕ λ) = card(τ ⊕ μ  τ ⊕ λ) = card(μ  λ) = d(μ, λ).

(1.9.2)

In particular τ = (1, ..., 1) transforms this last equality in d(μ, λ) = d(μ, λ).

(1.9.3)

Theorem 21. We suppose that h is compatible with the affine structure of Bn . Then (a) h is Lipschitz; (b) if h is bijective, then ∀μ ∈ Bn , ∀λ ∈ Bn , d(μ, λ) = d(h(μ), h(λ)).

(1.9.4)

Proof. Let μ ∈ Bn , λ ∈ Bn arbitrary. (a) We have card([μ, λ]) ≥ card(h([μ, λ])).

(1.9.5)

We infer 2d(μ,λ)

Theorem 20

(1.8.1)

=

(1.9.5)

card([μ, λ]) ≥ card(h([μ, λ]))

= card([h(μ), h(λ)])

Theorem 20 d(h(μ),h(λ))

=

2

,

thus (1.9.1) is true. (b) The bijectivity of h makes (1.9.5) be replaced by card([μ, λ]) = card(h([μ, λ])), thus 2d(μ,λ) = card([μ, λ]) = card(h([μ, λ])) = card([h(μ), h(λ)]) = 2d(h(μ),h(λ)) .

20

Boolean Systems

We get the truth of (1.9.4). Example 9. If h = θ τ (the translation with τ ∈ Bn ) then (1.9.4) is satisfied, this is the meaning of (1.9.2) and (1.9.3). Example 10. The function from Theorem 17, page 17 is bijective, compatible with the affine structure of Bn and Lipschitz. It fulfills (1.9.4).

2 Morphisms of generator functions The morphisms from  : Bn → Bn to  : Bm → Bm consist in two functions h, h : Bn → Bm with the property that a certain diagram is commutative. These morphisms, denoted (h, h ) :  →  or (h, h ) ∈ H om(, ), make some properties of  be transferred to . In Sections 2.1, 2.2 we give their definition and some examples. The composition of morphisms is a morphism, as we prove in Section 2.3. The isomorphisms (h, h ) ∈ I so(, ) are these morphisms (h, h ) ∈ H om(, ) for which h and h are bijections. They are introduced in Section 2.4 and two special cases of isomorphisms are highlighted in Sections 2.5, 2.6, when ,  are identical modulo the order of their coordinates, respectively when h is a translation. We get properties like: (h, h ) ∈ H om(, ) ⇐⇒ (h∗ , h ) ∈ H om(∗ ,  ∗ ), in Section 2.7 (Morphisms vs. duality); a natural morphism (1Bn , h ) ∈ H om(ν , ) is defined, ν ∈ Bn , in Section 2.8 (Morphisms vs. iterates); two natural morphisms (π, π) ∈ H om( × , ), (π  , π  ) ∈ H om( × , ) exist, in Section 2.9 (Morphisms vs. Cartesian product of functions), where π, π  are projections; morphisms (h, h ) ∈ H om(, ) bring the successors + + + − − μ+  , O (μ) in successors h(μ) , O (h(μ)) and the predecessors μ , O (μ) in predeces− − sors h(μ) , O (h(μ)), in Section 2.10 (Morphisms vs. successors and predecessors), where μ ∈ Bn ; morphisms (h, h ) ∈ H om(, ) bring the fixed points (μ) = μ in fixed points (h(μ)) = h(μ), in Section 2.11 (Morphisms vs. fixed points).

2.1 Definition Definition 20. A morphism (of generator functions) from  : Bn → Bn to  : Bm → Bm , denoted (h, h ) :  → , consists in two functions h, h : Bn → Bm having the property that ∀λ ∈ Bn , the diagram Bn

λ - n B

h ? Bm



h (λ)

h ? - Bm

is commutative. Notation 5. The set of the morphisms from  to  has the notation H om(, ). Remark 15. The terminology of morphism of generator functions makes sense in the context when (a) the functions ,  are called generator functions (of a system) and (b) other Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00008-8 Copyright © 2023 Elsevier Inc. All rights reserved.

21

22

Boolean Systems

similar morphisms exist also, i.e. the morphisms of flows, and a distinction is necessary between the two kinds of morphisms. Things will become more precise in the next chapters. Remark 16. The existence of a morphism (h, h ) :  →  means that some properties of  are transmitted to , thus  and  behave similarly.

2.2 Examples of morphisms Example 11. For any  : Bn → Bn , we have the morphism (1Bn , 1Bn ) :  −→ . The usual notation of this morphism is 1 . Example 12. We suppose that  : Bn → Bn is arbitrary and h : Bn → Bn is the constant null function: ∀μ ∈ Bn , h (μ) = (0, ..., 0) ∈ Bn . Then (1Bn , h ) : 1Bn →  is a morphism since for any μ ∈ Bn , λ ∈ Bn we can write 

(1Bn ◦ (1Bn )λ )(μ) = (1Bn )(μ) = μ = (0,...,0) (μ) = h (λ) (μ) 



= h (λ) (1Bn (μ)) = (h (λ) ◦ 1Bn )(μ). Example 13. We take  : Bn → Bn ,  : Bm → Bm arbitrary, h : Bn → Bm constant: ∃μ ∈ Bm , ∀μ ∈ Bn , h(μ) = μ and h : Bn → Bm constant null: ∀μ ∈ Bn , h (μ) = (0, ..., 0) ∈ Bm . We get ∀μ ∈ Bn , ∀λ ∈ Bn , 

h(λ (μ)) = μ =  (0,...,0) (μ ) =  h (λ) (h(μ)), resulting that (h, h ) :  →  is a morphism. Remark 17. One of the meanings of Example 13 is that for any  : Bn → Bn ,  : Bm → Bm we have H om(, ) = ∅. Example 14. For the functions ,  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 , μ2 ) = (μ1 , μ2 ⊕ 1), (μ1 , μ2 ) = (μ1 , μ1 μ2 ∪ μ1 μ2 ) = (μ1 , μ1 ⊕ μ2 ⊕ 1), we have the morphism (h, h ) ∈ H om(, ) given by h(μ1 , μ2 ) = (μ1 , μ1 ∪ μ2 ) = (μ1 , μ1 ⊕ μ2 ⊕ μ1 μ2 ) and h (μ1 , μ2 ) = (μ1 , μ2 ) : ∀μ ∈ B2 , ∀λ ∈ B2 , h(λ (μ)) = h(μ1 , (λ2 ⊕ 1)μ2 ⊕ λ2 (μ2 ⊕ 1)) = h(μ1 , μ2 ⊕ λ2 ) = (μ1 , μ1 ⊕ μ2 ⊕ λ2 ⊕ μ1 μ2 ⊕ μ1 λ2 ),  λ (h(μ)) =  λ (μ1 , μ1 ⊕ μ2 ⊕ μ1 μ2 ) = (μ1 , (λ2 ⊕ 1)(μ1 ⊕ μ2 ⊕ μ1 μ2 ) ⊕ λ2 (μ2 ⊕ μ1 μ2 ⊕ 1)) = (μ1 , λ2 μ1 ⊕ λ2 μ2 ⊕ λ2 μ1 μ2 ⊕ μ1 ⊕ μ2 ⊕ μ1 μ2 ⊕ λ2 μ2 ⊕ λ2 μ1 μ2 ⊕ λ2 ).

Chapter 2 • Morphisms of generator functions 23

2.3 Composition Theorem 22. The functions  : Bn → Bn ,  : Bm → Bm ,  : Bp → Bp , h, h : Bn → Bm , f, f  : Bm → Bp are given. The following implication holds: (h, h ) :  → , (f, f  ) :  →  =⇒ (f ◦ h, f  ◦ h ) :  → . Proof. This happens because the hypothesis of the theorem makes λ - n B

Bn h ? Bm



h (λ)

h ? - Bm

f

f ? f  (h (λ)) ?  - Bp Bp

all the diagrams commute for any λ ∈ Bn . Definition 21. The composition of the morphisms (h, h ) :  → , (f, f  ) :  →  is the morphism (f, f  ) ◦ (h, h ) :  →  defined by (f, f  ) ◦ (h, h ) = (f ◦ h, f  ◦ h ).

2.4 Isomorphisms Definition 22. If at Definition 20, page 21 h, h are bijections, then (h, h ) is called isomorphism, and ,  are said to be topologically equivalent, or conjugated; if, in addition,  = , then (h, h ) is called automorphism. Notation 6. The sets of isomorphisms from  to  and of automorphisms of  are denoted with I so(, ) and Aut (). Theorem 23. We consider the functions , ,  : Bn → Bn . We have (a) If (h, h ) ∈ I so(, ) and (f, f  ) ∈ I so(, ), then (f, f  ) ◦ (h, h ) ∈ I so(, ); (b) if (h, h ) ∈ I so(, ), then (h−1 , h−1 ) ∈ I so(, ); (c) Aut () is a group, where the multiplication is the composition of the isomorphisms, the identity is (1Bn , 1Bn ) :  → , and the inverse of (h, h ) is (h−1 , h−1 ). Proof. (a) (f, f  ) ◦ (h, h ) is a morphism from Theorem 22 and f ◦ h, f  ◦ h are bijections. (b) Let λ ∈ Bn , μ ∈ Bn arbitrary and we use the notations μ = h−1 (μ ),

(2.4.1)

24

Boolean Systems

λ = h−1 (λ ),

(2.4.2)

μ = λ (μ).

(2.4.3)

The hypothesis states that 

(h ◦ λ )(μ) = ( h (λ) ◦ h)(μ),

(2.4.4)

i.e. (2.4.3)

(2.4.4)



(2.4.2)



(2.4.1)



h(μ ) = h(λ (μ)) =  h (λ) (h(μ)) =  λ (h(μ)) =  λ (μ ),

(2.4.5)

thus we can write that 



(2.4.3)

(2.4.5)

(h−1 ◦  λ )(μ ) = h−1 ( λ (μ )) = h−1 (h(μ )) = μ = λ (μ) (2.4.2)

−1 (λ )

= h

(2.4.1)

−1 (λ )

(μ) = h

(h−1 (μ )).

As λ , μ were arbitrarily chosen, the truth of (b) follows. Example 15. The identity 1 :  →  is an automorphism. Remark 18. The isomorphic functions have an increased number of common properties. Several examples of isomorphisms will occur in our work, of which the synonymous functions and the symmetrical functions relative to translations are the topics of the next two sections.

2.5 Synonymous functions Theorem 24. The function  : Bn → Bn and the bijection s : {1, ..., n} → {1, ..., n} are given. We define js : Bn → Bn by ∀μ ∈ Bn , js (μ1 , ..., μn ) = (μs(1) , ..., μs(n) ). Then  : Bn → Bn ,  = js ◦  ◦ js −1 fulfills (js , js ) ∈ I so(, ). Proof. We remark first that js is bijection, thus the statement (js , js ) ∈ I so(, ) makes sense. In addition, we have js −1 = (js )−1 since ∀μ ∈ Bn , (js −1 ◦ js )(μ) = js −1 (js (μ)) = js −1 (μs(1) , ..., μs(n) ) = (μs −1 (s(1)) , ..., μs −1 (s(n)) ) = (μ1 , ..., μn ) = 1Bn (μ) = ((js )−1 ◦ js )(μ), and we multiply js −1 ◦ js = (js )−1 ◦ js at the right with (js )−1 .

Chapter 2 • Morphisms of generator functions 25

We fix now μ, λ ∈ Bn arbitrary, and we must prove that (js ◦  ◦ js −1 )js (λ) (js (μ)) = (js ◦ λ )(μ). We have (js ◦  ◦ js −1 )js (λ) (js (μ))  (js ◦  ◦ js −1 )1 (js (μ)), if λs(1) = 1, =( , ..., μs(1) , if λs(1) = 0  (js ◦  ◦ js −1 )n (js (μ)), if λs(n) = 1, ) μs(n) , if λs(n) = 0  (js ◦ )1 ((js )−1 ◦ js (μ)), if λs(1) = 1, =( , ..., μs(1) , if λs(1) = 0  (js ◦ )n ((js )−1 ◦ js (μ)), if λs(n) = 1, ) μs(n) , if λs(n) = 0   (js ◦ )1 (μ), if λs(1) = 1, (js ◦ )n (μ), if λs(n) = 1, =( , ..., ) μs(1) , if λs(1) = 0 μs(n) , if λs(n) = 0   s(1) (μ), if λs(1) = 1, s(n) (μ), if λs(n) = 1, =( , ..., ) μs(1) , if λs(1) = 0 μs(n) , if λs(n) = 0   n (μ), if λn = 1, 1 (μ), if λ1 = 1, = js ( , ..., ) = (js ◦ λ )(μ). μ1 , if λ1 = 0 μn , if λn = 0 Definition 23. The functions ,  : Bn −→ Bn are called synonymous, if the bijection s : {1, ..., n} → {1, ..., n} exists such that  = js ◦  ◦ js −1 . Remark 19. The interpretation of Theorem 24 is:  coincides with  modulo the order of its coordinates. Example 16. We give the following example.  : B3 → B3 , ∀μ ∈ B3 ,  s:

1 2 3 3 1 2



, s −1 :



(μ1 , μ2 , μ3 ) = (μ1 μ2 , μ3 , μ2 ),  1 2 3 ,  : B3 → B3 , ∀μ ∈ B3 , 2 3 1

(μ1 , μ2 , μ3 ) = js ((js −1 (μ))) = js ((μ2 , μ3 , μ1 )) = js (μ2 μ3 , μ1 , μ3 ) = (μ3 , μ2 μ3 , μ1 ). For any μ, λ ∈ B3 we have (js ◦ λ )(μ) = js (λ1 μ1 ∪ λ1 μ1 μ2 , λ2 μ2 ∪ λ2 μ3 , λ3 μ3 ∪ λ3 μ2 ) = (λ3 μ3 ∪ λ3 μ2 , λ1 μ1 ∪ λ1 μ1 μ2 , λ2 μ2 ∪ λ2 μ3 ), ( js (λ) ◦ js )(μ) =  (λ3 ,λ1 ,λ2 ) (μ3 , μ1 , μ2 ) = (λ3 μ3 ∪ λ3 μ2 , λ1 μ1 ∪ λ1 μ1 μ2 , λ2 μ2 ∪ λ2 μ3 ).

26

Boolean Systems

2.6 Symmetry relative to translations Definition 24. We call the functions ,  : Bn −→ Bn symmetrical relative to the translation with τ ∈ Bn , if h : Bn −→ Bn exists with the property that (θ τ , h ) ∈ I so(, ) and (θ τ , h ) = (1Bn , 1Bn ). Definition 25. The function  : Bn −→ Bn is said to be symmetrical relative to the translation with τ , if h : Bn −→ Bn exists such that (θ τ , h ) ∈ Aut () and (θ τ , h ) = (1Bn , 1Bn ). Remark 20. Note that the translations θ τ are bijections, therefore the statements (θ τ , h ) ∈ I so(, ), (θ τ , h ) ∈ Aut () make sense. Example 17. The functions ,  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 ∪ μ2 , μ1 ∪ μ2 ), (μ1 , μ2 ) = (μ1 μ2 , μ1 ∪ μ2 ) fulfill (θ (1,0) , 1B2 ) ∈ I so(, ) : ∀μ ∈ B2 , ∀λ ∈ B2 , θ (1,0) ((λ1 ,λ2 ) (μ1 , μ2 )) = θ (1,0) (λ1 μ1 ∪ λ1 (μ1 ∪ μ2 ), λ2 μ2 ∪ λ2 (μ1 ∪ μ2 )) = θ (1,0) (μ1 ∪ λ1 μ2 , μ2 ∪ λ2 μ1 ) = (μ1 ∪ λ1 μ2 , μ2 ∪ λ2 μ1 ) = (μ1 (λ1 ∪ μ2 ), μ2 ∪ λ2 μ1 ),  (λ1 ,λ2 ) (θ (1,0) (μ1 , μ2 )) =  (λ1 ,λ2 ) (μ1 , μ2 ) = (λ1 μ1 ∪ λ1 μ1 μ2 , λ2 μ2 ∪ λ2 (μ1 ∪ μ2 )) = (λ1 μ1 ∪ λ1 μ1 μ2 , μ2 ∪ λ2 μ1 ).

2.7 Morphisms vs. duality Remark 21. The translation with (1, ..., 1) ∈ Bn has the meaning of the complement: ∀μ ∈ Bn , θ (1,...,1) (μ) = (μ1 , ..., μn ) ⊕ (1, ..., 1) = (μ1 ⊕ 1, ..., μn ⊕ 1) = (μ1 , ..., μn ) = μ, and this helps proving that the duality of the functions is equivalent with the existence of the isomorphism (θ (1,...,1) , 1Bn ). Theorem 25. The functions ,  : Bn −→ Bn satisfy:  is the dual of  if and only if (θ (1,...,1) , 1Bn ) ∈ I so(, ). Proof. If. (θ (1,...,1) , 1Bn ) ∈ I so(, ) means that ∀μ ∈ Bn , ∀λ ∈ Bn , (θ (1,...,1) ◦ λ )(μ) =  λ (θ (1,...,1) (μ)),

Chapter 2 • Morphisms of generator functions 27

in other words λ (μ) =  λ (μ), λ (μ) =  λ (μ). For λ = (1, ..., 1) ∈ Bn , the last equation becomes  =  ∗ , i.e. 

Remark 8, page 7

=

( ∗ )∗ = ∗ .

Remark 22. As (θ (1,...,1) , 1Bn ) ∈ I so(, ∗ ), we get that (θ (1,...,1) , 1Bn )−1 = (θ (1,...,1) , 1Bn ) ∈ I so(∗ , ). An interesting situation is that of self-duality, when  = ∗ , for example the function (μ1 , μ2 ) = (μ1 , μ2 ) is in this situation, we have (θ (1,1) , 1B2 ) ∈ Aut (). Theorem 26. For  : Bn −→ Bn and  : Bm −→ Bm , the following equivalence holds: (h, h ) ∈ H om(, ) ⇐⇒ (h∗ , h ) ∈ H om(∗ ,  ∗ ). Proof. =⇒ We denote τ = (1, ..., 1) ∈ Bn and τ  = (1, ..., 1) ∈ Bm . We take into account that   (θ τ )−1 = θ τ and we consider the following diagram Bn

∗λ Q Q Qθτ Q Q Q s

h∗

λ - n B

Bn

h h∗  ?  h (λ) - m B Q  Q Qθτ Q Q  Q ? s  ∗h (λ) - Bm

h

3    θτ   ? Bm

- Bn   θτ     +

? Bm

where (h, h ) ∈ H om(, ), λ ∈ Bn are arbitrary. We have, see also Theorem 6, page 8: 





h∗ ◦ ∗λ = θ τ ◦ h ◦ θ τ ◦ ∗λ = θ τ ◦ h ◦ θ τ ◦ λ∗ = θ τ ◦ h ◦ λ ◦ θ τ hyp















= θ τ ◦  h (λ) ◦ h ◦ θ τ = θ τ ◦  h (λ) ◦ θ τ ◦ h∗ =  h (λ)∗ ◦ h∗ =  ∗h (λ) ◦ h∗ ,

i.e. (h∗ , h ) ∈ H om(∗ ,  ∗ ). Remark 23. The previous result may be generalized if we replace ∗ ,  ∗ with  isomorphic with  and   isomorphic with .

28

Boolean Systems

2.8 Morphisms vs. iterates Theorem 27. Let  : Bn → Bn and ν ∈ Bn . A morphism (h, h ) : ν →  is defined by h = 1Bn and ∀λ ∈ Bn , h (λ) = νλ. Proof. Indeed, ∀i ∈ {1, ..., n}, ∀μ ∈ Bn , ∀λ ∈ Bn , ⎧  μi , if λi = 0, ⎨ , if λ = 0, μ i i = (νi )λ (μ) = , if λi = 1 and νi = 0, = (νλ μ i i )(μ), ⎩ νi (μ), if λi = 1 i (μ), if λi = 1 and νi = 1 thus we can write: 

(1Bn ◦ (ν )λ )(μ) = νλ (μ) = (h (λ) ◦ 1Bn )(μ).

2.9 Morphisms vs. Cartesian product of functions Theorem 28. The functions  : Bn → Bn ,  : Bm → Bm ,  ×  : Bn+m → Bn+m are considered. Then π : Bn+m → Bn , π  : Bn+m → Bm defined by ∀μ ∈ Bn , ∀ν ∈ Bm , π(μ, ν) = μ, π  (μ, ν) = ν satisfy the property that (π, π) :  ×  → , (π  , π  ) :  ×  →  are morphisms. Proof. For any μ ∈ Bn , ν ∈ Bm , λ ∈ Bn and δ ∈ Bm , we infer π(( × )(λ,δ) (μ, ν))

Theorem 8, page 10

=

π((λ ×  δ )(μ, ν))

= π(λ (μ),  δ (ν)) = λ (μ) = π(λ,δ) (π(μ, ν)), and this proves the first statement of the theorem. The second statement is proved similarly. Theorem 29. We consider the functions  : Bn → Bn ,  : Bm → Bm ,  : Bp → Bp , ϒ : Bq → Bq and the morphisms (h, h ) :  → , (g, g  ) :  → ϒ. Then the functions f, f  : Bn+m → Bp+q defined in the following way: ∀μ ∈ Bn , ∀ν ∈ Bm ,  hi (μ), if i ∈ {1, ..., p}, fi (μ, ν) = gi−p (ν), if i ∈ {p + 1, ..., p + q},  fi (μ, ν) =

hi (μ), if i ∈ {1, ..., p},  gi−p (ν), if i ∈ {p + 1, ..., p + q}

satisfy that (f, f  ) :  ×  →  × ϒ is a morphism.

Chapter 2 • Morphisms of generator functions 29

Proof. Let μ ∈ Bn , ν ∈ Bm , λ ∈ Bn , and δ ∈ Bm arbitrary. We can write that f (( × )(λ,δ) (μ, ν)) = f ((λ ×  δ )(μ, ν)) = f (λ (μ),  δ (ν)) 







= (h(λ (μ)), g( δ (ν))) = ( h (λ) (h(μ)), ϒ g (δ) (g(ν))) = ( h (λ) × ϒ g (δ) )(h(μ), g(ν)) 



= ( × ϒ)(h (λ),g (δ)) (h(μ), g(ν)) = ( × ϒ)f

 (λ,δ)

(f (μ, ν)).

2.10 Morphisms vs. successors and predecessors Theorem 30. Let the functions  : Bn → Bn ,  : Bm → Bm and μ ∈ Bn . (a) If (h, h ) ∈ H om(, ), then + h(μ+  ) ⊂ h(μ) ,

(2.10.1)

+ + h(O (μ)) ⊂ O (h(μ)),

(2.10.2)

− h(μ−  ) ⊂ h(μ) ,

(2.10.3)

− − (μ)) ⊂ O (h(μ)); h(O

(2.10.4)

(b) in case that (h, h ) ∈ I so(, ), we have + h(μ+  ) = h(μ) ,

(2.10.5)

+ + (μ)) = O (h(μ)), h(O

(2.10.6)

− h(μ−  ) = h(μ) ,

(2.10.7)

− − (μ)) = O (h(μ)). h(O

(2.10.8)

n λ Proof. (a) (2.10.1): We take δ ∈ h(μ+  ) arbitrary, thus λ ∈ B exists such that δ = h( (μ)).  (λ) + h (h(μ)), i.e. δ ∈ h(μ) . This means that δ =  − (μ)). This is equivalent with saying that ξ, λ, λ , ..., (2.10.4): We take an arbitrary δ ∈ h(O    n λ λ λ λ ∈ B exist with ( ◦  ◦ ... ◦  )(ξ ) = μ and δ = h(ξ ). We can write: 







h(μ) = (h ◦ (λ ◦ λ ◦ ... ◦ λ ))(ξ ) = ((h ◦ λ ) ◦ (λ ◦ ... ◦ λ ))(ξ ) 











= (( h (λ) ◦ h) ◦ (λ ◦ ... ◦ λ ))(ξ ) = ( h (λ) ◦ (h ◦ λ ) ◦ ... ◦ λ )(ξ ) 















= (( h (λ) ◦  h (λ ) ) ◦ h ◦ ... ◦ λ )(ξ ) = ... = (( h (λ) ◦  h (λ ) ◦ ...) ◦ (h ◦ λ ))(ξ ) 



















= (( h (λ) ◦  h (λ ) ◦ ... ◦  h (λ ) ) ◦ h)(ξ ) = ( h (λ) ◦  h (λ ) ◦ ... ◦  h (λ ) )(h(ξ )) 









= ( h (λ) ◦  h (λ ) ◦ ... ◦  h (λ ) )(δ), − i.e. δ ∈ O (h(μ)).

30

Boolean Systems

(b) As  and  are isomorphic, we have m = n. + (2.10.5): taking into account (2.10.1), we must prove h(μ)+  ⊂ h(μ ).  + Let δ ∈ h(μ) arbitrary. This means the existence of λ ∈ Bn with δ =  λ (h(μ)), and with  the notation λ = h−1 (λ ) we have δ =  h (λ) (h(μ)) = h(λ (μ)), thus δ ∈ h(μ+  ). − − (h(μ)) ⊂ h(O (μ)). (2.10.8): by taking into account (2.10.4), we must still prove that O −   We take an arbitrary δ ∈ O (h(μ)), and this means the existence of λ, λ , ..., λ ∈ Bn such   that ( λ ◦  λ ◦ ... ◦  λ )(δ) = h(μ). We denote ν = h−1 (δ), ξ = h−1 (λ), ξ  = h−1 (λ ), ..., ξ  = h−1 (λ ) and we have 













h(μ) = ( λ ◦  λ ◦ ... ◦  λ )(δ) = ( h (ξ ) ◦  h (ξ ) ◦ ... ◦  h (ξ ) )(h(ν)) 

















= ( h (ξ ) ◦  h (ξ ) ◦ ... ◦ ( h (ξ ) ◦ h))(ν) = ( h (ξ ) ◦  h (ξ ) ◦ ... ◦ (h ◦ ξ ))(ν) 













... = ( h (ξ ) ◦ ( h (ξ ) ◦ h) ◦ ... ◦ ξ )(ν) = ( h (ξ ) ◦ (h ◦ ξ ) ◦ ... ◦ ξ )(ν) 









= (( h (ξ ) ◦ h) ◦ ξ ◦ ... ◦ ξ )(ν) = ((h ◦ ξ ) ◦ ξ ◦ ... ◦ ξ )(ν) 



= h((ξ ◦ ξ ◦ ... ◦ ξ )(ν)), i.e. 







μ = (ξ ◦ ξ ◦ ... ◦ ξ )(ν) = (ξ ◦ ξ ◦ ... ◦ ξ )(h−1 (δ)). − − The last equality states that h−1 (δ) ∈ O (μ), therefore δ ∈ h(O (μ)).

2.11 Morphisms vs. fixed points Theorem 31. The functions  : Bn −→ Bn ,  : Bm −→ Bm are given, together with μ ∈ Bn and (h, h ) ∈ H om(, ). If (μ) = μ then 

∀λ ∈ Bn ,  h (λ) (h(μ)) = h(μ), and if h is bijective, then (h(μ)) = h(μ). Proof. For arbitrary λ ∈ Bn we have 

 h (λ) (h(μ)) = h(λ (μ)) = h(μ) and if h is bijective, then λ ∈ Bn exists with h (λ ) = (1, ..., 1), when 



(h(μ)) =  h (λ ) (h(μ)) = h(μ).

3 State portraits The state portrait G of the function  : Bn → Bn is a directed graph that completely characterizes the timeless asynchronous behavior of . The definition of the state portrait G , its relation with  and several examples are the topics of Sections 3.1–3.4. The state subportraits are introduced in Section 3.5 and the isomorphisms of state portraits are introduced in Section 3.6. Interesting questions here concern the characterization with morphisms H om(, ) of the situations when G is a state subportrait of G , respectively when G is isomorphic with G . The concepts of indegree and outdegree are addressed in Section 3.7. In Section 3.8, the paths and the path-connectedness are introduced. Such concepts will be reiterated later, in a timeful systemic framework. Section 3.9 introduces the Hamiltonian and the Eulerian paths.

3.1 Preliminaries Theorem 32. We consider the function  : Bn −→ Bn . The following equalities hold: {(μ, λ (μ))|μ, λ ∈ Bn , μ = λ (μ)} = {(μ, ν)|μ ∈ Bn , ν ∈ μ+  {μ}} = {(μ, ν)|μ ∈ Bn , ν ∈ (μ, (μ)]} = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  (μ)}. i∈A

Proof. We fix an arbitrary μ ∈ Bn . In order to prove {(μ, λ (μ))|μ, λ ∈ Bn , μ = λ (μ)} ⊂ {(μ, ν)|μ ∈ Bn , ν ∈ μ+  {μ}}, let (μ , μ ) ∈ {(μ, λ (μ))|μ, λ ∈ Bn , μ = λ (μ)} arbitrary. Then λ ∈ Bn exists with the property that μ = λ (μ ) and μ = μ . We infer that μ ∈ μ+  {μ }. We prove {(μ, ν)|μ ∈ Bn , ν ∈ μ+  {μ}} ⊂ {(μ, ν)|μ ∈ Bn , ν ∈ (μ, (μ)]}. We take an arbitrary point (μ , μ ) ∈ {(μ, ν)|μ ∈ Bn , ν ∈ μ+  {μ}}. We have from Theorem 10, page 11 that μ+  {μ } = [μ , (μ )]  {μ } = (μ , (μ )], thus μ ∈ (μ , (μ )]. The inclusion {(μ, ν)|μ ∈ Bn , ν ∈ (μ, (μ)]} ⊂ {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  (μ)} i∈A

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00009-X Copyright © 2023 Elsevier Inc. All rights reserved.

31

32

Boolean Systems

is proved by taking (μ , μ ) ∈ {(μ, ν)|μ ∈ Bn , ν ∈ (μ, (μ)]} arbitrary, thus μ ∈ (μ , (μ )]. This implies, from Remark 5, page 6, the existence of A with μ = μ ⊕  ε i and ∅ = A ⊂ i∈A

μ  (μ ). The inclusion {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  (μ)} i∈A λ

⊂ {(μ,  (μ))|μ, λ ∈ Bn , μ = λ (μ)} is proved in the following manner. We take an arbitrary (μ , μ ) ∈ {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = i∈A

A ⊂ μ  (μ)}, thus A exists such that μ = μ ⊕  ε i and ∅ = A ⊂ μ  (μ ). We define i∈A

λ ∈ Bn by ∀i ∈ {1, ..., n},

 λi =

1, if i ∈ A, 0, otherwise,

and we infer ∀i ∈ {1, ..., n},     μi ⊕ 1, if λi = 1 and i ∈ μ  (μ ), μi ⊕ 1, if i ∈ A,  = μi = μi , otherwise μi , otherwise  i (μ ), if λi = 1 and μi = i (μ ), = = λi (μ ). μi , otherwise Obviously μ = μ .

3.2 State portraits Notation 7. We use the notation P ({1, ..., n}) = {A|A ⊂ {1, ..., n}} for the set of the subsets of {1, ..., n}. Definition 26. A directed graph is an ordered pair G = (V , E), where E ⊂ V × V . The elements of V are called vertices, nodes, or points, and the elements of E are called arrows, directed edges, or directed arcs. Definition 27. A state portrait (of dimension n) is a directed graph GH = (Bn , E H ), where E H = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ H (μ)} i∈A

(3.2.1)

is defined by means of a function H : Bn → P ({1, ..., n}). Notation 8. The set of the state portraits of dimension n is denoted with Spn . Definition 28. The state portrait of  : Bn → Bn is defined as G = (Bn , E ), with1 E = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  (μ)}, i∈A

1

See also Theorem 32, page 31.

(3.2.2)

Chapter 3 • State portraits

33

i.e. in (3.2.1) we have H (μ) = μ  (μ). Notation 9. For any H : Bn → P ({1, ..., n}),  : Bn → Bn , we denote with H : Bn → Bn , H : Bn → P ({1, ..., n}) the functions ∀μ ∈ Bn , H (μ) = μ ⊕

 εi ,

i∈H (μ)

H (μ) = μ  (μ).

(3.2.3) (3.2.4)

Remark 24. The following statements are true G = GH ,

(3.2.5)

GH = GH

(3.2.6)

because, from the way that H and H were defined, we have E = E H and also because ∀μ ∈ Bn , H (μ) = μ  (μ ⊕

 ε i ) = μ  H (μ)

i∈H (μ)

(3.2.7)

implies E H = EH . Remark 25. The terminology of ‘state portrait’, or rather ‘phase portrait’ is specific to dynamical systems theory. In a binary context the syntagms ‘state transition diagram’, or ‘state transition graph’ are preferred by many authors. Remark 26. In (3.2.1) we may have the existence of μ ∈ Bn such that H (μ) = ∅. Then no nonempty set A fulfills A ⊂ H (μ), and consequently ∀μ ∈ Bn , (μ, μ ) ∈ / E H . The limit situan H tion of this type is when ∀μ ∈ B , H (μ) = ∅ and E = ∅ ( = 1Bn agrees with this situation, since ∀μ ∈ Bn , H (μ) = μ  (μ) = μ  μ = ∅).

3.3 State portraits vs. generator functions Notation 10. We denote the set of the Bn → Bn functions with Fn . Theorem 33. We define the functions U : Fn → Spn , ∀ ∈ Fn , U () = G ,

(3.3.1)

W (GH ) = H .

(3.3.2)

and W : Spn → Fn , ∀GH ∈ Spn ,

The following statements hold: (a) For any H : Bn → P ({1, ..., n}) and any  : Bn → Bn , HH = H,

(3.3.3)

34

Boolean Systems

H = ,

(3.3.4)

(b) U ◦ W = 1Spn and W ◦ U = 1Fn . Proof. (a) Let H : Bn → P ({1, ..., n}) arbitrary. For any μ ∈ Bn , we have (3.2.4)

(3.2.3)

HH (μ) = μ  H (μ) = μ  (μ ⊕

 ε i ) = H (μ).

i∈H (μ)

We consider now  : Bn → Bn arbitrary, for which ∀μ ∈ Bn , (3.2.3)

H (μ) = μ ⊕



i∈H (μ)

(3.2.4)

εi = μ ⊕



ε i = (μ).

i∈μ(μ)

(b) For any GH ∈ Spn , (3.3.2)

(3.3.1)

(U ◦ W )(GH ) = U (W (GH )) = U (H ) = GH

(3.2.6)

= GH ,

and for any  ∈ Fn , we can write (3.3.1)

(3.2.5)

(3.3.2)

(3.3.4)

(W ◦ U )() = W (U ()) = W (G ) = W (GH ) = H = . Remark 27. The previous proof of (b) contains the interesting statement: ∀G ∈ Spn , W (G ) = . Remark 28. Theorem 33 makes us conclude that the functions  : Bn −→ Bn and the state portraits G may be identified, and this is a timeless interpretation of the state portraits. Remark 29. Another way of interpreting the existence in G of an arrow from μ to μ = λ (μ) = μ is the increase of time. Remark 30. In a state portrait, if an arrow exists from μ to μ and μ = λ (μ): – we must have λ = (0, ..., 0), otherwise we get μ = (0,...,0) (μ) = μ, contradiction, – we must have also a coordinate i ∈ {1, ..., n} with i (μ) = μi , otherwise μ is a fixed point of  and ∀i ∈ {1, ..., n}, ∀λ ∈ Bn ,   μi , if λi = 0, μi , if λi = 0, = = μi μi = λi (μ) = i (μ), if λi = 1 μi , if λi = 1 representing a contradiction too. Notation 11. We denote the arrow (μ, μ ) ∈ E with the symbol μ → μ . Definition 29. Let  : Bn −→ Bn and μ ∈ Bn . The coordinates2 i ∈ μ  (μ), i.e. these coordinates that satisfy i (μ) = μi , are called unstable, or excited, or enabled, and the 2

Two things are understood here by the word coordinate: i ∈ {1, ..., n} and μi , where μ = (μ1 , ..., μi , ..., μn ) ∈ Bn , but this will create no confusions.

Chapter 3 • State portraits

35

coordinates i ∈ / μ  (μ), i.e. these coordinates that satisfy i (μ) = μi , are called stable, or not excited, or disabled. Notation 12. In state portraits we underline μi the unstable coordinates of μ ∈ Bn , and we do not underline μi the stable coordinates of μ. Remark 31. The state portraits do not contain arrows of the form (μ, μ), due to the request A = ∅ in (3.2.1) and (3.2.2). This request, that does not interfere with the truth of Theorem 33, will be kept until Definition 178, page 412, when the full state portraits will be introduced.

3.4 Examples Example 18. The state portrait of the identity 1B2 : B2 −→ B2 is the following. (0, 0)

(1, 0)

(0, 1)

(1, 1)

All the points μ ∈ B2 are fixed points of 1B2 , so there is no arrow and all the coordinates μi are stable. Vice versa, the only function  : B2 −→ B2 whose state portrait is the previous one, i.e. that fulfills (0, 0) = (0, 0), (1, 0) = (1, 0), (1, 1) = (1, 1), (0, 1) = (0, 1), is the identity 1B2 . Example 19. Here is the state portrait of the constant function  : B2 −→ B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (1, 0).

(3.4.1)

- (1, 0) 3 6       - (1, 1) (0, 1) (0, 0) 6

We notice that 22 −1 = 3 arrows start from (0, 1), since it has 2 unstable coordinates, 21 −1 = 1 arrows start from (0, 0), (1, 1), since they have one unstable coordinate and no arrows start (or 20 − 1 = 0 arrows start) from (1, 0), since it has no unstable coordinate, being a fixed point. Vice versa, the only function  : B2 −→ B2 with this state portrait is function (3.4.1). Example 20. The function  : B2 −→ B2 , ∀μ ∈ B2 ,

36

Boolean Systems

(μ1 , μ2 ) = (μ1 ∪ μ2 , μ1 μ2 )

(3.4.2)

has its state portrait drawn below. (0, 0) 6

(1, 0)  3      - (1, 1) (0, 1) We see that  has three fixed points, (0, 0), (1, 0), (1, 1) with no unstable coordinate and no arrow starting from them. As (0, 1) = (1, 0), the point (0, 1) has both coordinates unstable, and there are 22 − 1 = 3 arrows starting from it. Vice versa, the only function  : B2 −→ B2 with the previous state portrait, i.e. fulfilling (0, 0) = (0, 0), (1, 0) = (1, 0), (1, 1) = (1, 1), (0, 1) = (1, 0) is function (3.4.2). Example 21. We give now the state portrait of the dual function of (μ1 , μ2 ) = (μ1 ∪ μ2 , μ1 μ2 ) from (3.4.2), i.e. ∗ (μ1 , μ2 ) = (μ1 μ2 , μ1 ∪ μ2 ). (1, 1) 6

(0, 1)  3      - (0, 0) (1, 0) We see that the state portrait of ∗ results from the state portrait of  by complementing all the binary coordinates, while the arrows and the underlined coordinates remain the same.

3.5 State subportrait Definition 30. We consider the functions ,  : Bn → Bn , and their state portraits G = (Bn , E ), G = (Bn , E ). If E ⊂ E , the state portrait G is called state subportrait of G and we denote G ⊂ G . Remark 32. Note that in the definition of the state subportrait we must have the same sets of points Bn in order to make G , G comparable. Example 22. 1Bn : Bn → Bn and θ (1,...,1) : Bn → Bn give G1Bn ⊂ G ⊂ Gθ (1,...,1) for any  : Bn → Bn . Example 23. In Example 20 we have a state subportrait of the state portrait from Example 19.

Chapter 3 • State portraits

37

Example 24. The function ∀μ ∈ B2 , (μ) = (μ1 ∪ μ2 , μ2 ) with the state portrait (0, 0)

(1, 0)

(0, 1)

- (1, 1)

satisfies itself G ⊂ G , where (μ1 , μ2 ) = (1, 0) is the one from Example 19. Problem 2. If the functions ,  : Bn → Bn satisfy G ⊂ G , is there a natural morphism (h, h ) :  → ?

3.6 Isomorphisms. Duality Definition 31. The state portraits G = (Bn , E ), G = (Bn , E ) of ,  : Bn → Bn are isomorphic (or topologically equivalent, or topologically conjugated) if a bijection h : Bn → Bn exists such that (μ, ν) ∈ E iff (h(μ), h(ν)) ∈ E . In this case h is called isomorphism from G to G and we denote h : G → G . Theorem 34. Let  : Bn → Bn . The state portraits G , G∗ are isomorphic, with h : Bn → Bn defined by ∀μ ∈ Bn , h(μ) = μ. Proof. We know from Theorem 32, page 31 that E = {(μ, ν)|μ ∈ Bn , ν ∈ (μ, (μ)]}, E∗ = {(μ, ν)|μ ∈ Bn , ν ∈ (μ, ∗ (μ)]}, and we take an arbitrary (μ, ν) ∈ E . This implies the existence of λ ∈ Bn , such that ν = λ (μ) and μ = ν. We infer ν = λ (μ) = λ (μ) = (λ )∗ (μ)

Theorem 6, page 8

=

(∗ )λ (μ)

and μ = ν, thus (μ, ν) ∈ E∗ . The inverse implication is proved similarly. Example 25. The dual functions from Examples 20, 21 have isomorphic state portraits, with the bijective h : B2 → B2 given by ∀μ ∈ B2 , h(μ) = μ. Definition 32. The isomorphic state portraits3 G , G∗ are called dual. Theorem 35. The following statement is true for ,  : Bn → Bn : if (h, h ) ∈ I so(, ), then h : Bn → Bn fulfills (μ, ν) ∈ E iff (h(μ), h(ν)) ∈ E . 3

Where the isomorphism h : G → G∗ is the bijection h : Bn → Bn , ∀μ ∈ Bn , h(μ) = μ.

38

Boolean Systems

Proof. We suppose that (h, h ) :  →  is isomorphism and let μ, λ ∈ Bn such that (μ, λ (μ)) ∈ E . This means that μ = λ (μ), thus h(μ) = h(λ (μ)), and we have (h(μ),  h(λ (μ))) = (h(μ),  h (λ) (h(μ))) ∈ E .  Conversely, we suppose that (h(μ),  h (λ) (h(μ))) = (h(μ), h(λ (μ))) ∈ E . As far as h(μ) = h(λ (μ)), we get μ = λ (μ), thus (μ, λ (μ)) ∈ E . Problem 3. If h : Bn → Bn bijective fulfills that (μ, ν) ∈ E iff (h(μ), h(ν)) ∈ E , then do we have the existence of h : Bn → Bn such that (h, h ) ∈ I so(, )?

3.7 Indegree, outdegree, balanced state portraits Definition 33. For a point μ ∈ Bn of a state portrait, the indegree deg− (μ) and the outdegree deg+ (μ) of μ are the natural numbers defined by deg− (μ) = card(μ− ) − 1, deg+ (μ) = card(μ+ ) − 1. If for any μ we have deg− (μ) = deg+ (μ), the state portrait is called balanced. Example 26. In the state portrait (0, 1) 6 ? (1, 1) 6

- (0, 0)

? (1, 0) we have deg− (1, 1) = 2, deg+ (1, 1) = 3, and the state portrait is not balanced. The state portrait of the identity from Example 18, page 35 is balanced, with ∀μ ∈ B2 , deg− (μ) = deg+ (μ) = 0. The state portrait (0, 0) 6 (1, 0) 

- (0, 1)

? (1, 1)

is also balanced with ∀μ ∈ B2 , deg− (μ) = deg+ (μ) = 1.

Chapter 3 • State portraits

39

Remark 33. The state portrait G of  : Bn → Bn has the property that for any μ ∈ Bn , deg− (μ) ∈ {0, ..., 2n − 1}, and deg+ (μ) ∈ {2k − 1|k ∈ {0, ..., n}}: if for example (μ) = μ then deg+ (μ) = 0 and if (μ) = μ, then deg+ (μ) = 2n − 1.

3.8 Path, path-connectedness Definition 34. A path (or a walk) in a state portrait (or in a directed graph) is a sequence of arrows μ0 → μ1 → ... → μk . We say that the path starts in μ0 , it ends in μk and that its length is k. If μk = μ0 , then the path is said to be closed and if μ0 , μ1 , ..., μk are all distinct, then the path is called simple. A simple closed path is by definition a closed path μ0 → μ1 → ... → μk = μ0 such that μ0 , μ1 , ..., μk−1 are distinct. Example 27. In the state portrait (0, 0)

(1, 0) 6

? (0, 1)

? (1, 1)

we notice the existence of the simple closed path of length 2: (1, 0) → (1, 1) → (1, 0). Definition 35. A state portrait is called acyclic if it does not contain closed paths, and cyclic otherwise. Example 28. The state portraits from Examples 26, 27 are cyclic, and the state portraits from Examples 18, 19, 20, 21, 24 are acyclic. Definition 36. The distinct points μ, μ ∈ Bn are path-connected, if there is a path μ → ... → μ . By convention, every point is considered to be connected to itself by a path of length zero. G is path-connected if all the points are path-connected. Notation 13. Let G = (Bn , E ) the state portrait of  : Bn → Bn and X ⊂ Bn nonempty. We denote with G |X the directed graph (X, E |X ), where E |X = {(μ, λ (μ))|μ ∈ X, λ ∈ Bn , λ (μ) ∈ X, μ = λ (μ)}. Definition 37. A path-connected component of G = (Bn , E ) is a directed subgraph G |X = (X, E |X ), where X ⊂ Bn , X = ∅, with the property that ∀μ ∈ X, ∀μ ∈ X, if μ = μ , then a path μ → ... → μ exists. Example 29. The second state portrait from Example 26 is path-connected and B2 is a path-connected component. In the state portrait

40

Boolean Systems

(0, 0) 6

(1, 0) 6

? (0, 1)

? (1, 1)

{(0, 0), (0, 1)} and {(1, 0), (1, 1)} are path-connected components.

3.9 Hamiltonian path, Eulerian path Definition 38. A Hamiltonian path is a path that visits every point exactly once. Definition 39. An Eulerian path is a path that contains every arrow. Remark 34. The Hamiltonian paths and the Eulerian paths may be closed or not. In order that a Hamiltonian path be closed, we must accept by definition that in the closed simple path μ0 → μ1 → ... → μk , the point μ0 = μk is visited exactly once. Example 30. In Example 26, the second state portrait, a Hamiltonian closed path and also an Eulerian closed path are for example (1, 0) → (0, 0) → (0, 1) → (1, 1) → (1, 0). In the following state portrait (0, 0) 6 (1, 0)

- (0, 1)

? (1, 1)

Hamiltonian closed paths and Eulerian closed paths do not exist, but (1, 0) → (0, 0) → (0, 1) → (1, 1) is a Hamiltonian path and an Eulerian path at the same time.

4 Signals The signals, introduced in Section 4.1, are the functions x : N → Bn . They are the models of the electrical signals from digital electronics. Their initial value and final value, initial time and final time are the topic of Section 4.2. The dual x ∗ of x is introduced in Section 4.3. The dual signals have many similar properties. The (coordinate-wise) monotonicity of the signals from Section 4.4 is an interesting feature that occurs in the study of the Boolean asynchronous systems. For example the combinational systems with one level and constant input have coordinate-wise monotonic states. The orbit and the omega-limit set of a signal x are defined in Sections 4.5 and 4.6.   The forgetful function σ k is the one that associates to each signal x the signal σ k (x) :  N → Bn , k  ∈ N, defined by ∀k ∈ N, σ k (x)(k) = x(k + k  ). Section 4.7 shows some possibilities of using it in the characterization of the final values, the orbits and the omega-limit sets of the signals. (x) : N → The image of a signal x : N → Bn via the function  : Bn → Bm is the signal  m (x)(k) = (x(k)). This is the topic of Section 4.8. B , defined by ∀k ∈ N,  The periodicity of the signals is another major topic, which is briefly addressed in Section 4.9. We prove proprieties like: the sum, the difference and the multiples of periods of a signal x are periods.

4.1 Definition Definition 40. A (discrete time, n-dimensional) signal is a function x : N → Bn , N  k → x(k) ∈ Bn . N is called the time set, and k is the time (instant). Notation 14. The set of the signals is denoted with S (n) . For n = 1, we usually denote S instead of S (1) . Remark 35. What we call signal in Definition 40 represents the idealization of an electrical signal of an asynchronous circuit from electronics. It is understood that (a) the two levels, low and high (0 and 1), (b) the number n itself, (c) the discrete time N Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00010-6 Copyright © 2023 Elsevier Inc. All rights reserved.

41

42

Boolean Systems

are approximations of the reality. We can easily accept that working with real numbers or with several logical values at (a), taking a higher dimension n > n at (b), or real time at (c) could improve the analysis, or perhaps make us get lost in non-essential details. Everything is a matter of choice. Remark 36. In equations, the signals will occur either as functions: x or perhaps x(·) indicating a missing variable, or as sequences of values x = μ, μ , μ , ..., 0

1

2

x = μ, μ , μ , ..., where μ, μ , μ , ... ∈ Bn . Sometimes the notations x(k) ∈ Bn , k ∈ N or α k ∈ Bn , k ∈ N will be used.

4.2 Initial value and final value, initial time and final time Definition 41. Let x ∈ S (n) . (a) The tuple x(0) is called the initial value of x; (b) if ∃k  ∈ N, ∀k ≥ k  , x(k) = x(k  ), we say that the final value of x exists, and the tuple x(k  ) is called the final value of x. Notation 15. The usual notation of the final value of x is lim x(k). k→∞

Definition 42. Given x, the initial time (instant) is by definition 0, and if lim x(k) exists, then any k  with ∀k ≥ k  , x(k) = x(k  ) is called final time (instant).

k→∞

Remark 37. lim x(k) coincides with the limit of x when k → ∞, indeed. k→∞

Example 31. The signal x = 0, 1, 0, 1, 0, ... ∈ S has the initial value 0 and no final value. The final value of x = 0, 1, 1, 1, ... ∈ S is lim x(k) = 1. k→∞

4.3 Duality Definition 43. The dual of the signal x ∈ S (n) is by definition x ∗ ∈ S (n) , x ∗ = x. Remark 38. The final value of x exists if and only if the final value of x ∗ exists. If these final values exist, then lim x(k) = lim x ∗ (k).

k→∞

k→∞

Chapter 4 • Signals

43

Remark 39. The dual signals have very similar properties. We shall not mention frequently during the exposure that x is monotonic if and only if x ∗ is monotonic, which is the relation between the orbits and the omega-limit sets of x and x ∗ , the fact that x is periodic if and only if x ∗ is periodic etc.; we shall just keep in mind dualities.

4.4 Monotonicity Definition 44. The signal x ∈ S (n) is called (coordinate-wise) monotonic, if ∀i ∈ {1, ..., n}, one of xi (0) ≤ xi (1) ≤ xi (2) ≤ ...

(4.4.1)

xi (0) ≥ xi (1) ≥ xi (2) ≥ ...

(4.4.2)

holds. If xi fulfills (4.4.1), it is called (monotonically) increasing, and if xi fulfills (4.4.2), it is called (monotonically) decreasing. Theorem 36. (a) x ∈ S is increasing if and only if one of the following statements is true: ∃μ ∈ B, ∀k ∈ N, x(k) = μ, 



∃k ≥ 1, x(k) =

0, if k ∈ {0, ..., k  − 1}, 1, if k ≥ k  .

(4.4.3)

(4.4.4)

(b) x ∈ S is decreasing if and only if one of (4.4.3) and 



∃k ≥ 1, x(k) =

1, if k ∈ {0, ..., k  − 1}, 0, if k ≥ k 

(4.4.5)

is true. (c) x ∈ S is increasing and decreasing at the same time if and only if (4.4.3) is true. (d) x ∈ S is monotonic implies that lim x(k) exists. k→∞

Proof. Obvious. Remark 40. The existence of lim x(k), see item (d) of the previous theorem, does not imply k→∞

monotonicity, of course.

4.5 Orbit, orbital equivalence Definition 45. The orbit of x ∈ S (n) is the set of its values O(x) = {x(k)|k ∈ N}.

44

Boolean Systems

Definition 46. We say that the signals x, y ∈ S (n) are orbitally equivalent, denoted x ∼ y, if the sequences 0 = i0 < i1 < i2 < ..., and 0 = j0 < j1 < j2 < ... exist such that ∀k ∈ N, x(ik ) = x(ik + 1) = ... = x(ik+1 − 1) = y(jk ) = y(jk + 1) = ... = y(jk+1 − 1).

(4.5.1)

Remark 41. The switching time instants1 in electronics are not very well predictable/ known. Orbital equivalence identifies the signals that switch at perhaps different time instances, but run through the same sequences of values. Remark 42. Orbital equivalence is an equivalence indeed. Example 32. The signals x, y ∈ S defined by x = 0, 1, 0, 1, ..., y = 0, 0, 1, 1, 0, 0, 1, 1, ... are orbitally equivalent. Theorem 37. If x ∼ y, then O(x) = O(y). Proof. We suppose that x and y are orbitally equivalent, thus the sequences 0 = i0 < i1 < i2 < ... and 0 = j0 < j1 < j2 < ... exist with the property that ∀k ∈ N, (4.5.1) is true. Then (4.5.1)

O(x) = {x(ik )|k ∈ N} = {y(jk )|k ∈ N} = O(y).

4.6 Omega-limit set, omega-limit equivalence Notation 16. For a signal x ∈ S (n) and μ ∈ Bn , we denote Txμ = {k|k ∈ N, x(k) = μ}. Theorem 38. The set ω(x) = {μ|μ ∈ Bn , Txμ is infinite}

(4.6.1)

is nonempty and ω(x) ⊂ O(x). Proof. For any μ ∈ Bn , we are in one of three situations (a) Txμ = ∅, (b) Txμ is finite, nonempty (c) Txμ is infinite. If we put Bn under the form n

Bn = {μ1 , ..., μ2 }, 1

k ≥ 1 is a switching time instant of x if x(k − 1) = x(k).

Chapter 4 • Signals

45

the equation Txμ1 ∨ ... ∨ Txμ2n = N, where the right hand term is infinite, shows that μi exist with Txμi infinite. We get that ω(x) is nonempty. The inclusion ω(x) ⊂ O(x) follows from the fact that O(x) = {μ|μ ∈ Bn , Txμ = ∅}. Definition 47. The set ω(x) that was defined at (4.6.1) is called the (omega-)limit set of x, and the points μ ∈ ω(x) are called (omega-)limit points (of x). Theorem 39. For any x ∈ S (n) , we have ∃k  ∈ N, ∀k  ≥ k  , O(x) = {x(k)|0 ≤ k ≤ k  },

(4.6.2)

∃k  ∈ N, ∀k  ≥ k  , ω(x) = {x(k)|k ≥ k  }.

(4.6.3)

Proof. (4.6.2): As O(x) is a finite set, O(x) = {μ1 , ..., μp }, where μ1 , ..., μp ∈ Bn , p ≥ 1, we have the existence of k  ∈ N with the property that {0, ..., k  } ∧ Txμ1 = ∅, ..., {0, ..., k  } ∧ Txμp = ∅. We infer ∀k  ≥ k  , {x(k)|k ∈ {0, ..., k  }} = O(x). (4.6.3): We can suppose without losing the generality that from the values μ1 , ..., μp of  x, the first p  ≤ p are taken infinitely many times: ω(x) = {μ1 , ..., μp }, and Txμ1 , ..., Tx p are μ



infinite, while the values μp +1 , ..., μp are taken finitely many times, i.e. Tx p +1 , ..., Txμp are μ

finite. If p  = p, then Txμ1 , ..., Txμp are all infinite. The number k  ∈ N exists with the property {k  , k  + 1, ...} ∧ Txμp +1 = ∅, ..., {k  , k  + 1, ...} ∧ Txμp = ∅ whenever p  < p, otherwise we put k  = 0. Then ∀k  ≥ k  , ω(x) = {x(k)|k ≥ k  }. Theorem 40. For any x ∈ S (n) , the following statements are true: (a) ∃ lim x(k) ⇐⇒ card(ω(x)) = 1; k→∞

(b) lim x(k) = μ ⇐⇒ ω(x) = {μ}. k→∞

Proof. (a) =⇒ If k  ∈ N exists such that ∀k ≥ k  , x(k) = x(k  ), then ω(x) = {x(k  )} and card(ω(x)) = 1. ⇐= If card(ω(x)) = 1, then exactly one μ ∈ Bn exists with the property that Txμ is infinite and, from Theorem 39, k  ∈ N exists with ∀k  ≥ k  , {μ} = {x(k)|k ≥ k  }, therefore lim x(k) = k→∞

μ. Remark 43. Let the signal x. If O(x) = ω(x), the time instant k  ≥ 1 exists that determines two time intervals: {0, 1, ..., k  − 1} when x has a ‘transient’ behavior, it takes values in any of O(x)  ω(x), ω(x) and {k  , k  + 1, ...} when x enters a ‘permanent’ behavior, it takes values

46

Boolean Systems

in ω(x) only. If O(x) = ω(x),2 x has a ‘permanent’ behavior from the initial time k  = 0. For example the periodic signals, in particular the constant signals are in this situation. Definition 48. The signals x, y ∈ S (n) are said to be (omega-)limit equivalent, denoted x ≈ y, if two sequences 0 ≤ i0 < i1 < i2 < ..., and 0 ≤ j0 < j1 < j2 < ... exist such that ∀k ∈ N, x(ik ) = x(ik + 1) = ... = x(ik+1 − 1) = y(jk ) = y(jk + 1) = ... = y(jk+1 − 1).

(4.6.4)

Remark 44. The difference between Definition 48 and Definition 46, page 44 is that 0 ≤ i0 , 0 ≤ j0 in the first case and 0 = i0 , 0 = j0 in the second case. Remark 45. The omega-limit equivalence identifies the signals that, at certain time instances, start running through the same sequences of values. Remark 46. The omega-limit equivalence is an equivalence indeed. Example 33. The signals x, y ∈ S which are of the form x = ..., 0, 1, 0, 1, ..., y = ..., 0, 1, 1, 0, 0, 1, 1, ... are omega-limit equivalent. Example 34. If for x, y ∈ S (n) and k  ∈ N we have ∀k ≥ k  , x(k − k  ) = y(k), then x ≈ y. This happens because we can define the sequences 0 ≤ i0 < i1 < ..., 0 ≤ j0 < j1 < ... by ∀k ∈ N, ik = k, jk = k  + k and we see that relation (4.6.4) is true. Theorem 41. If x ≈ y, then ω(x) = ω(y). Proof. The hypothesis states the existence of 0 ≤ i0 < i1 < i2 < ... and 0 ≤ j0 < j1 < j2 < ... with the property that ∀k ∈ N, (4.6.4) is true. From Theorem 39, page 45, we have the existence of k  ∈ N with ω(x) = {x(ik )|k ≥ k  } = {y(jk )|k ≥ k  } = ω(y). 2

x is called Poisson stable in this case.

Chapter 4 • Signals

47

4.7 The forgetful function 

Definition 49. The forgetful (or shift) function σ k : S (n) −→ S (n) is defined for any k  ∈ N by ∀x ∈ S (n) , ∀k ∈ N, 

σ k (x)(k) = x(k + k  ). Remark 47. The letter  n is missing in the notation of the forgetful function, thus the distinct forgetful functions σ k : S (m) −→ S (m) , σ k : S (n) −→ S (n) , ... have, abusively, the same notation. 

Remark 48. The name of forgetful function is justified by the fact that for k  ≥ 1, σ k (x) forgets the values x(0), ..., x(k  − 1) of x. Theorem 42. We consider the signal x ∈ S (n) and k  , k  ∈ N. We have: σ 0 = 1S (n) , 







σ k ◦ σ k = σ k +k , 

lim σ k (x)(k) exists ⇐⇒ lim x(k) exists,

k→∞

k→∞



lim σ k (x)(k) = lim x(k).

k→∞

k→∞

k

Proof. The existence of lim σ (x)(k) means the existence of k1 such that ∀k ≥ k1 , 

k→∞



σ k (x)(k) = σ k (x)(k1 ) i.e. ∀k ≥ k1 , x(k + k  ) = x(k1 + k  ), thus ∀k ≥ k  + k1 , x(k) = x(k  + k1 ), wherefrom we get the existence of lim x(k). The inverse implication is obvious also. From k→∞

the previous facts, we can write: 



lim x(k) = x(k  + k1 ) = σ k (x)(k1 ) = lim σ k (x)(k).

k→∞

k→∞

Theorem 43. For x ∈ S (n) , k  ∈ N and μ ∈ Bn , we get k  (x)

Tσμ

= {k − k  |k ∈ Txμ ∧ {k  , k  + 1, ...}}.

Proof. The following equivalencies hold: k  (x)



σ k ∈ Txμ ⇐⇒ x(k) = μ ⇐⇒ σ k (x)(k − k  ) = μ ⇐⇒ k − k  ∈ Tμ

,

for any k ≥ k  . Theorem 44. For any x ∈ S (n) and any k  ∈ N, the following diagram commutes: 

O(σ k (x)) ∪  ω(σ k (x))

⊂ O(x) ∪ = ω(x)

48

Boolean Systems

Proof. We get 



O(σ k (x)) = {σ k (x)(k)|k ∈ N} = {x(k + k  )|k ∈ N} = {x(k)|k ≥ k  } ⊂ {x(k)|k ∈ N} = O(x). 



On the other hand, σ k (x) ≈ x, thus from Theorem 41, page 46 we infer that ω(σ k (x)) = ω(x).

4.8 The image of a signal via a function  : S (n) −→ S (m) the function that is defined Notation 17. Let  : Bn → Bm . We denote with  by ∀x ∈ S (n) , ∀k ∈ N, (x)(k) = (x(k)). 

(4.8.1)

(x) ∈ S (m) is the signal given by ∀k ∈ N, (4.8.1) holds. In particular, for any x ∈ S (n) ,  (x) is called the image of x via . Definition 50. The signal  Example 35. We consider the functions πi : Bn → B, i ∈ {1, ..., n} defined by ∀μ ∈ Bn , πi (μ) = μi . The image of x ∈ S (n) via πi is the coordinate xi ∈ S. (x)(k) exists and we have Theorem 45. (a) If lim x(k) exists, then lim  k→∞

k→∞

(x)(k); ( lim x(k)) = lim  k→∞

k→∞

(x)(k) exists, then lim x(k) exists and (b) if  is bijective and lim  k→∞

k→∞

(x)(k)). lim x(k) = −1 ( lim 

k→∞

k→∞

Proof. (a) If ∃k  ∈ N, ∀p ≥ k  , x(p) = x(k  ), then ∀p ≥ k  , (x)(k). (x)(k  ) =  (x)(p) = lim  ( lim x(k)) = (x(p)) = (x(k  )) =  k→∞

k→∞

Theorem 46. The forgetful function acts on the image of x via  like this: 



(x)) =  (σ k (x)). ∀k  ∈ N, σ k ( Proof. Let k  ∈ N, k ∈ N arbitrary. We get: 





(x))(k) =  (x)(k + k  ) = (x(k + k  )) = (σ k (x)(k)) =  (σ k (x))(k). σ k ( 

(x) Theorem 47. If  : Bn → Bm and x ∈ S (n) then for any μ ∈ Bn we have Txμ ⊂ T (μ) .

Chapter 4 • Signals

49

(x)(k) = Proof. Let k ∈ Txμ arbitrary, thus x(k) = μ. This means that (x(k)) = (μ), i.e.  (x) 

(μ), in other words k ∈ T(μ) . Theorem 48. We have (x)) = (O(x)), O(

(4.8.2)

(x)) = (ω(x)). ω(

(4.8.3)

(x)) ⊂ (O(x)). We take an arbitrary μ ∈ O( (x)), i.e. k ∈ N Proof. We prove (4.8.2): O(  exists such that μ = (x)(k) = (x(k)). This shows that μ ∈ (O(x)). (x)). For an arbitrary μ ∈ (O(x)), we have the existence of k ∈ N such (O(x)) ⊂ O( (x)(k). This means that μ ∈ O( (x)). that μ = (x(k)) =  (x)   (x)) arbitrary, meaning that T We prove Eq. (4.8.3): ω((x)) ⊂ (ω(x)). Let μ ∈ ω( μ is infinite. A sequence p0 < p1 < p2 < ... exists then such that ∀k ∈ N, (x(pk )) = μ . In this situation, a subsequence p0 < p1 < p2 < ... exists, together with μ ∈ −1 (μ ) (the set −1 (μ ) is finite and nonempty), having the property that ∀k ∈ N, x(pk ) = μ. We have μ ∈ ω(x) and (μ) = μ , thus μ ∈ (ω(x)). (x)). We take μ ∈ (ω(x)) arbitrary. This fact means that μ ∈ ω(x) exists (ω(x)) ⊂ ω( (x)  with Txμ infinite and μ = (μ). But the inclusion Txμ ⊂ T(μ) stated in Theorem 47 shows (x)  (x)). that T(μ) is infinite, in other words μ = (μ) ∈ ω(

∗ : S (n) → S (m) , Definition 51. For  : Bn → Bm ,  : Bn → Bn , k ∈ N and λ ∈ Bn , we define  (k) λ (n) (n) (n)  ,  : S → S by ∀x ∈ S ,  ∗ (x) =  ∗ (x ∗ ),   (k) (x),  (k) (x) =   λ (x).  λ (x) =   Theorem 49. The functions  : Bn −→ Bn ,  : Bm −→ Bm are given and the morphism (h, h ) :  −→ . For any λ ∈ Bn , the diagram S (n)

λ  - S (n)

 h ? S (m)

 h (λ)  -

 h ? S (m)

commutes. Proof. Indeed, for any λ ∈ Bn , x ∈ S (n) and k ∈ N, we can write: λ )(x)(k) = h( λ (x)(k)) = h(λ (x(k))) =  h (λ) (h(x(k))) λ )(x)(k) = ( h◦ ( h◦

50

Boolean Systems

 h (λ) ◦   h (λ) ◦  =  h (λ) ( h(x)(k)) = ( h)(x)(k) = ( h)(x)(k). 





Remark 49. The previous property allows us to say that the morphism (h, h ) :  −→   −→  . induces a morphism ( h, h ) : 

4.9 Periodicity Definition 52. The signal x ∈ S (n) is called eventually periodic if ∃k  ∈ N, ∃p ≥ 1, ∀k ≥ k  , x(k) = x(k + p).

(4.9.1)

In (4.9.1), k  is called limit of periodicity of x and p is called period of x. If the eventually periodic signal x accepts a limit of periodicity k  = 0, then it is called periodic. Definition 53. If x is eventually periodic, the omega-limit set ω(x) is called limit cycle, and if x is periodic, then it is called cycle. Notation 18. The set of the periods of the eventually periodic signal x is denoted with Px . Example 36. An example of signal x ∈ S which is not eventually periodic is x =  0 , 1, 0, 0 , 1, 0, 0, 0, 1, ...    1

2

3

Example 37. The constant signal x ∈ S (n) equal with μ ∈ Bn is periodic with p = 1. Example 38. The signal x = (0, 0), (0, 1), (1, 1), (0, 1), (1, 1), (0, 1), ... has the limit of periodicity k  = 1, the period p = 2 and the limit cycle ω(x) = {(0, 1), (1, 1)}. Note that σ 1 (x) is periodic, with the period p = 2 and with O(σ 1 (x)) = ω(σ 1 (x)) = {(0, 1), (1, 1)}. Theorem 50. If x is periodic, then O(x) = ω(x). Proof. We prove that O(x) ⊂ ω(x) and let μ ∈ O(x) arbitrary. We suppose that p ≥ 1 is a period of x, thus k  ∈ N exists with the property that x(k  ) = μ and {k  + kp|k ∈ N} ⊂ Txμ . As Txμ is infinite, μ ∈ ω(x). The inclusion ω(x) ⊂ O(x) is obvious. Theorem 51. If x is eventually periodic, then (a) a least k  ∈ N exists such that ω(x) = {x(k)|k ≥ k  },

(4.9.2)

Chapter 4 • Signals

∃p ≥ 1, ∀k ≥ k  , x(k) = x(k + p)

51

(4.9.3)

hold,  (b) ∀k  ≥ k  , σ k (x) is periodic and Px = Pσ k (x) . 

Proof. (b) Let k  ≥ k  arbitrary. The signal y = σ k (x) satisfies, for any p ∈ Px and any k ∈ N, that y(k) = x(k + k  ) = x(k + k  + p) = y(k + p) (because k + k  ≥ k  ) i.e. p ∈ Pσ k (x) . We have obtained that Px ⊂ Pσ k (x) . The inclusion Pσ k (x) ⊂ Px is shown similarly. Theorem 52. (a) If x ∈ S (n) is eventually periodic and k  , p are these from Definition 52, then ∀k ≥ k  , ω(x) = {x(i)|i ∈ {k, k + 1, ..., k + p − 1}}. (b) We suppose that x is periodic with the period p; then ∀k ∈ N, O(x) = {x(i)|i ∈ {k, k + 1, ..., k + p − 1}}. Proof. (a) We have ω(x) = {x(i)|i ≥ k  }, thus for any k ≥ k  we infer ω(x) = {x(i)|i ≥ k}. As the values of x repeat with the period p : x(k) = x(k + p), x(k + 1) = x(k + p + 1), ..., x(k + p − 1) = x(k + 2p − 1), ... we conclude that ω(x) = {x(i)|i ∈ {k, k + 1, ..., k + p − 1}}. (b) The previous result written for the limit of periodicity k  = 0 makes us replace ω(x) with O(x). Theorem 53. If p, p  are periods of the (eventually) periodic signal x ∈ S (n) , then p + p  is period, kp, ∀k ≥ 1 are periods and if p > p  , then p − p  is period also; p

≥ 1 exists such that the set Px of the periods of x is of the form Px = {

p, 2

p , 3

p, ...}. Proof. We suppose that ∀k ≥ k  , x(k) = x(k + p),

(4.9.4)

∀k ≥ k  , x(k) = x(k + p  )

(4.9.5)

(we can choose the same limit of periodicity for p and p  , for example we can take k  = max{kp , kp  }). The first statement of the theorem results from (4.9.5)

(4.9.4)

∀k ≥ k  , x(k + (p + p  )) = x((k + p) + p  ) = x(k + p) = x(k). In the special case p = p  , we infer that 2p is a period and, by induction on k ≥ 1, we get that kp is period for arbitrary k. We put now l = k + p  in (4.9.5). We infer ∀l ≥ k  + p  , x(l − p  ) = x(l),

(4.9.6)

52

Boolean Systems

thus (4.9.4)

(4.9.6)

∀l ≥ k  + p  , x(l + (p − p  )) = x((l − p  ) + p) = x(l − p  ) = x(l). For the last statement of the theorem, we denote with p

the least element of Px . We obviously have {

p, 2

p , 3

p, ...} ⊂ Px . If the equality holds in the previous inclusion then the p, 2

p , 3

p, ...} = ∅ theorem is proved, thus we can suppose against all reason that Px  {

p, 2

p, 3

p , ...}. Since p

= min(Px ), we have the existence of k ≥ 1 with and let T ∈ Px  {

k

p < T < (k + 1)

p . On one hand we have obtained that T − k

p is period, but on the other hand we infer T − k

p 1, then μ1 ∪ ... ∪ μp = (1, ..., 1), 1

In discrete time, the computation functions are signals and at this moment it is not very clear why we do not call the computation functions, simply, signals. In continuous time, the two classes of functions are distinct, as we shall see in Appendix A, page 387. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00011-8 Copyright © 2023 Elsevier Inc. All rights reserved.

53

54

Boolean Systems

∀k ∈ N, ∃N ∈ N, supp α k ∨ supp α k+1 ∨ ... ∨ supp α k+N = {1, ..., n},

(5.1.2)

∀i ∈ {1, ..., n}, the sets {k|k ∈ N, αik = 1} are infinite.

(5.1.3)

Proof. (5.1.1)=⇒(5.1.2) The hypothesis states that the sets Tαμ1 , ..., Tαμp are infinite and ∀i ∈ j

{1, ..., n}, ∃j ∈ {1, ..., p}, μi = 1, thus the sets {k|k ∈ N, αik = 1} are infinite. We can define for each i ∈ {1, ..., n} and each k ∈ N the numbers 

tik = min{k  |k  ≥ k, αik = 1}. Then N = max{t1k , ..., tnk } fulfills the request supp α k ∨ supp α k+1 ∨ ... ∨ supp α k+N = {1, ..., n}. (5.1.2)=⇒(5.1.3) We suppose against all reason that (5.1.3) is false, i.e. i ∈ {1, ..., n} exists such that {k|k ∈ N, αik = 1} is finite. This means the existence of k  ∈ N such that ∀k ≥ k  , αik =    0, therefore i ∈ / supp α k ∨ supp α k +1 ∨ supp α k +2 ∨ ... representing a contradiction with hypothesis (5.1.2). (5.1.3)=⇒(5.1.1) We suppose against all reason the falsity of (5.1.1), i.e. ∃i ∈ {1, ..., n} p such that for the infinite sets Tαμ1 , ..., Tαμp we have μ1i = ... = μi = 0. In this situation, the set {k|k ∈ N, αik = 1} is finite, contradiction. Definition 55. If the computation function α : N −→ Bn fulfills one of (5.1.1), (5.1.2), (5.1.3) it is called progressive. Notation 19. The set of progressive computation functions α : N −→ Bn is denoted with n . For n = 1, we write  instead of 1 . Remark 51. Previously, progressiveness refers to the progress of time. Example 39. The functions α, β ∈ , α = 1, 1, 1, ... 0 1 2

β = 0, 1, 0, 1, 0, 1, ... 0 1 2 3 4 5

are periodical, with ω(α) = {1}, ω(β) = {0, 1}. Example 40. We define α ∈  by α = 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0 , 0 , 0 , 1 , ... 0 1 2 3 4 5 6 7 8 9 10 11 12 13

This progressive function is not periodic and it has the remarkable property that ∀L ∈ N, ∃k  ∈ N, ∀k ∈ {k  , k  + 1, ..., k  + L}, α k = 0 i.e. intervals exist of arbitrary length where it is null. Another manner of characterizing this computation function is to note that (5.1.2) is true, but the stronger property

Chapter 5 • Computation functions. Progressiveness

55

∃N ∈ N, ∀k ∈ N, supp α k ∨ supp α k+1 ∨ ... ∨ supp α k+N = {1, ..., n} is false. Example 41. We fix n ≥ 1 arbitrary. For any k ∈ N, we have that k1 ∈ N and j ∈ {0, 1, ..., n − 1} exist such that k = k1 n + j , and this allows defining the computation function α : N −→ Bn by α k = (0, 0, ..., 1 , ..., 0). j +1

We infer that ω(α) = {(1, 0, ..., 0), (0, 1, ..., 0), ..., (0, 0, ..., 1)}, thus α ∈ n since (5.1.1) is satisfied. Remark 52. We have n =  ... × , i.e. α : N → Bn is progressive if and only if all  ×  n

α1 , ..., αn : N → B are progressive. This is best seen by taking a look at (5.1.3). Similarly, n+m = n × m . Theorem 55. For any α : N → Bn , ∀k ∈ N, α ∈ n ⇐⇒ σ k (α) ∈ n . Proof. Let k ∈ N arbitrary, fixed. We have p

α ∈ n ⇐⇒ ∀i ∈ {1, ..., n}, the sets {p|p ∈ N, αi = 1} are infinite p

⇐⇒ ∀i ∈ {1, ..., n}, the sets {p|p ≥ k, αi = 1} are infinite ⇐⇒ σ k (α) ∈ n . 

Corollary 2. We take β 0 , ..., β k −1 ∈ Bn , k  ≥ 1 arbitrary, and the computation function α : N −→ Bn . Then γ : N −→ Bn defined by ∀k ∈ N,  k β , if k ∈ {0, ..., k  − 1}, γk =  α k−k , if k ≥ k  fulfills the equivalence α ∈ n ⇐⇒ γ ∈ n . 

Proof. α = σ k (γ ) and we use Theorem 55.

5.2 Morphisms of progressive computation functions Definition 56. The functions n → m are called morphisms of progressive computation functions. Example 42. Four  →  functions are: ∀α ∈ , h(α) = α 0 , α 1 , α 2 , ...

56

Boolean Systems

h (α) = α 1 , α 0 , α 3 , α 2 , α 5 , ... h (α) = α 2 , α 0 , α 1 , α 5 , α 3 , α 4 , α 8 , ... h (α) = α 1 , α 2 , α 3 , ... Note that h is the identity and that h, h , h are invertible, for example h−1 = h . On the other hand, h (α) is simply σ 1 (α). Definition 57. Let the function h : Bn → Bm . It defines2 a function h(α) ∈ (Bm )N , (Bn )N  α −→  in the following way: for any α : N −→ Bn and any k ∈ N,  h(α)k = h(α k ).

(5.2.1)

h(α) ∈ m , we say that h is compatible with the progresIf h has the property that ∀α ∈ n ,  siveness of the computation functions, and also that h induces a morphism  h : n → m . Notation 20. We denote by n,m the set of these functions: h(α) ∈ m }. n,m = {h|h : Bn → Bm , ∀α ∈ n ,  The set n,n is usually denoted with n . Remark 53. h ∈ n,m means that h : Bn → Bm defines a (Bn )N → (Bm )N function with the property that to the progressive functions α, there correspond progressive images  h(α). h : n → m is a morphism. The functions h , h , h from Example 42 Obviously ∀h ∈ n,m ,  h, h =  h or h = have an ‘anticipatory’ character, showing that no h ∈ 1 exists with h =   h. We infer that the morphisms n → m of progressive computation functions are two kinds, induced and not induced by functions h ∈ n,m . Example 43. The identity 1Bn together with the following functions h : Bn → Bn define functions from n : ∀μ ∈ Bn , h(μ) = (1, 1, ..., 1),

(5.2.2)

h(μ) = (μ1 , μ1 ∪ μ2 , ..., μ1 ∪ μ2 ∪ ... ∪ μn ),

(5.2.3)

h(μ) = (μ1 , μ1 , ..., μ1 ),

(5.2.4)

h(μ) = (μs(1) , μs(2) , ..., μs(n) ),

(5.2.5)

where s : {1, ..., n} −→ {1, ..., n} is bijective. 2

Like in the case of Notation 17 and Definition 50, page 48.

Chapter 5 • Computation functions. Progressiveness

57

Example 44. If n < m, the function Bn  μ −→ h(μ) = (μ1 , ..., μn , 1, ..., 1) ∈ Bm satisfies h ∈ n,m and if n > m, then the function Bn  μ −→ h(μ) = (μ1 , ..., μm ) ∈ Bm satisfies h ∈ n,m . Theorem 56. (a) If h ∈ n,m , then h(1, ..., 1) = (1, ..., 1),       n

m

(b) for any g ∈ n,m and h ∈ m,p , we have h ◦ g ∈ n,p , (c) if g : n → m , h : m → p are morphisms, then h ◦ g : n → p is morphism. Proof. (a) We consider the function α ∈ n , ∀k ∈ N, α k = (1, ..., 1). We get h(1, ..., 1),       n

n

h(1, ..., 1), h(1, ..., 1), ... ∈ m , and the only constant progressive computation function is       n

n

β ∈ m , ∀k ∈ N, β k = (1, ..., 1), thus h(1, ..., 1) = (1, ..., 1).          m

n

m

Definition 58. A bijective morphism h : n → n is called isomorphism. Example 45. The identity 1Bn defines the isomorphism 1 Bn of progressive computation functions. For any bijection s : {1, ..., n} −→ {1, ..., n}, (5.2.5) defines an isomorphism  h too.   h and h from Example 42 are also isomorphisms.

5.3 Special cases of progressive computation functions Notation 21. Let L ≥ 1. We denote with ≤L n the set of functions α ∈ n that fulfill ∀i ∈ {1, ..., n}, 

∀k ∈ N, ∃k  ∈ {k, k + 1, ..., k + L}, αik = 1. We put ≤L instead of ≤L 1 . Notation 22. For L ≥ 1, we denote with L n the set of progressive functions α ∈ n for which ∀i ∈ {1, ..., n}, ∃ki ∈ {0, ..., L − 1}, {k|k ∈ N, αik = 1} = {ki , ki + L, ki + 2L, ...}. L The usual notation of L 1 is  . ≤L implies the truth of ≤L = Remark 54. The fact that α ∈ ≤L n if and only if α1 , ..., αn ∈  n ≤L ≤L ≤L L ≤L ≤L ... ×  , n+m = n × m and in a similar way we have L ... × L, n =  ×   ×  n

L L L n+m = n × m .

n

58

Boolean Systems

Theorem 57. For any α : N → Bn , ∀k ∈ N, k ≤L α ∈ ≤L n ⇐⇒ σ (α) ∈ n , k L α ∈ L n ⇐⇒ σ (α) ∈ n .

Proof. We prove the first equivalence, and let α : N → Bn arbitrary, fixed.  =⇒ We fix k ∈ N arbitrarily. If α ∈ ≤L n , i.e. if ∀i ∈ {1, ..., n}, ∀p ∈ N, ∃k ∈ {p, p + 1, ..., p +  k k  L}, αi = 1, then ∀i ∈ {1, ..., n}, ∀p ≥ k, ∃k ∈ {p, p + 1, ..., p + L}, αi = 1, and we obtain that σ k (α) ∈ ≤L n . ⇐= We take k = 0. Remark 55. The meaning of the three sets of computation functions ≤L L n ⊂ n ⊂ n

is that they give three models of computation of the Boolean functions: the fixed delay model, the bounded delay model and the unbounded delay model. Other non-equivalent ways of introducing these models exist.

6 Flows and equations of evolution The flow generated by  : Bn → Bn is a function φ : n × Bn × N → Bn whose value is called state, x(k) = φ α (μ, k): α ∈ n shows when and how  is iterated, and μ ∈ Bn is the initial value of the state. The flows are defined in Section 6.1 of this chapter. The reachability of μ , representing the property φ α (μ, k) = μ that the state takes the value μ , is the topic of Section 6.2. Several examples of flows are given in Section 6.3. The properties from Section 6.4 are the adaptation to the present framework of the properties of consistency, composition and causality of the transition function which are contained in the definition of a dynamical system from [10], page 11. Since the authors consider, at the same page, that the attributes ‘dynamical’, ‘non-anticipatory’ and ‘causal’ have approximately the same meaning, we conclude that the property of causality to be introduced may be also called non-anticipation. We mention however that in [10] the systems had an input, unlike here where, for the moment, it is convenient to omit this aspect, therefore causality referred there to the input, unlike here where it refers to the computation function.1 The input controls the state and the computation function shows when and how the next state is computed. The solutions of the equations of evolution which are introduced in Section 6.5 are the states x and this gives a slightly different point of view on these switching phenomena. The flows with constant generator functions are briefly studied in Section 6.6. They are important in the analysis of the combinational systems.

6.1 Flows Definition 59. We consider the function  : Bn → Bn . The flow (or evolution function, or state transition function, or next state function) φ : n × Bn × N → Bn is defined by ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N,  μ, if k = 0, α (6.1.1) φ (μ, k) = k−1 α (φ α (μ, k − 1)), if k ≥ 1. Function  is called the generator function of φ, and we use to say that φ is generated by . The signal x ∈ S (n) given by ∀k ∈ N, x(k) = φ α (μ, k)

(6.1.2)

1

If we would work with input systems (or control systems), then causality would refer to both the input and the computation function. This is the case at page 306. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00012-X Copyright © 2023 Elsevier Inc. All rights reserved.

59

60

Boolean Systems

is called state (function), or trajectory2 ; k is the present time, x(k) is the present (value of the) state, 0 is the initial time, and μ = x(0) is the initial (value of the) state. Remark 56. Note the fact that the flow φ and its generator function  are denoted with the same Greek letter, lower case and upper case. This rule will be frequently used: ψ, ; γ , ; etc. Notation 23. The set of the flows with n-dimensional generator functions is denoted with F ln : F ln = {φ| ∈ Fn }. Remark 57. For any α ∈ n , μ ∈ Bn , k ≥ 1 and i ∈ {1, ..., n}, one of φiα (μ, k) = φiα (μ, k − 1), φiα (μ, k) = i (φ α (μ, k − 1)) is true. This happens for αik−1 = 0, respectively for αik−1 = 1, when φiα (μ, k) takes the previous value, respectively the desired value. Remark 58. The function , applied to the argument μ ∈ Bn , is computed (synchronously) on all its coordinates: i (μ), i ∈ {1, ..., n}. For any λ ∈ Bn , the function λ applied to μ computes (asynchronously) these coordinates i for which λi = 1 and it does not compute these coordinates i for which λi = 0. The flows (6.1.1) generalize the synchronous computations from the dynamical systems theory, since the constant sequence α k = (1, ..., 1) ∈ Bn , k ∈ N belongs to n and it gives for any μ ∈ Bn that φ α (μ, ·) = μ, (μ), ((μ)),... 0

1

2

Another interpretation of the flows (6.1.1) coming from the dynamical systems theory is that they refer to systems with variable structure: instead of a function  that iterates, k we have several functions α , k ∈ N that iterate. This is a special case of variable structure, k since the functions α have all of them the origin in . Remark 59. We give now the meaning of progressiveness: α ∈ n shows that each coordinate i , i ∈ {1, 2, ..., n} is computed infinitely many times as k runs in N. Remark 60. Systems theory exists making use of several time axes. We can get to such interpretations, if we associate each αik = 1 with the time advancement of the i-th time axis. Recap: we have associated Definition 59 with asynchronicity, with the variable structure and with the existence of n time axes. 2

Or orbit; we use the concept of orbit for the set of the values of this function.

Chapter 6 • Flows and equations of evolution

61

Theorem 58. We define the functions U : Fn → F ln , ∀ ∈ Fn , U () = φ, and V : F ln → Fn , ∀φ ∈ F ln , ∀μ ∈ Bn , V (φ)(μ) = φ α (μ, 1), where α ∈ n is arbitrary, such that α 0 = (1, ..., 1). (a) ∀φ ∈ F ln , V (φ) = , (b) U ◦ V = 1F ln and V ◦ U = 1Fn . Proof. (a) We take φ ∈ F ln , α ∈ n , μ ∈ Bn arbitrary, such that α 0 = (1, ..., 1). We get 0

V (φ)(μ) = φ α (μ, 1) = α (μ) = (μ). (b) For any φ ∈ F ln , (a)

(U ◦ V )(φ) = U (V (φ)) = U () = φ etc. Corollary 3. Any flow φ ∈ F ln has a unique generator function  ∈ Fn . Proof. This follows from the bijectivity of U . Notation 24. It will be convenient in general to use the notations O α (μ) = O(φ α (μ, ·)) and ωα (μ) = ω(φ α (μ, ·)) for the orbits and the omega-limit sets of the states. Theorem 59. If φ, φ ∗ are the flows generated by , ∗ then, for arbitrary α ∈ n , μ ∈ Bn , and k ∈ N, we have (φ ∗ )α (μ, k) = φ α (μ, k). Proof. The proof is made by induction on k ≥ 0. The equation holds for k = 0, and we suppose that it is true for k. We get: (φ ∗ )α (μ, k + 1) = (∗ )α ((φ ∗ )α (μ, k)) = (α )∗ (φ α (μ, k)) k

k

= α (φ α (μ, k)) = φ α (μ, k + 1). k

Theorem 60. ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , we can write (φ ∗ )α (μ,·)

Tμ

φ α (μ,·)

= Tμ

α α O ∗ (μ) = O (μ),

,

(6.1.3) (6.1.4)

62

Boolean Systems

α α ω ∗ (μ) = ω(φ (μ, ·)).

(6.1.5)

If X ⊂ Bn is some set, we have used at (6.1.4), (6.1.5) the notation X = {x|x ∈ X}. Proof. These follow from Theorem 59.

6.2 Reachability Definition 60. We consider the function  : Bn → Bn . The point μ ∈ Bn is called reachable (or accessible) from (the initial value) μ ∈ Bn at (the time instant) k ∈ N under (the computation function) α ∈ n if φ α (μ, k) = μ is true. Remark 61. In Definition 60 there are four parameters α, μ, μ and k, which may be either fixed or variable. The word fixed means that, given γ ∈ n , ν ∈ Bn , ν  ∈ Bn , k  ∈ N, reachability is considered for either of α = γ , μ = ν, μ = ν  , k = k  , and the word variable means that in the definition of reachability we use the quantifiers ∃α, ∀α, ∃μ, ∀μ, ∃μ , ∀μ , ∃k, ∀k. We give the example ∃α ∈ n , φ α (ν, k  ) = ν  , α exists such that ν  is reachable from ν at time instant k  , and also ∀α ∈ n , ∃μ ∈ Bn , ∀μ ∈ Bn , ∃k ∈ N, φ α (μ, k) = μ . An exception exists here, it is interesting to consider several forms of variability of k, in addition to the previous ones, for example ∀α ∈ n , ∃k  ∈ N, ∀μ ∈ Bn , ∃k ∈ {0, ..., k  }, φ α (ν, k) = μ expresses the idea of reachability from ν at k ≤ k  , while ∃μ ∈ Bn , ν  ∈ ωγ (μ) shows that the access from a certain μ to ν  is made repeatedly, the set {k|k ∈ N, φ γ (μ, k) = ν  } is infinite. A special case here is {k|k ∈ N, φ γ (μ, k) = ν  } = {k  , k  + 1, k  + 2, ...}, when the last reachability property with k variable becomes ∃μ ∈ Bn , lim φ γ (μ, k) = ν  . k→∞

6.3 Examples Example 46. The function  : Bn −→ Bn ,  = 1Bn gives for any α ∈ n , μ ∈ Bn and k ∈ N, that

Chapter 6 • Flows and equations of evolution

φ α (μ, k) = μ.

63

(6.3.1)

In Example 18, page 35, we have drawn the state portrait of the identity 1B2 . The constancy of φ in (6.3.1) is accompanied by the absence in the state portrait of any arrow. Example 47. If the function  : B → B is the complement: ∀μ ∈ B, (μ) = μ, then ∀α ∈ , ∀μ ∈ B, ∀k ∈ N,  φ α (μ, k) =

μ ⊕ α0

μ, if k = 0, ⊕ ... ⊕ α k−1 , if k ≥ 1.

We infer from the state portrait of this function 0

- 1

that for arbitrary α ∈  and μ ∈ B, the orbital equivalence φ α (μ, ·) ∼ μ, μ, μ, μ, μ, ... holds. Example 48. We define  : B2 −→ B2 by ∀μ ∈ B2 , (μ1 , μ2 ) = (μ2 , μ1 ). For α = (1, 1), (1, 1), (1, 1), ... we get  (1, 0), if k is even, α φ ((1, 0), k) = (0, 1), if k is odd,  (1, 0), if k = 0, and for α = ∈ 2 we have arbitrary, if k ≥ 1  φ α ((1, 0), k) =

(1, 0), if k = 0, (0, 0), if k ≥ 1

since (0, 0) is a fixed point of . The state portrait of  is the following one. (0, 0)  k 3 Q  Q  Q  Q  Q  Q (1, 0) - (0, 1) Q  Q  Q  Q  Q  s + Q  (1, 1)

64

Boolean Systems

6.4 Consistency, causality and composition Theorem 61. (Consistency) A function  : Bn → Bn is given, together with α ∈ n and μ ∈ Bn . We have φ α (μ, 0) = μ. Theorem 62. (Causality, or non-anticipation) We consider the function  : Bn → Bn and let α ∈ n , β ∈ n , μ ∈ Bn , k  ≥ 1 arbitrary. If ∀k ∈ {0, ..., k  − 1}, αk = β k , then ∀k ∈ {0, ..., k  }, φ α (μ, k) = φ β (μ, k). Proof. The proof is made by induction on k  ≥ 1. For k  = 1, the hypothesis states that α 0 = β 0 , and we see the truth of φ α (μ, 0) = μ = φ β (μ, 0), 0

0

φ α (μ, 1) = α (μ) = β (μ) = φ β (μ, 1). We suppose that the result is true for k  and we get k

k

φ α (μ, k  + 1) = α (φ α (μ, k  )) = β (φ β (μ, k  )) = φ β (μ, k  + 1). Theorem 63. (Composition) ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , ∀k  ∈ N, φ α (μ, k  ) = μ =⇒ ∀k ∈ N, φ α (μ, k + k  ) = φ σ

k  (α)

(μ , k).

(6.4.1)

Proof. We suppose that φ α (μ, k  ) = μ and we use the induction on k ∈ N. For k = 0 the equality holds, thus we can suppose that it is true for k. We get φ α (μ, k + k  + 1) = α = σ

k  (α)k

k+k 

(φ σ

(φ α (μ, k + k  )) = α

k  (α)

(μ , k)) = φ σ

k  (α)

k+k 

(φ σ

k  (α)

(μ , k))

(μ , k + 1).

Remark 62. Equivalently, (6.4.1) can be written as: 

σ k (φ α (μ, ·))(k) = φ α (μ, k + k  ) = φ σ

k  (α)

(φ α (μ, k  ), k),

with arbitrary α ∈ n , μ ∈ Bn , k ∈ N, and k  ∈ N. Remark 63. We notice something else also, from the previous remark: that φ α (μ, ·) ≈ φ σ

k  (α)

(φ α (μ, k  ), ·).

(6.4.2)

Chapter 6 • Flows and equations of evolution

65

Corollary 4. Let  : Bn → Bn , α ∈ n , μ ∈ Bn and k  ∈ N. We have: ωα (μ) = ωσ

k  (α)

(φ α (μ, k  )).

Proof. This happens because ωα (μ) = ω(φ α (μ, ·)) (6.4.2)

= ω(φ σ

k  (α)

Theorem 44, page 47

=

(φ α (μ, k  ), ·)) = ωσ



ω(σ k (φ α (μ, ·)))

k  (α)

(φ α (μ, k  )).

Theorem 64. (Composition) (a) Let p ≥ 1 and α 0 , α 1 , ..., α p ∈ n , μ0 , μ1 , ..., μp ∈ Bn ,  k0 , k1 , ..., kp−1 ≥ 1 such that φ α (μ0 , k0 ) = μ1 , 0

φ α (μ1 , k1 ) = μ2 , 1

... φα

p−1

 (μp−1 , kp−1 ) = μp .

If we define γ ∈ n in the following way: ⎧ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪ ⎪  ⎪ ⎪ α 1,k−k0 , if k ∈ {k0 , ..., k0 + k1 − 1}, ⎪ ⎪ ⎪ ⎨ ...  γk = p−1,k−k0 −...−kp−2  , if k ∈ {k0 + ... + kp−2 , α ⎪ ⎪ ⎪ ⎪   ⎪ + ... + k − 1}, ..., k ⎪ 0 p−1 ⎪ ⎪  ⎩ α p,k−k0 −...−kp−1  , if k ≥ k0 + ... + kp−1 , then we have

φ γ (μ0 , k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(6.4.3)

φ α (μ0 , k), if k ∈ {0, ..., k0 }, 1 φ α (μ1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p−1   ), if k ∈ {k0 + ... + kp−2 , φ α (μp−1 , k − k0 − ... − kp−2   ..., k0 + ... + kp−1 }, p   φ α (μp , k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 . 0

(b) We consider α 0 , α 1 , ..., α p , ... ∈ n , μ0 , μ1 , ..., μp , ... ∈ Bn , k0 , k1 , ..., kp , ... ≥ 1 such that φ α (μ0 , k0 ) = μ1 ,

(6.4.4)

φ α (μ1 , k1 ) = μ2 ,

(6.4.5)

0

1

66

Boolean Systems

... φ α (μp , kp ) = μp+1 , p

(6.4.6)

... For the computation function γ ∈ n defined as: ⎧ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪  ⎪ 1,k−k ⎪ 0 , if k ∈ {k  , ..., k  + k  − 1}, ⎪ α ⎨ 0 0 1 ... γk =   ⎪ ⎪  ⎪ α p,k−k0 −...−kp−1 , if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, ⎪ ⎪ ⎩ ... we have

φ γ (μ0 , k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(6.4.7)

φ α (μ0 , k), if k ∈ {0, ..., k0 }, φ (μ1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p   ), if k ∈ {k0 + ... + kp−1 , φ α (μp , k − k0 − ... − kp−1   ..., k0 + ... + kp }, ... 0

α1

Proof. (b) The proof is by induction on p ∈ N. For p = 0, we can write that ∀k ∈ {0, ..., k0 − 1}, (6.4.7)

γ k = α 0,k , thus Theorem 62, page 64 implies ∀k ∈ {0, ..., k0 }, 0

φ γ (μ0 , k) = φ α (μ0 , k), therefore (6.4.4)

φ γ (μ0 , k0 ) = φ α (μ0 , k0 ) = μ1 . 0

The hypothesis of the induction is  ) = φα φ γ (μ0 , k0 + ... + kp−1

p−1

 (μp−1 , kp−1 ) = μp .

(6.4.8)

We see that ∀k ∈ {0, ..., kp − 1}, (σ

 k0 +...+kp−1

(6.4.7)

(γ ))k = α p,k ,

wherefrom ∀k ∈ {0, ..., kp },  ) φ γ (μ0 , k + k0 + ... + kp−1

Theorem 63

=

φσ

 k0 +...+kp−1 (γ )

 (φ γ (μ0 , k0 + ... + kp−1 ), k)

(6.4.9)

Chapter 6 • Flows and equations of evolution

(6.4.8)

= φσ

 k0 +...+kp−1 (γ )

(μp , k)

(6.4.9), Theorem 62

=

67

p

φ α (μp , k).

We infer that (6.4.6)

 + kp ) = φ α (μp , kp ) = μp+1 . φ γ (μ0 , k0 + ... + kp−1 p

Remark 64. We say that γ defined at (6.4.3) represents the ‘concatenation’ of α 0 , ..., α p , and (6.4.7) represents an infinite version of concatenation.

6.5 Equations of evolution Theorem 65. We consider the function  : Bn −→ Bn and the equations ∀k ∈ N, x(k) = φ α (μ, k), k

(6.5.1)

y(k + 1) = α (y(k)),

(6.5.2)

⎧ ⎪ ⎨ z1 (k + 1) = 1 (z(k))α1k ∪ z1 (k)α1k , ... ⎪ ⎩ zn (k + 1) = n (z(k))αnk ∪ zn (k)αnk .

(6.5.3)

For any α ∈ n and μ ∈ Bn , if y(0) = μ and z1 (0) = μ1 , ..., zn (0) = μn , we have x = y = z. Proof. We use the induction on k. Definition 61. Any of (6.5.2) and (6.5.3) is called equation of evolution (of ). Remark 65. A slight difference exists between (6.5.1) and (6.5.2), (6.5.3); we may want to call (6.5.1) equation of evolution too. Notation 25. We denote with Eq the equation of evolution of , and with Eqn the set of the equations of evolution of the functions : Eqn = {Eq | ∈ Fn }. Theorem 66. We define the functions U : Fn → Eqn , ∀ ∈ Fn , U () = Eq , and V : Eqn → Fn , ∀Eq ∈ Eqn , ∀μ ∈ Bn , V (Eq )(μ) = x(1).

(6.5.4)

In (6.5.4), x is the solution of Eq with the initial value μ and α ∈ n arbitrary, such that α 0 = (1, .., 1).

68

Boolean Systems

(a) ∀Eq ∈ Eqn , V (Eq ) = , (b) U ◦ V = 1Eqn and V ◦ U = 1Fn . Proof. (a) We consider Eq. (6.5.2) as Eq , where α is arbitrary with α 0 = (1, ..., 1). We fix μ ∈ Bn arbitrary itself. The solution x satisfies x(0) = μ and in addition we get 0

V (Eq )(μ) = x(1) = α (μ) = (μ). (b) For an arbitrary Eq , we have (a)

(U ◦ V )(Eq ) = U (V (Eq )) = U () = Eq , etc. Corollary 5. For any equation Eq ∈ Eqn a unique  ∈ Fn exists such that Eq = Eq . Proof. This is a consequence of the bijectivity of U .

6.6 Flows with constant generator functions Theorem 67. If  : Bn −→ Bn is constant: ∃μ ∈ Bn , ∀μ ∈ Bn , (μ) = μ , then ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, ∀i ∈ {1, ..., n}, 

φiα (μ, k) = μi



p

αi ∪ μi

p∈{0,...,k−1}

p

αi .

(6.6.1)

p∈{0,...,k−1}

In particular, φ is coordinate-wise monotonic and lim φ α (μ, k) = μ . k→∞

Proof. Let α ∈ n , μ ∈ Bn arbitrary and we make the proof by induction on k. For k = 0, ∀i ∈ {1, ..., n}, we have φiα (μ, 0) = μi



p

αi ∪ μi

p∈{0,...,−1}



αi = μi 0 ∪ μi 0 = μi , p

p∈{0,...,−1}

see Definition 6, page 3, thus (6.6.1) is true. We suppose now its truth for arbitrary k. For any i ∈ {1, ..., n}, we infer ⎧   p p ⎪ ⎨ μi αi ∪ μi αi , if αik = 0, α φi (μ, k + 1) = p∈{0,...,k−1} p∈{0,...,k−1} ⎪ ⎩ i (φ α (μ, k)), if αik = 1,

Chapter 6 • Flows and equations of evolution

=

⎧ ⎪ ⎨ μi ⎪ ⎩





p

αi ∪ μi

p∈{0,...,k−1}

= μi αik ∪ (μi

p∈{0,...,k−1}





p

αi ∪ μi



p∈{0,...,k}





p

αi ) ∪ μi

p

αi αik

p∈{0,...,k−1}



αi = μi

Obviously when k → ∞, we have that

p

αi )αik

p∈{0,...,k−1}

p

p∈{0,...,k}

therefore φiα (μ, k) → μi monotonically.



p

αi ∪ μi

p∈{0,...,k−1}

= μi

p

αi , if αik = 0,

μi , if αik = 1,

p∈{0,...,k−1}

= μi (αik ∪ αik

69

p∈{0,...,k}



p

αi → 1,

p∈{0,...,k}



p

αi ∪ μi

p

αi .

p∈{0,...,k}



p

αi → 0 monotonically,

p∈{0,...,k}

Remark 66. Another way of writing (6.6.1) is ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, ⎧ ⎪ μ, if k = 0, ⎨ 0 α φ α (μ, k) =  (μ), if k = 1, ⎪ 0 ⎩ α ∪...∪α k−1 (μ), if k ≥ 2.

7 Systems Several equivalent perspectives are offered: by the functions , by the state portraits G , by the flows φ and by the equations of evolution Eq . Here ‘equivalent’ means the existence of bijective functions between Fn , Spn , F ln and Eqn . A system is defined as any of , G , φ and Eq , and these aspects, together with explanations referring to the attributes of the systems, are given in the first two sections. A subsystem  : Bn → Bn of  : Bp → Bp is a system with the property that n ≤ p and, modulo the order of the coordinates, ∀ν ∈ Bp , ∀i ∈ {1, ..., n}, i (ν1 , ..., νn , ..., νp ) = i (ν1 , ..., νn ).  reproduces the properties of , restricted to the coordinates 1, ..., n. The subsystems are introduced in Section 7.3. The Cartesian product  =  ×  of the systems , , which is the topic of Section 7.4, refers to the special case when  has two subsystems,  and .

7.1 Several equivalent perspectives Remark 67. We have previously used the notations (a) Fn for the set of the functions  : Bn → Bn , (b) Spn for the set of the state portraits G = (Bn , {(μ, λ (μ))|μ, λ ∈ Bn , μ = λ (μ)}, (c) F ln for the set of the flows φ : n × Bn × N → Bn , (d) Eqn for the set of the equations of evolution Eq . We have also previously defined: Fn 6 U V ? Spn

Fn

Fn 6 V

U

? F ln

U 

6 V  ? Eqn

(i) the functions U : Fn → Spn , ∀ ∈ Fn , U () = G , V : Spn → Fn , ∀G ∈ Spn , V (G ) =  that fulfill (see Theorem 33, page 33 and Remark 27, page 34) U ◦ V = 1Spn , V ◦ U = 1Fn ; (ii) the functions U  : Fn → F ln , ∀ ∈ Fn , U  () = φ, Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00013-1 Copyright © 2023 Elsevier Inc. All rights reserved.

71

72

Boolean Systems

V  : F ln → Fn , ∀φ ∈ F ln , ∀μ ∈ Bn , V  (φ)(μ) = φ α (μ, 1), where α ∈ n is arbitrary with α 0 = (1, ..., 1) ∈ Bn ; they satisfy (see Theorem 58, page 61) U  ◦ V  = 1F ln and V  ◦ U  = 1Fn ; (iii) the functions U  : Fn → Eqn , ∀ ∈ Fn , U  () = Eq , and V  : Eqn → Fn , where ∀Eq ∈ Eqn , V  (Eq ) : Bn −→ Bn is given by ∀μ ∈ Bn , V  (Eq )(μ) = x(1);

(7.1.1)

in (7.1.1), x is the solution of Eq with the initial value x(0) = μ, and α ∈ n is arbitrary with α 0 = (1, ..., 1) ∈ Bn . U  and V  satisfy (see Theorem 66, page 67) U  ◦ V  = 1Eqn and V  ◦ U  = 1Fn . These bijections allow us to identify the elements of Fn , Spn , F ln , Eqn .

7.2 Definition Definition 62. An n-dimensional Boolean, universal, regular, asynchronous, nondeterministic, non-initialized, discrete time, autonomous system (or network, or circuit)1 with unbounded delays, shortly a system, is any of (a) a function  : Bn −→ Bn , (b) a state portrait G = (Bn , E ), (c) a flow φ : n × Bn × N −→ Bn , (d) an equation Eq . Function  is the generator function of the system, G is its state portrait, φ is its flow and Eq is its equation of evolution; signal x(k) = φ α (μ, k) is the state, α ∈ n is the computation function (of ), μ ∈ Bn is the initial (value of the) state, and k ∈ N is the time (instant). Remark 68. Here are some explanations concerning the attributes of a system that have occurred in Definition 62: – n-dimensional refers to Bn and n , – Boolean vaguely refers to the binary Boolean algebra B, i.e. discrete space, – universal indicates that the state space is all of Bn ; in certain circumstances of invariance of a nonempty set X ⊂ Bn , the possibility of restricting the state space to X exists, – regularity means the existence of a function  that generates the system, in the sense that the state x is the result of the iterations of , – asynchronicity is the property that the coordinates 1 , ..., n of the generator function are not computed at the same time, and the fact that these coordinates are computed independently on each other makes us say also that the structure of the system is variable. 1

Or switching circuit.

Chapter 7 • Systems

73

Indeed, we can write instead of  x(k) = the equation ∀k ∈ N,

α 

x(k) =

k−1

μ, if k = 0, (x(k − 1)), if k ≥ 1,

μ, if k = 0, f (x(k − 1), k), if k ≥ 1,

(7.2.1)

where by definition ∀ν ∈ Bn , ∀k ≥ 1, f (ν, k) = α

k−1

(ν),

(7.2.2)

– non-determinism is the property that, at a certain time instant k, x(k) can take several values. In our framework this is caused by the dependence of x on μ and α, which are not known, given, in particular: – non-initialization is the property that x(0) ∈ Bn is arbitrary, – the discrete time set is N. We shall suggest in Appendix A, page 387, a possibility of replacing the discrete time with the continuous time R, – autonomy is considered in general to be the property that the generator function is constant vs. time, and there is no input. In our case the absence of the input holds, but the request of asynchronicity as previously stated is in contradiction, see (7.2.2), with time independence. We can consider that autonomy refers here to the absence of the input only, and make the distinction: autonomous system – input system in order to indicate the absence and the presence of the input. Remark 69. We stress on the fact that the equations Eq must be solved and discussed for different values of α and μ. But the durations of computation (the delays that occur in the computation) of 1 , ..., n – are not known, since any technology would indicate in such occasions values like t0 ± t1 % (we have used on purpose a common notation for the real numbers), – are not constant, they depend on temperature, and moreover the duration of the switch from 0 to 1 is different from the duration of the switch from 1 to 0 etc. When using α ∈ n , these durations of computation are subject to no restriction and this is called, see Remark 55, page 58, the unbounded delay model of computation of the Boolean functions. The previous ±t1 % term brings us to a more sophisticated and more realistic bounded delay model of computation, corresponding in discrete time to α ∈ ≤L n . And making t1 = 0, i.e. a perfect knowledge of the durations of computation of 1 , ..., n gives the fixed delay model of computation, that corresponds in discrete time to α ∈ L n. Remark 70. We shall call system any of , G , φ, Eq . Definition 63. The systems generated by the dual functions , ∗ : Bn → Bn are called dual.

74

Boolean Systems

7.3 Subsystem Theorem 68. The system  : Bp → Bp and the coordinates j1 , ..., jn ∈ {1, ..., p} are given, j1 < ... < jn , n ≤ p. The following statements are equivalent: (a) ∀i ∈ {1, ..., n}, {k|k ∈ {1, ..., p}, ∃ν ∈ Bp , ji (ν ⊕ ε k ) = ji (ν)} ⊂ {j1 , ..., jn }, (b) the system  : Bn → Bn exists with the property that ∀i ∈ {1, ..., n}, ∀ν ∈ Bp , ji (ν1 , ..., νj1 , ..., νjn , ..., νp ) = i (νj1 , ..., νjn ).

(7.3.1)

Proof. (a)=⇒(b) If n = p the implication is obvious, so that we suppose in the rest of the proof that n < p. We suppose against all reason that defining the functions i : Bn → B, i ∈ {1, ..., n} that fulfill (7.3.1) is not possible, and this means the existence of ν ∈ Bp , ν  ∈ Bp and i ∈ {1, ..., n} with νj1 = νj 1 , ... νjn = νj n and ji (ν1 , ..., νj1 , ..., νjn , ..., νp ) = ji (ν1 , ..., νj1 , ..., νjn , ..., νp ).

(7.3.2)

We know from (7.3.2) that l1 , ..., lq ∈ {1, ..., p}  {j1 , ..., jn } exist such that (ν1 , ..., νj1 , ..., νjn , ..., νp ) = (ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ⊕ ... ⊕ ε lq , ji (ν1 , ..., νj1 , ..., νjn , ..., νp ) = ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ⊕ ... ⊕ ε lq ).

(7.3.3)

We infer ji (ν1 , ..., νj1 , ..., νjn , ..., νp ) = ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ),

(7.3.4)

otherwise we obtain a contradiction with (a), ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ) = ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ⊕ ε l2 ),

(7.3.5)

otherwise we obtain a contradiction with (a) again, ... ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ⊕ ... ⊕ ε lq−1 ) = ji ((ν1 , ..., νj1 , ..., νjn , ..., νp ) ⊕ ε l1 ⊕ ... ⊕ ε lq ),

(7.3.6)

Chapter 7 • Systems

75

for the same reason. The conjunction of statements (7.3.4), (7.3.5), ..., (7.3.6) contradicts (7.3.3), thus the supposition that  like at (b) cannot be defined which has generated (7.3.3) is proved to be false. (b)=⇒(a) The hypothesis states that j1 , ..., jn depend on the values of νj1 , ..., νjn only, i.e. (a) holds. Definition 64. The system  : Bn → Bn is called a subsystem of  : Bp → Bp if n ≤ p and, modulo the order of the coordinates, ∀ν ∈ Bp , ∀i ∈ {1, ..., n}, i (ν1 , ..., νn , ..., νp ) = i (ν1 , ..., νn ). The fact that  is a subsystem of  is denoted by  ⊂ . Remark 71. We shall prefer in general presenting the fact that  is a subsystem of  under the form:  : Bn → Bn ,  : Bn+m → Bn+m and ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ).

(7.3.7)

We see two abuses here: (a) the special case when  and  have the same dimension (m = 0) might seem to be excluded, which is not the case, (b) the words ‘modulo the order of the coordinates’ are missing. We shall keep them in mind. In fact we keep in mind that synonymous functions are identified, see Definition 23, page 25. Theorem 69. Let  : Bn+m → Bn+m , its subsystem  : Bn → Bn that fulfills ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, (7.3.7) is true and we fix μ ∈ Bn , ν ∈ Bm , α ∈ n , β ∈ m arbitrary. Then ∀k ∈ N, ∀i ∈ {1, ..., n}, we have (α,β)

γi

((μ, ν), k) = φiα (μ, k).

Proof. The proof is made by induction on k ∈ N. For k = 0, we have ∀i ∈ {1, ..., n}, (α,β)

γi

((μ, ν), 0) = μi = φiα (μ, 0).

We suppose that the statement is true for k. We obtain ∀i ∈ {1, ..., n},  (α,β) γi ((μ, ν), k), if αik = 0, (α,β) (α,β)k (α,β) γi ((μ, ν), k + 1) = i (γ ((μ, ν), k)) = (α,β) ((μ, ν), k)), if αik = 1 i (γ  hyp

=

(7.3.7)

=

φiα (μ, k), if αik = 0, (α,β) (α,β) i (φ1α (μ, k), ..., φnα (μ, k), γn+1 ((μ, ν), k), ..., γn+m ((μ, ν), k)), if αik = 1 

k φiα (μ, k), if αik = 0, = αi (φ α (μ, k)) = φiα (μ, k + 1). k α α i (φ1 (μ, k), ..., φn (μ, k)), if αi = 1

76

Boolean Systems

Remark 72. If  ⊂ , then a morphism (π, π) ∈ H om(, ) exists, where π is the projection of the coordinates of  on the coordinates of , in other words the following diagram is commutative Bn+m

 (λ,δ)-

Bn+m

π

π ? ? λ - n B Bn for any λ ∈ Bn , δ ∈ Bm . The reason for this is suggested by Theorem 28, page 28. Example 49. We have drawn at (a), (b) the state portraits of the functions  : B3 → B3 , ∀ξ ∈ B3 , (ξ1 , ξ2 , ξ3 ) = (ξ2 , ξ1 , ξ1 ξ2 ∪ ξ3 ), and  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ2 , μ1 ), where  ⊂ . (0, 0, 0)  (1, 0, 0) Q Q k 6 Q Q Q Q Q Q Q Q ? s Q Q (0, 1, 0) (1, 1, 0) (0, 0, 1)  (1, 0, 1) (0, 0)  (1, 0) Q k Q Q Q k 6 6 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ? s Q Q? s ? Q Q - (1, 1, 1) - (1, 1) (0, 1, 1) (0, 1) (a)

(b)

Note how the state portrait G from (b) is duplicated in the state portrait G from (a): an arrow (1, 1, 0) → (1, 1, 1) exists between these two duplicates, due to 3 .

7.4 Cartesian product Theorem 70. Let  : Bn → Bn ,  : Bm → Bm and  : Bn+m → Bn+m ,  =  × . Then γ = (φ, ψ). Proof. The proof is made by induction on k ∈ N and it is similar with the proof of Theorem 69, page 75. This follows from the fact that  ⊂ ,  ⊂  hold.

Chapter 7 • Systems

77

Remark 73. We interpret the Cartesian product  =  ×  as a system of dimension n + m where  generates the first n coordinates and  generates the last m coordinates. We can split the computation function in (α, β), representing the first n and the last m coordinates, and also the initial value of the state in (μ, ν), representing the first n and the last m coordinates; then Theorem 70 states that the first n and the last m coordinates of the state are x(k) = φ α (μ, k) and y(k) = ψ β (ν, k). Problem 4. Let  : Bn → Bn for which the following (state space decomposition) problem is formulated: to be found  : Bn → Bn isomorphic with  such that  1 : Bn1 → Bn1 , ...,  p : Bnp → Bnp exist with  =  1 × ... ×  p , where n1 + ... + np = n. Theorem 71. We consider  : Bn+m → Bn+m ,  : Bn → Bn ,  : Bm → Bm with  =  × , α ∈ n , β ∈ m , μ ∈ Bn and ν ∈ Bm . We have for any μ ∈ Bn , ν  ∈ Bm that γ (α,β) ((μ,ν),·)

T(μ ,ν  )

φ α (μ,·)

= Tμ

ψ β (ν,·)

∧ Tν 

and in addition (α,β)

O (α,β)

ω

β

α (μ, ν) = {(φ α (μ, k), ψ β (ν, k))|k ∈ N} ⊂ O (μ) × O (ν),

φ α (μ,·)

(μ, ν) = {(μ , ν  )|(μ , ν  ) ∈ Bn × Bm , Tμ

ψ β (ν,·)

∧ Tν 

is infinite}

β α (μ) × ω (ν). ⊂ ω

Proof. We use the fact that, from Theorem 70, we have ∀k ∈ N, γ (α,β) ((μ, ν), k) = (φ α (μ, k), ψ β (ν, k)). The first statement of the theorem is proved like this: γ (α,β) ((μ,ν),·)

T(μ ,ν  )

= {k|k ∈ N, γ (α,β) ((μ, ν), k) = (μ , ν  )}

= {k|k ∈ N, φ α (μ, k) = μ and ψ β (ν, k) = ν  } φ α (μ,·)

= {k|k ∈ N, φ α (μ, k) = μ } ∧ {k|k ∈ N, ψ β (ν, k) = ν  } = Tμ

ψ β (ν,·)

∧ Tν 

.

Remark 74. To be compared the second statement of the theorem with the timeless equality (1.7.17)page 15 .

8 Morphisms of flows Two kinds of morphisms are considered in this work: the morphisms of generator functions (h, h ) :  →  from Chapter 2, where  : Bn → Bn ,  : Bm → Bm , and the morphisms of flows (h, h ) : φ → ψ which are addressed in this chapter. Their definition is given in Section 8.1.  ) : φ → ψ whenever h ∈ n,m , The morphisms (h, h ) :  →  induce morphisms (h, h and this is discussed in Section 8.2. The connections between the two kinds of morphisms are the topic of Section 8.3. The composition of the morphisms, the isomorphisms and the symmetry relative to translations from Sections 8.4, 8.5, 8.6 are treated analogously with the same topics from Chapter 2. A morphism (h, h ) : γ → ψ is compatible with the subsystems  ⊂ , ϒ ⊂  if it induces a morphism (h1 , h1 ) : φ → υ and this is discussed in Section 8.7. Sections 8.8–8.13 point out the behavior of the morphisms of flows vs. duality, orbits and omega-limit sets, Cartesian products, successors and predecessors, limits, orbital and omega-limit equivalence. A section dedicated to the action of the morphisms of flows on fixed points is postponed for Chapter 10. Section 8.14 introduces the pseudo-morphisms. The main idea represented by this con cept is that of replacing the equality of the signals h(φ α (μ, k)) = ψ h (α) (h(μ), k) defining  the morphisms (h, h ) : φ → ψ with the orbital equivalence h(φ α (μ, ·)) ∼ ψ h (α) (h(μ), ·). The reason of doing so is the fact that the exact moments in time when the flows change their values are not known.

8.1 Definition Definition 65. Let the functions  : Bn → Bn ,  : Bm → Bm , h : Bn → Bm and h : n → m . We say that the couple (h, h ) defines a morphism from φ to ψ, denoted (h, h ) : φ → ψ, if ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, 

h(φ α (μ, k)) = ψ h (α) (h(μ), k).

(8.1.1)

Notation 26. The set of morphisms from φ to ψ has the notation H om(φ, ψ). Remark 75. We can define ∀α ∈ n the function φ α : Bn × N → Bn by Bn × N  (μ, k) −→ φ α (μ, k) ∈ Bn . The fact that (h, h ) ∈ H om(φ, ψ) means then the commutativity of the diaBoolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00014-3 Copyright © 2023 Elsevier Inc. All rights reserved.

79

80

Boolean Systems

gram φα

Bn × N

- Bn

h × 1N

h ? ? ψ h (α) - Bn Bn × N

for any α ∈ n , where Bn × N  (μ, k) −→ (h × 1N )(μ, k) = (h(μ), k) ∈ Bn × N. This commutative diagram recalls the one of Definition 20, page 21. Remark 76. If a morphism (h, h ) : φ → ψ exists, we expect that some properties of φ are transmitted to ψ. Example 50. For the function  : Bn → Bn we show that (1Bn , 1 Bn ) : φ → φ is a morphism: n ∀α ∈ n , ∀μ ∈ B , ∀k ∈ N, 

1Bn (φ α (μ, k)) = φ α (μ, k) = φ 1Bn (α) (1Bn (μ), k). Example 51. We suppose that  : Bn → Bn is arbitrary, that h : Bn → Bm is the constant function ∃μ ∈ Bm , ∀μ ∈ Bn , h(μ) = μ and let h : n → m be arbitrary too. We take  = 1Bm and we have that ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, 



h(φ α (μ, k)) = μ = ψ h (α) (μ , k) = ψ h (α) (h(μ), k). Relation (8.1.1) is true, therefore (h, h ) ∈ H om(φ, ψ). Theorem 72. The functions  : Bn → Bn ,  : Bm → Bm are given, and the morphism (h, h ) ∈ H om(φ, ψ). We suppose that the function g ∈ n,m exists, having the property that ∀α ∈ n , h (α) =  g (α).

(8.1.2)

Then (h, g) ∈ H om(, ). Proof. We notice first of all that Eq. (8.1.2) makes sense because g ∈ n,m . Let now μ ∈ Bn , λ ∈ Bn , α ∈ n arbitrary, such that α 0 = λ. We infer (8.1.1)

0



h(λ (μ)) = h(α (μ)) = h(φ α (μ, 1)) = ψ h (α) (h(μ), 1) g (α) = ψ (h(μ), 1) =  g(α ) (h(μ)) =  g(λ) (h(μ)). 0

Chapter 8 • Morphisms of flows

81

8.2 Induced morphisms Theorem 73. The functions  : Bn → Bn ,  : Bm → Bm and h : Bn → Bm , h ∈ n,m are given.  ) ∈ H om(φ, ψ). If (h, h ) ∈ H om(, ), then (h, h Proof. We must prove that ∀k ∈ N, 

h(φ α (μ, k)) = ψ h (α) (h(μ), k)

(8.2.1)

is true, for arbitrary α ∈ n and μ ∈ Bn . We note in the beginning that (8.2.1) makes sense  (α) ∈ m . Eq. (8.2.1) is true for k = 0, as both members are equal because h ∈ n,m , thus h with h(μ), therefore we can suppose that it is true for k. We infer: 

k

k

h(φ α (μ, k + 1)) = h(α (φ α (μ, k))) =  h (α ) (h(φ α (μ, k))) (8.2.1)



k





=  h (α ) (ψ h (α) (h(μ), k)) = ψ h (α) (h(μ), k + 1).

Definition 66. In case that (h, h ) ∈ H om(, ) and h ∈ n,m , we say that the morphism  ) ∈ H om(φ, ψ). (h, h ) of generator functions induces the morphism of flows (h, h Remark 77. The set H om(φ, ψ) can be empty. Note that the nonemptyness of H om(, ) was caused by the remark that h constant and h null gave always a morphism (h, h ) ∈ H om(, ), Example 13, page 22. But the null function h does not belong to n,m , thus (h, h ) :  →  induces no morphism of flows. Example 52. We have already given at Example 50 the case of (1Bn , 1Bn ) :  −→  that induces the morphism (1Bn , 1 Bn ) : φ −→ φ as far as 1Bn ∈ n . Example 53. The functions , , h, h : B2 → B2 are defined by ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 ∪ μ2 , μ1 ∪ μ2 ), (μ1 , μ2 ) = (μ1 , μ1 ∪ μ2 ), h(μ1 , μ2 ) = (μ1 μ2 , μ2 ),

(8.2.2)

h (μ1 , μ2 ) = (μ1 , μ2 )

(8.2.3)

(0, 0)

- (0, 1)

(0, 0)

- (0, 1)

(1, 0)

? (1, 1)

(1, 0)

(1, 1)

(a)

(b)

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Boolean Systems

 ) ∈ and the state portraits of ,  were drawn at (a), (b). In order to show that (h, h 2 H om(φ, ψ), we have to notice first that ∀λ ∈ B , the diagram B2

λ - 2 B

h ? B2



h (λ)

h ? - B2

is commutative, i.e. (h, h ) ∈ H om(, ). As h ∈ 2 , the conclusion follows. We want to show now the mechanism making the commutativity of the previous diagram hold. As h = 1B2 , λ ,  λ compute the same values, except the fact that (0, 1) =  μ, if μ = (1, 1), (0, 1) = (1, 1) = (0, 1). The point is that ∀μ ∈ B2 , h(μ) = h acts as (0, 1), if μ = (1, 1), identity except the fact that h(0, 1) = h(1, 1) = (0, 1), i.e. it hides the computation in (0, 1), that  makes differently from . Example 54. The same mechanism works for ,  : B2 → B2 defined by ∀μ ∈ B2 , (μ1 , μ2 ) = (μ2 , μ1 ∪ μ2 ), (μ1 , μ2 ) = (μ1 μ2 , μ1 ∪ μ2 ) and h, h from (8.2.2), (8.2.3), see the following state portraits, with ,  at (a), (b). - (0, 1)

(0, 0) 6

? (1, 1)

(1, 0) (a)

- (0, 1)

(0, 0) 6 (1, 0)

(1, 1) (b)

 ) ∈ H om(φ, ψ) is true. We have (h, h ) ∈ H om(, ) and h ∈ 2 , therefore (h, h

8.3 Morphisms of generator functions vs. morphisms of flows Remark 78. We consider the generator functions  : Bn −→ Bn ,  : Bm −→ Bm and the flows φ, ψ. Two kinds of morphisms have been defined. The morphisms of generator functions (h, h ) :  →  consist in two functions h, h : Bn −→ Bm having the property that

Chapter 8 • Morphisms of flows

83

∀λ ∈ Bn , the diagram Bn h ? Bm

λ - n B h  ?  h (λ) - m B

is commutative. And the morphisms of flows (h, h ) : φ → ψ consist in two functions h : Bn −→ Bm , h : n → m with the property that ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, 

ψ h (α) (h(μ), k) = h(φ α (μ, k)). (a) We suppose that (h, h ) :  →  is given. In the special case when h ∈ n,m , (h, h )  ) : φ → ψ, see Theorem 73, page 81. This means that morphisms induces a morphism (h, h of generator functions exist which cannot define morphisms of flows. (b) The morphism of flows (h, h ) : φ → ψ is considered now. In the special case when g , (h, h ) defines the morphism of generator functions (h, g) : g ∈ n,m exists such that h =   → , see Theorem 72, page 80. We conclude that morphisms of flows exist that cannot define morphisms of generator functions. Remark 79. The rhythm of the exposure is faster if we do not pay very much attention to these differences between the two kinds of morphisms. This means that the morphisms are interpreted as being tools of investigation, but not purpose of the investigation. This means also that the door is left open for giving more attention to these differences.

8.4 Composition Theorem 74. The functions  : Bn → Bn ,  : Bm → Bm ,  : Bp → Bp , h : Bn → Bm , f : Bm → Bp are given. (a) For h ∈ n,m , f  ∈ m,p the implication  ◦ h ) : φ → γ  ) : φ → ψ, (f, f ) : ψ → γ =⇒ (f ◦ h, f (h, h

holds, (b) if h : n → m , f  : m → p , the following implication holds: (h, h ) : φ → ψ, (f, f  ) : ψ → γ =⇒ (f ◦ h, f  ◦ h ) : φ → γ . Proof. (a) The hypothesis and Theorem 72, page 80 imply the fact that (h, h ) :  → , (f, f  ) :  →  are morphisms; this together with Theorem 22, page 23 shows that (f ◦ h, f  ◦ h ) :  →  is morphism. The assumption that h ∈ n,m , f  ∈ m,p implies  ◦ h ) : φ → γ results f  ◦ h ∈ n,p , from Theorem 56, page 57. And the fact that (f ◦ h, f from Theorem 73, page 81.

84

Boolean Systems

(b) For arbitrary α ∈ n , β ∈ m , μ ∈ Bn , ν ∈ Bm , the hypothesis states ∀k ∈ N, 

h(φ α (μ, k)) = ψ h (α) (h(μ), k), f (ψ β (ν, k)) = γ f

 (β)

(f (ν), k).

We infer ∀k ∈ N, 

f (h(φ α (μ, k))) = f (ψ h (α) (h(μ), k)) = γ f

 (h (α))

(f (h(μ)), k).

Definition 67. The composition of the morphisms (h, h ) : φ → ψ, (f, f  ) : ψ → γ is the morphism (f, f  ) ◦ (h, h ) : φ → γ defined by (f, f  ) ◦ (h, h ) = (f ◦ h, f  ◦ h ).

8.5 Isomorphisms Definition 68. In case that in Definition 65, page 79, h and h are bijections, the morphism (h, h ) : φ → ψ is called isomorphism from φ to ψ, and φ, ψ are called topologically equivalent (or conjugated); and if, in addition, φ = ψ, then (h, h ) is called automorphism. Notation 27. The sets of isomorphisms φ → ψ and automorphisms φ → φ are denoted with I so(φ, ψ), Aut (φ). Theorem 75. We consider the functions , ,  : Bn → Bn and the flows φ, ψ, γ that they generate. We infer (a) if (h, h ) ∈ I so(φ, ψ) and (g, g  ) ∈ I so(ψ, γ ), then (g, g  ) ◦ (h, h ) ∈ I so(φ, γ ), (b) if (h, h ) ∈ I so(φ, ψ), then (h−1 , h−1 ) ∈ I so(ψ, φ), (c) Aut (φ) is a group relative to the composition of the isomorphisms, where the identity  −1 −1 ). is (1Bn , 1 Bn ) : φ → φ, and the inverse of (h, h ) is (h , h Proof. (b) Let α ∈ n , μ ∈ Bn , k ∈ N arbitrary, fixed. In Eq. (8.1.1)page 79 , we put β = h (α), ν = h(μ) and we obtain −1 (β)

h(φ h

(h−1 (ν), k)) = ψ β (ν, k),

i.e. −1 (β)

h−1 (ψ β (ν, k)) = φ h

(h−1 (ν), k).

It has resulted that (h−1 , h−1 ) ∈ I so(ψ, φ). Example 55. If  : Bn → Bn and the bijection s : {1, ..., n} → {1, ..., n} are given, Theorem 24, page 24 states that, for h, h : Bn → Bn defined by ∀μ ∈ Bn , h(μ1 , ..., μn ) = h (μ1 , ..., μn ) = (μs(1) , ..., μs(n) ), the function  : Bn → Bn ,  = js ◦  ◦ js −1 fulfills (h, h ) ∈ I so(, ). As  ) ∈ I so(φ, ψ). h ∈ n , we get that (h, h

Chapter 8 • Morphisms of flows

85

Example 56. An easy way to construct topologically conjugated flows (this example anticipates the symmetry relative to translations section) is shown below, - (0, 1) (0, 0) Q Q Q Q Q ? s ? Q (1, 0) (1, 1)

- (0, 0) (0, 1) Q Q Q Q Q ? s ? Q (1, 1) (1, 0)

(a)

(b)

where at (a), (b) we have drawn the state portraits of the functions ∀μ ∈ B2 , (μ1 , μ2 ) = (1, μ1 ∪ μ2 ), (μ1 , μ2 ) = (1, μ1 μ2 ), and we consider also ∀μ ∈ B2 , h(μ1 , μ2 ) = (μ1 , μ2 ), h (μ1 , μ2 ) = (μ1 , μ2 ). The state portrait from (b) results by a translation with (0, 1) of the points of the state portrait from (a): indeed, h(μ1 , μ2 ) = (μ1 , μ2 ) ⊕ (0, 1) = θ (0,1) (μ1 , μ2 ) and all the arrows and the  ) ∈ excited coordinates are the same. We have that (h, h ) ∈ H om(, ), h ∈ 2 thus (h, h  −1 −1 −1 H om(φ, ψ). In addition, h, h are both bijections and (h , h ) ∈ H om(, ), h ∈ 2 are true also, thus ,  are topologically conjugated and φ, ψ are topologically conjugated too. Problem 5. Do topologically equivalent generator functions ,  exist such that φ, ψ are not topologically equivalent? This would mean that the only isomorphisms (h, h ) ∈ I so(, ) / n . satisfy h ∈ Problem 6. Do topologically equivalent flows φ, ψ exist such that ,  are not topologically equivalent? This would mean that ∀(h, h ) ∈ I so(φ, ψ), ∀g ∈ n , we have h =  g. Example 57. We do not know the answer to Problem 5, but we give the following example. , , h, h : B2 −→ B2 are defined by ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 μ2 , μ2 ), (μ1 , μ2 ) = (μ1 μ2 , μ2 ), h(μ1 , μ2 ) = h (μ1 , μ2 ) = (μ1 , μ2 ),

86

Boolean Systems

and we notice first of all that h, h are bijections, and they coincide with their inverses. We can write ∀μ ∈ B2 , ∀λ ∈ B2 , h((λ1 ,λ2 ) (μ1 , μ2 )) = h(λ1 μ1 ∪ λ1 μ1 μ2 , μ2 ) = (λ1 μ1 ∪ λ1 μ1 μ2 , μ2 ) 

= (λ1 μ1 ∪ λ1 μ1 μ2 , λ2 μ2 ∪ λ2 μ2 ) =  (λ1 ,λ2 ) (μ1 , μ2 ) =  h (λ1 ,λ2 ) (h(μ1 , μ2 )),  ) ∈ i.e. (h, h ) ∈ I so(, ). As far as h ∈ / 2 , we get that (h, h / I so(φ, ψ). Problem 7. How can we characterize the situation when one of ,  : Bn −→ Bn is constant and I so(, ) = ∅ or I so(φ, ψ) = ∅?

8.6 Symmetry relative to translations Definition 69. The flows φ, ψ generated by ,  : Bn −→ Bn are called symmetrical relative to the translation with τ ∈ Bn , if h : n → n exists such that (θ τ , h ) ∈ I so(φ, ψ) and (θ τ , h ) = (1Bn , 1 Bn ). Definition 70. We say that the flow φ generated by  : Bn −→ Bn is symmetrical relative to the translation with τ , if h : n → n exists having the property that (θ τ , h ) ∈ Aut (φ), where (θ τ , h ) = (1Bn , 1 Bn ). Remark 80. The translations are bijections, thus the statements (θ τ , h ) ∈ I so(φ, ψ) and (θ τ , h ) ∈ Aut (φ) make sense. Example 58. The function (μ1 , μ2 ) = (μ2 , μ1 ) has the following state portrait (0, 0) 6 (0, 1) 

- (1, 0)

? (1, 1)

If we translate in it with τ = (1, 0) the points of B2 , we get (1, 0) 6 (1, 1) 

- (0, 0)

? (0, 1)

representing the state portrait of (μ1 , μ2 ) = (μ2 , μ1 ).  and  are symmetrical relative to the translation with (1, 0), i.e. (θ (1,0) , 1B2 ) ∈ I so(, ) takes place. Indeed, for any μ, λ ∈ B2 we can write: (θ (1,0) ◦ λ )(μ) = θ (1,0) (λ1 μ1 ⊕ λ1 1 (μ), λ2 μ2 ⊕ λ2 2 (μ))

Chapter 8 • Morphisms of flows

87

= ((λ1 ⊕ 1)μ1 ⊕ λ1 (μ2 ⊕ 1) ⊕ 1, (λ2 ⊕ 1)μ2 ⊕ λ2 μ1 ) = (μ1 ⊕ μ1 λ1 ⊕ μ2 λ1 ⊕ λ1 ⊕ 1, μ2 ⊕ μ2 λ2 ⊕ μ1 λ2 ) and on the other hand ( λ ◦ θ (1,0) )(μ) =  λ (μ1 ⊕ 1, μ2 ) = (λ1 (μ1 ⊕ 1) ⊕ λ1 1 (μ1 ⊕ 1, μ2 ), λ2 μ2 ⊕ λ2 2 (μ1 ⊕ 1, μ2 )) = ((λ1 ⊕ 1)(μ1 ⊕ 1) ⊕ λ1 μ2 , (λ2 ⊕ 1)μ2 ⊕ λ2 μ1 ) = (μ1 ⊕ 1 ⊕ μ1 λ1 ⊕ λ1 ⊕ λ1 μ2 , μ2 ⊕ μ2 λ2 ⊕ μ1 λ2 ) i.e. θ (1,0) ◦ λ =  λ ◦ θ (1,0) . As 1B2 ∈ 2 , the flows φ, ψ are also symmetrical relative to the translation with (1, 0). Example 59. We continue the previous reasoning with the function (μ1 , μ2 ) = (μ1 , μ2 ). (0, 1) 

- (1, 1)

(0, 0) 

- (1, 0)

We translate each point of B2 with (0, 1) and we get (0, 0) 

(0, 1) 

-

(1, 0)

- (1, 1)

representing the state portrait of the same function . We prove that (θ (0,1) , 1B2 ) ∈ Aut () in the following way: ∀μ ∈ B2 , ∀λ ∈ B2 , (θ (0,1) ◦ λ )(μ) = θ (0,1) (λ1 μ1 ⊕ λ1 μ1 , μ2 ) = (λ1 μ1 ⊕ λ1 μ1 , μ2 ) and (λ ◦ θ (0,1) )(μ) = λ (μ1 , μ2 ) = (λ1 μ1 ⊕ λ1 μ1 , μ2 ). We have obtained that θ (0,1) ◦ λ = λ ◦ θ (0,1) . The flow φ is symmetrical relative to the translation with (0, 1).

88

Boolean Systems

8.7 Morphisms compatible with the subsystems 



Notation 28. The functions h : Bn → Bn , g : Bn+m → Bm are given, for which we denote   with h  g : Bn+m → Bn +m the function ∀μ ∈ Bn , ∀ν ∈ Bm , (h  g)(μ, ν) = (h(μ), g(μ, ν)). Similarly for the functions h : n → n , g  : n+m → m , where we denote with h  g  : n+m → n +m the function ∀α ∈ n , ∀β ∈ m , (h  g  )(α, β) = (h (α), g  (α, β)). Remark 81. Recall that, given the functions  : Bn+m → Bn+m ,  : Bn → Bn , we have called  a subsystem of , with the notation  ⊂ , if ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ). In this situation, Theorem 28, page 28 shows the existence of a morphism (π, π) :  → , where π(μ, ν) = μ, and Theorem 73, page 81, shows that (π,  π ) : γ → φ, as far as π ∈ n+m,n . Definition 71. We consider the systems  : Bn+m → Bn+m ,  : Bn → Bn ,  ⊂  and  :       Bn +m → Bn +m , ϒ : Bn → Bn , ϒ ⊂ .   (a) If for h, h : Bn → Bn , g, g  : Bn+m → Bm we have (h  g, h  g  ) ∈ H om(, ) and (h, h ) ∈ H om(, ϒ), we say that the morphism (h  g, h  g  ) is compatible with the subsystems , ϒ.   (b) Let h : Bn → Bn , h : n → n , g : Bn+m → Bm , g  : n+m → m . In case that (h  g, h  g  ) ∈ H om(γ , ψ) and (h, h ) ∈ H om(φ, υ), the morphism (h  g, h  g  ) is called compatible with the subsystems φ, υ. Problem 8. (a) In which conditions can we decompose (f, f  ) ∈ H om(, ) as f = h  g, f  = h  g  such that (h  g, h  g  ) is compatible with , ϒ? (b) When is the decomposition of (f, f  ) ∈ H om(γ , ψ) as f = h  g, f  = h  g  possible, such that the morphism (h  g, h  g  ) is compatible with the subsystems φ, υ? Remark 82. In the previous definition, item (a) is equivalent with the commutativity of the diagrams  (λ,δ) - n+m B

Bn+m hg 

?

Bn +m





(h g  )(λ,δ)

hg ?

- Bn +m

Chapter 8 • Morphisms of flows

89

λ - n B

Bn h

h ? ?   ϒ h (λ) - n B Bn for any λ ∈ Bn , δ ∈ Bm , and in fact the diagrams Bn

λ k Q Q Qπ Q Q Q Bn+m hg

h



?

Bn +m   π    ?+   Bn

- Bn  3  π   

 (λ,δ) - n+m B





hg ?

(h g  )(λ,δ)

h

- Bn +m Q Q Qπ Q Q  (λ) h Q ? s ϒ - Bn 





are all commutative. We have denoted with π : Bn+m → Bn , π  : Bn +m → Bn the projections on the first n, respectively on the first n coordinates. And item (b) means that (h(φ α (μ, k)), g(γ (α,β) ((μ, ν), k))) = (h  g)(γ (α,β) ((μ, ν), k)) 







= ψ (h g )(α,β) ((h  g)(μ, ν), k) = ψ (h (α),g (α,β)) ((h(μ), g(μ, ν)), k), 

h(φ α (μ, k)) = υ h (α) (h(μ), k) are true for all α ∈ n , β ∈ m , μ ∈ Bn , ν ∈ Bm , k ∈ N. Example 60. ,  : B2 → B2 , , ϒ : B → B are given by ∀μ ∈ B, ∀ν ∈ B, (μ, ν) = (μ, μν), (μ, ν) = (μ, μ ∪ ν), (μ) = μ, ϒ(μ) = μ, and h, h : B → B, g, g  : B2 → B are defined as ∀μ ∈ B, ∀ν ∈ B, h(μ) = μ,

90

Boolean Systems

h (μ) = 0, g(μ, ν) = ν, g  (μ, ν) = ν. For any μ ∈ B, ν ∈ B, λ ∈ B, δ ∈ B, we have (h  g)(μ, ν) = (μ, ν), (h  g  )(λ, δ) = (0, δ) and (h  g)( (λ,δ) (μ, ν)) = (h  g)(μ, δν ∪ δμν) = (μ, δν ∪ δμν) = (μ, δμ ∪ ν), 



 (h g )(λ,δ) ((h  g)(μ, ν)) =  (0,δ) (μ, ν) = (μ, δ ν ∪ δ(μ ∪ ν)) = (μ, δμ ∪ ν), h(λ (μ)) = h(μ) = μ, 

ϒ h (λ) (h(μ)) = ϒ 0 (μ) = μ. We conclude that (h  g, h  g  ) ∈ H om(, ) is compatible with the subsystems , ϒ of   g  ) ∈ H om(γ , ψ) makes no sense. , , but (h  g, h Example 61. We change slightly the previous example to ∀μ ∈ B, ∀ν ∈ B, (μ, ν) = (μ, μν), (μ, ν) = (μ, μ ∪ ν), (μ) = μ, ϒ(μ) = μ, and h, h : B → B, g, g  : B2 → B are defined in the following manner ∀μ ∈ B, ∀ν ∈ B, h(μ) = μ, h (μ) = μ, g(μ, ν) = ν, g  (μ, ν) = ν. In this case we obtain that (h  g, h  g  ) ∈ H om(, ) is compatible with the subsystems   g  ) ∈ H om(γ , ψ) is also compatible with the subsystems φ, υ of , ϒ of , , and (h  g, h   γ , ψ. The fact that (h  g )(λ, δ) = (λ, δ) has fixed the problem from Example 60.

Chapter 8 • Morphisms of flows

91

8.8 Morphisms vs. duality Theorem 76. For the functions ,  : Bn −→ Bn , the following statement is true: ψ = φ ∗ if and only if (θ (1,...,1) , 1 Bn ) ∈ I so(φ, ψ). Proof. If. For arbitrary α ∈ n and μ ∈ Bn , the fact that (θ (1,...,1) , 1 Bn ) ∈ I so(φ, ψ) implies ∀k ∈ N, θ (1,...,1) (φ α (μ, k)) = ψ α (θ (1,...,1) (μ), k), thus we obtain ψ α (μ, k) = φ α (μ, k)

Theorem 59, page 61

=

(φ ∗ )α (μ, k).

Theorem 77. Let  : Bn −→ Bn and  : Bm −→ Bm . The following equivalence is true: (h, h ) ∈ H om(φ, ψ) ⇐⇒ (h∗ , h ) ∈ H om(φ ∗ , ψ ∗ ). Proof. =⇒ We fix α ∈ n , β ∈ m , μ ∈ Bn , ν ∈ Bm , k ∈ N arbitrary. We know from Theorem 59, page 61 that φ α (μ, k) = (φ ∗ )α (μ, k),

(8.8.1)

ψ β (ν, k) = (ψ ∗ )β (ν, k),

(8.8.2)

and the hypothesis (h, h ) ∈ H om(φ, ψ) means the truth of 

h(φ α (μ, k)) = ψ h (α) (h(μ), k).

(8.8.3)

We infer (8.8.1)

(8.8.3)

h∗ ((φ ∗ )α (μ, k)) = h((φ ∗ )α (μ, k)) = h(φ α (μ, k)) = ψ h (α) (h(μ), k) 



(8.8.2)

= ψ h (α) (h∗ (μ), k) = (ψ ∗ )h (α) (h∗ (μ), k), 

i.e. (h∗ , h ) ∈ H om(φ ∗ , ψ ∗ ). Remark 83. Similarly with Remark 23, page 27, we see that the flows φ, φ ∗ are isomorphic, and the flows ψ, ψ ∗ are isomorphic too (Theorem 76). The previous theorem can be generalized to morphisms of isomorphic flows.

8.9 Morphisms vs. orbits and omega-limit sets Theorem 78. The functions  : Bn → Bn ,  : Bm → Bm and the morphism (h, h ) ∈ H om(φ, ψ) are given. We infer ∀α ∈ n , ∀μ ∈ Bn , h (α)

α (μ)) = O h(O

(h(μ)),

(8.9.1)

h (α) α (μ)) = ω (h(μ)). h(ω

(8.9.2)



92

Boolean Systems

Proof. (8.9.1): We fix α ∈ n , μ ∈ Bn arbitrary, for which α (μ)) = h(O(φ α (μ, ·))) h(O

Theorem 48, page 49

=

h (α)



= O(ψ h (α) (h(μ), ·)) = O

O( h(φ α (μ, ·)))

(h(μ)).

The proof of (8.9.2) is similar.

8.10 Morphisms vs. Cartesian products Theorem 79. Let the functions  : Bn → Bn ,  : Bm → Bm , ϒ : Bp → Bp ,  : Bq → Bq , h : Bn → Bp , h : n → p , g : Bm → Bq , g  : m → q and we suppose that (h, h ) : φ → υ, (g, g  ) : ψ → γ are morphisms. We denote with (φ, ψ) the flow generated by  × , with (υ, γ ) the flow generated by ϒ × , with (h, g) : Bn+m → Bp+q the function ∀μ ∈ Bn , ∀ν ∈ Bm , (h, g)(μ, ν) = (h(μ), g(ν)),

(8.10.1)

and with (h , g  ) : n+m → p+q the function ∀α ∈ n , ∀β ∈ m , (h , g  )(α, β) = (h (α), g  (β)).

(8.10.2)

Then ((h, g), (h , g  )) : (φ, ψ) → (υ, γ ) is a morphism. Proof. We fix α ∈ n , β ∈ m , μ ∈ Bn , ν ∈ Bm , k ∈ N arbitrary and we can write from the hypothesis: 

h(φ α (μ, k)) = υ h (α) (h(μ), k),

(8.10.3)



g(ψ β (ν, k)) = γ g (β) (g(ν), k).

(8.10.4)

We infer: (h, g)((φ, ψ)(α,β) ((μ, ν), k)) (8.10.1)

=

Theorem 70, page 76

=

(h(φ α (μ, k)), g(ψ β (ν, k))) Theorem 70, page 76

=

(8.10.1),(8.10.2)

=

(8.10.3),(8.10.4)

= 

(h, g)(φ α (μ, k), ψ β (ν, k)) 



(υ h (α) (h(μ), k), γ g (β) (g(ν), k))



(υ, γ )(h (α),g (β)) ((h(μ), g(ν)), k) 



(υ, γ )(h ,g )(α,β) ((h, g)(μ, ν), k).

8.11 Morphisms vs. successors and predecessors Theorem 80. We consider the generator functions  : Bn → Bn ,  : Bm → Bm and the point μ ∈ Bn . (a) If (h, h ) ∈ H om(φ, ψ), then + h(μ+  ) ⊂ h(μ) ,

(8.11.1)

Chapter 8 • Morphisms of flows

93

+ + (μ)) ⊂ O (h(μ)), h(O − h(μ−  ) ⊂ h(μ) , − − (μ)) ⊂ O (h(μ)); h(O

(8.11.2)

(b) if (h, h ) ∈ I so(φ, ψ), we infer + h(μ+  ) = h(μ) , + + (μ)) = O (h(μ)), h(O − h(μ−  ) = h(μ) , − − h(O (μ)) = O (h(μ)).

(8.11.3)

Proof. (a) We prove (8.11.1), but please compare this proof with the proof of Theorem 30, page 29, inclusion (2.10.1)page 29 , because the two proofs cannot be different. n Let δ ∈ h(μ+  ) arbitrary, meaning the existence of λ ∈ B with the property that δ = λ h( (μ)). The hypothesis states the truth of ∀k ∈ N, 

h(φ α (μ, k)) = ψ h (α) (h(μ), k), 

where we choose α ∈ n arbitrary, with α 0 = λ. We infer δ = h(φ α (μ, 1)) = ψ h (α) (h(μ), 1) =  0  h (α) (h(μ)), i.e. δ ∈ h(μ)+ . (b) We prove (8.11.3) by taking into account (8.11.2), but please compare this proof with the proof of Theorem 30, page 29, Eq. (2.10.8)page 29 . − − − (h(μ)) ⊂ h(O (μ)). Let δ ∈ O (h(μ)), i.e. ∃α ∈ n , ∃k ∈ N with We prove that O α  ψ (δ, k) = h(μ). (h, h ) is isomorphism, therefore we obtain −1 (α)

h−1 (ψ α (δ, k)) = φ h

(h−1 (δ), k) = μ.

− − (μ), in other words δ ∈ h(O (μ)). This equation means that h−1 (δ) ∈ O

8.12 Morphisms vs. limits Theorem 81. For  : Bn −→ Bn ,  : Bm −→ Bm , h : Bn −→ Bm and h : n → m , we suppose that (h, h ) : φ → ψ is a morphism. Then ∀α ∈ n , ∀μ ∈ Bn , if lim φ α (μ, k) exists, then k→∞



lim ψ h (α) (h(μ), k) exists and

k→∞



lim ψ h (α) (h(μ), k) = h( lim φ α (μ, k)).

k→∞

k→∞

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Boolean Systems

Proof. We fix α ∈ n , μ ∈ Bn arbitrary. If k  ∈ N exists such that ∀k ≥ k  , φ α (μ, k) = φ α (μ, k  ), then in the equality 

h(φ α (μ, k)) = ψ h (α) (h(μ), k), both terms are constant for k ≥ k  .

8.13 Morphisms vs. orbital and omega-limit equivalence Theorem 82. Let the functions  : Bn → Bn ,  : Bm → Bm . If (h, h ) : φ −→ ψ is a morphism, then for any α, β ∈ n , μ ∈ Bn , φ α (μ, ·) ∼ φ β (μ, ·) implies 



ψ h (α) (h(μ), ·) ∼ ψ h (β) (h(μ), ·), i.e. the morphisms of flows are compatible with the orbital equivalence. At the same time, φ α (μ, ·) ≈ φ β (μ, ·) implies 



ψ h (α) (h(μ), ·) ≈ ψ h (β) (h(μ), ·), meaning that the morphisms of flows are compatible with the omega-limit equivalence. Proof. In the first case, we have the existence of the sequences 0 = i0 < i1 < i2 < ..., 0 = j0 < j1 < j2 < ... such that for k ∈ N arbitrary, fixed: φ α (μ, ik ) = φ α (μ, ik + 1) = ... = φ α (μ, ik+1 − 1) = φ β (μ, jk ) = φ β (μ, jk + 1) = ... = φ β (μ, jk+1 − 1). We infer h(φ α (μ, ik )) = h(φ α (μ, ik + 1)) = ... = h(φ α (μ, ik+1 − 1)) = h(φ β (μ, jk )) = h(φ β (μ, jk + 1)) = ... = h(φ β (μ, jk+1 − 1)), therefore 





ψ h (α) (h(μ), ik ) = ψ h (α) (h(μ), ik + 1) = ... = ψ h (α) (h(μ), ik+1 − 1) 





= ψ h (β) (h(μ), jk ) = ψ h (β) (h(μ), jk + 1) = ... = ψ h (β) (h(μ), jk+1 − 1).

Chapter 8 • Morphisms of flows

95

8.14 Pseudo-morphisms Definition 72. We consider  : Bn → Bn and  : Bm → Bm . A pseudo-morphism from φ to ψ, denoted by (h, h ) : φ  ψ, is given by the functions h : Bn → Bm , h : n → m with the property that ∀α ∈ n , ∀μ ∈ Bn , 

h(φ α (μ, ·)) ∼ ψ h (α) (h(μ), ·).

(8.14.1)

om(φ, ψ). Notation 29. The set of the pseudo-morphisms from φ to ψ has the notation H Example 62. We have drawn the state portraits of the functions ,  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 ∪ μ2 , μ1 ), (μ1 , μ2 ) = (μ1 μ2 , μ1 ) (0, 0)

- (0, 1)

- (1, 1)

- (1, 0)

- (0, 1)

- (0, 0)

(a) (1, 0)

- (1, 1) (b)

2 at (a) and (b). We notice that (θ (1,0) , 1 B2 ) ∈ H om(φ, ψ) : ∀α ∈ 2 , ∀μ ∈ B , ∀k ∈ N,

ψ α (θ (1,0) (μ), k) = θ (1,0) (φ α (μ, k)). But we can see that ∀α ∈ 2 , ∀β ∈ 2 , ∀μ ∈ B2 , ψ α (μ, ·) ∼ ψ β (μ, ·), and we say that ψ (or ) is race-free (it satisfies the technical condition of proper operaom(φ, ψ), tion).1 In such conditions we have for arbitrary h : 2 → 2 that (θ (1,0) , h ) ∈ H 2 i.e. ∀α ∈ 2 , ∀μ ∈ B , 

θ (1,0) (φ α (μ, ·)) ∼ ψ h (α) (θ (1,0) (μ), ·) is true. Problem 9. To be studied how the results on morphisms of flows are generalized to pseudomorphisms.

1

φ is in the same situation.

9 Nullclines The nullclines, defined in Section 9.1, are the orbits having the property that all their points have a coordinate i ∈ {1, ..., n} which is not excited. Several examples are given in Section 9.2. The properties of the nullclines from Section 9.3 include the fact that if O α (μ) has the i-th coordinate not excited, then ∀λ ∈ O α (μ), λi = μi , the nullclines have the not excited coordinate constant. Section 9.4 considers the special case when all the orbits O α (μ) are nullclines, for α variable in n and μ variable in Bn .

9.1 Definition Notation 30. Let  : Bn → Bn , and we denote with N Ci ⊂ Bn , i ∈ {1, ..., n} the sets N Ci = {μ|μ ∈ Bn , i (μ) = μi }. Definition 73. A nullcline is an orbit O α (μ), where α ∈ n and μ ∈ Bn , with the property that ∃i ∈ {1, ..., n}, O α (μ) ⊂ N Ci . Remark 84. A nullcline O α (μ) ⊂ N Ci is the orbit of a system with the property that the coordinate i of the state is always stable: running in the state portrait of  through the points of the orbit, no point has the coordinate i underlined.

9.2 Examples Example 63. The limit situation ∀i ∈ {1, ..., n}, N Ci = ∅ is that of  : Bn → Bn , ∀μ ∈ Bn , (μ) = μ. And the other limit situation, ∀i ∈ {1, ..., n}, N Ci = Bn is that of the identity 1Bn : Bn → Bn . Example 64. In the following example (0, 0)

- (0, 1)

(1, 0)

- (1, 1)

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00015-5 Copyright © 2023 Elsevier Inc. All rights reserved.

97

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Boolean Systems

we have N C1 = B2 , N C2 = {(0, 1), (1, 1)} and there are, when α runs in 2 and μ runs in B2 , four nullclines: {(0, 0), (0, 1)}, {(1, 0), (1, 1)} ⊂ N C1 , {(0, 1)}, {(1, 1)} ⊂ N C1 ∧ N C2 . Example 65. The nullclines can be closed paths too. In this example (0, 0)



(1, 0) 

- (0, 1)

- (1, 1)

we see that N C1 = B2 , N C2 = ∅ and there are two nullclines {(0, 0), (0, 1)}, {(1, 0), (1, 1)} ⊂ N C1 . Example 66. The system (0, 0, 0) 6 (0, 1, 0) 

- (0, 0, 1)

? (0, 1, 1)

(1, 0, 0) 6

(1, 1, 1) 6

? (1, 0, 1)

? (1, 1, 0)

gives another example when the nullclines are closed paths. We get N C1 = B3 , N C2 = {(0, 0, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}, N C3 = {(0, 0, 1), (0, 1, 0)} and we conclude that {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)} ⊂ N C1 , {(1, 0, 0), (1, 0, 1)}, {(1, 1, 0), (1, 1, 1)} ⊂ N C1 ∧ N C2 . Example 67. The orbits in the previous examples of this section did not depend on α. We give now an example - (0, 0, 1) (0, 0, 0) Q Q Q Q Q ? s Q (0, 1, 0) (0, 1, 1)

- (1, 0, 1) (1, 0, 0) Q Q Q Q Q ? s Q (1, 1, 0) (1, 1, 1)

when the orbits are included in N C1 = B3 and they depend on α. In addition, {(0, 0, 1)}, {(0, 1, 0)}, {(0, 1, 1)}, {(1, 0, 1)}, {(1, 1, 0)}, {(1, 1, 1)} ⊂ N C1 ∧ N C2 ∧ N C3 .

Chapter 9 • Nullclines

99

9.3 Properties Theorem 83.  : Bn → Bn is given and let i ∈ {1, ..., n} with μ ∈ N Ci . Then ∀δ ∈ μ+ , δi = μi ,

(9.3.1)

Proof. We take an arbitrary δ ∈ μ+ . This means the existence of λ ∈ Bn with δ = λ (μ). We infer  μi , if λi = 0, δi = = μi . i (μ), if λi = 1 Remark 85. The statement of Theorem 83 relative to μ− is false, in the sense that if μ ∈ N Ci , then an arbitrary δ ∈ μ− may satisfy δi = μi . In order to see this, we refer to Example 67 where μ = (0, 0, 1) ∈ N C3 , δ = (0, 0, 0) ∈ (0, 0, 1)− , and (0, 0, 0)3 = (0, 0, 1)3 . Theorem 84. For any α ∈ n , μ ∈ Bn , and i ∈ {1, ..., n}, the following statements are equivalent: O α (μ) ⊂ N Ci ,

(9.3.2)

∀k ∈ N, i (φ α (μ, k)) = φiα (μ, k).

(9.3.3)

Proof. (9.3.2)=⇒(9.3.3) Let k ∈ N arbitrary, and we have φ α (μ, k) ∈ O α (μ). As φ α (μ, k) ∈ N Ci , we infer i (φ α (μ, k)) = φiα (μ, k). (9.3.3)=⇒(9.3.2) We suppose against all reason that (9.3.2) is false, i.e. some δ ∈ O α (μ) exists such that δ ∈ / N Ci . This means the existence of k ∈ N with δ = φ α (μ, k). We get that α α i (φ (μ, k)) = φi (μ, k), contradiction. Corollary 6.  : Bn → Bn , α ∈ n , μ ∈ Bn and i ∈ {1, ..., n} are given. If O α (μ) ⊂ N Ci , then ∀k ∈ N, φiα (μ, k) = μi . Proof. This follows from Theorem 84. Theorem 85. ∀μ ∈ Bn , ∀i ∈ {1, ..., n} we have the equivalence of O + (μ) ⊂ N Ci ,

(9.3.4)

∀δ ∈ O + (μ), δi = μi .

(9.3.5)

Proof. (9.3.4)=⇒(9.3.5) For an arbitrary δ ∈ O + (μ), we get the existence of α ∈ n , k ∈ N with the property that δ = φ α (μ, k). If k = 0 then δi = μi , thus we can suppose that k ≥ 1. We (9.3.4)

infer, since O α (μ) ⊂ O + (μ) ⊂ N Ci :  δi = φiα (μ, k) =

φiα (μ, k − 1), if αik−1 = 0, i (φ α (μ, k − 1)), if αik−1 = 1

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Theorem 84, (9.3.3)

=

φiα (μ, k − 1) = ... = φiα (μ, 0) = μi .

(9.3.5)=⇒(9.3.4) We suppose against all reason the falsity of (9.3.4), meaning the exis/ N Ci , i.e. i (φ α (μ, k)) = φiα (μ, k). Then (φ α (μ, k)), tence of α ∈ n , k ∈ N with φ α (μ, k) ∈ α + φ (μ, k) ∈ O (μ) and i (φ α (μ, k)) = μi or φiα (μ, k) = μi , contradiction with (9.3.5). Remark 86. The implications ∀k ∈ N, φiα (μ, k) = μi =⇒ O α (μ) ⊂ N Ci , ∀λ ∈ Bn , λi (μ) = μi =⇒ μ+ ⊂ N Ci , stated for α ∈ n , μ ∈ Bn and i ∈ {1, ..., n}, are both false. A counterexample for the first statement results from Example 67, where we note the existence of α ∈ 3 such that ∀k ∈ / N C2 . The state portrait N, φ2α ((0, 0, 0), k) = 0, O α (0, 0, 0) = {(0, 0, 0), (0, 0, 1)}, and (0, 0, 0) ∈ - (0, 1, 0) (0, 0, 0) Q Q Q Q Q ? s Q (1, 0, 0) (1, 1, 0)

- (0, 1, 1)

(1, 0, 1)

(0, 0, 1)

(1, 1, 1)

gives also a counterexample for the second statement. We have there ∀λ ∈ B3 , λ3 (0, 0, 0) = 0 and (0, 1, 0) ∈ (0, 0, 0)+ , but 3 (0, 1, 0) = 1, thus (0, 0, 0)+ ⊂ N C3 .

9.4 Special case: NCi = Bn Remark 87. If i ∈ {1, ..., n} exists such that N Ci = Bn , then any orbit is a nullcline: ∀μ ∈ Bn , ∀α ∈ n , O α (μ) ⊂ N Ci . Remark 88. Note that the possibility N Ci = N Cj = Bn , i = j exists, see Example 63, page 97. Notation 31. We denote for a set A ⊂ Bn and i ∈ {1, ..., n}: A0i = {μ|μ ∈ A, μi = 0}, A1i = {μ|μ ∈ A, μi = 1}. In particular, we have the two half-spaces of Bn : (Bn )0i = {μ|μ ∈ Bn , μi = 0},

Chapter 9 • Nullclines

101

(Bn )1i = {μ|μ ∈ Bn , μi = 1}. Theorem 86. We suppose that i ∈ {1, ..., n} exists such that ∀μ ∈ Bn , i (μ) = μi , in other words N Ci = Bn . Then ∀α ∈ n , ∀μ ∈ (Bn )0i , O α (μ) ⊂ (Bn )0i ,

(9.4.1)

∀α ∈ n , ∀μ ∈ (Bn )1i , O α (μ) ⊂ (Bn )1i .

(9.4.2)

Proof. In order to prove (9.4.1), we take α ∈ n , μ ∈ (Bn )0i arbitrary and, as far as O α (μ) ⊂ N Ci , we apply Corollary 6. Remark 89. The theorem shows the way that N Ci = Bn partitions Bn in two sets, (Bn )0i and (Bn )1i that contain, together with a point μ, all the orbits starting from μ. The property is called invariance and will be studied later. Remark 90. Reread Examples 64–67, page 97 where we have N C1 = B2 , N C1 = B2 , N C1 = B3 , N C1 = B3 . Theorem 87. We consider the functions ,  : Bn → Bn with  = λ , where λ ∈ Bn and we ask that λ = (1, ..., 1). Then (a) ∀i ∈ {1, ..., n}  supp λ, N C,i = Bn , (b) we have ∀α ∈ n , ∀μ ∈ Bn ,  μ α O (μ) ⊂ (Bn )i i . i∈{1,...,n}supp λ

Proof. (a) For any i ∈ {1, ..., n}  supp λ and any μ ∈ Bn we have λi = 0 and i (μ) = λi (μ) = μi . This proves that Bn ⊂ N C,i , i.e. (a) holds. (b) We fix α ∈ n , μ ∈ Bn arbitrary, and we suppose that λi = 0 is true, with i ∈ {1, ..., n} α (μ) ⊂ (Bn )μi . arbitrary, fixed too. We have from Theorem 86 that O i

10 Fixed points We first give in Section 10.1 several properties which are equivalent with the fact that (μ) = μ. The main purpose of Section 10.2 is that of presenting two important theorems concerning the fixed points: the final value of a state, if it exists, is a fixed point of the generator function; and an accessible fixed point of the generator function is the final value of the state. The morphisms of flows, as stated in Section 10.3, bring fixed points in fixed points.

10.1 Definition Theorem 88. Let  : Bn → Bn and μ ∈ Bn . The following statements are equivalent: (μ) = μ,

(10.1.1)

μ ∈ N C1 ∧ ... ∧ N Cn ,

(10.1.2)

μ+ = {μ},

(10.1.3)

O + (μ) = {μ},

(10.1.4)

∀α ∈ n , ∀k ∈ N, φ α (μ, k) = μ,

(10.1.5)

∀α ∈ n , O α (μ) = {μ},

(10.1.6)

∃α ∈ n , ∀k ∈ N, φ α (μ, k) = μ,

(10.1.7)

∃α ∈ n , O α (μ) = {μ}.

(10.1.8)

Proof. The scheme of the proof is

⇑

=⇒ (10.1.5) ⇓ (10.1.7)

(10.1.3)

⇐= (10.1.2)

(10.1.4)

=⇒ (10.1.6) ⇓ =⇒ (10.1.8) ⇓ ⇐= (10.1.1)

(10.1.1)=⇒(10.1.2) If 1 (μ) = μ1 , ..., n (μ) = μn , then μ ∈ N C1 , ..., μ ∈ N Cn . Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00016-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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Boolean Systems

(10.1.2)=⇒(10.1.3) The hypothesis (10.1.2) implies 1 (μ) = μ1 , ..., n (μ) = μn , therefore (μ) = μ. We have μ+ = [μ, (μ)] = [μ, μ] = {μ}. (10.1.3)=⇒(10.1.4) This results from Theorem 10, page 11. (10.1.4)=⇒(10.1.5) We prove the implication by induction on k ∈ N. If k = 0, then k k−1 0 φ α (μ, 0) = μ, so that we take k ≥ 1 and α 0 , ..., α k ∈ Bn arbitrary. As (α ◦α ◦...◦α )(μ) ∈ O + (μ) = {μ}, we compute k

φ α (μ, k + 1) = (α ◦ α

k−1

0

◦ ... ◦ α )(μ) = μ.

(10.1.5)=⇒(10.1.6), (10.1.6)=⇒(10.1.8) and (10.1.5)=⇒(10.1.7), (10.1.7)=⇒(10.1.8) are obvious. (10.1.8)=⇒(10.1.1) A computation function α ∈ n exists with ∀k ∈ N, φ α (μ, k) = μ, and we can prove by induction on k that ∀k ∈ N, k

α (μ) = μ.

(10.1.9)

We infer ∀i ∈ {1, ..., n}, ∃k ∈ N, αik = 1 for which k

i (μ) = αi (μ)

(10.1.9)

=

μi .

Definition 74. A point that is not fixed (i.e. if (μ) = μ) is called ordinary.

10.2 Fixed points vs. final values. Rest position Theorem 89. The following fixed point property is true ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , lim φ α (μ, k) = μ =⇒ (μ ) = μ . k→∞

Proof. We fix α ∈ n , μ ∈ Bn , μ ∈ Bn arbitrary and the hypothesis states that k  ∈ N exists with ∀k ≥ k  , φ α (μ, k) = μ . We get that ∀k ∈ N, φσ

k  (α)

(μ , k) = φ σ

k  (α)

(φ α (μ, k  ), k) = φ α (μ, k + k  ) = μ .

The statement (10.1.7) is true thus, from Theorem 88, implication (10.1.7)=⇒(10.1.1), we get the truth of (10.1.1) under the form (μ ) = μ . Remark 91. Theorem 89 shows that the final values of the states φ α (μ, ·) are fixed points of .

Chapter 10 • Fixed points

105

Theorem 90. We have ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , ((μ ) = μ and ∃k  ∈ N, φ α (μ, k  ) = μ ) =⇒ ∀k ≥ k  , φ α (μ, k) = μ . Proof. We take α ∈ n , μ ∈ Bn , μ ∈ Bn arbitrary, fixed such that (μ ) = μ

(10.2.1)

and the hypothesis asks also the existence of k  ∈ N with φ α (μ, k  ) = μ .

(10.2.2)

Statement (10.2.1) interpreted as (10.1.1), and Theorem 88, implication (10.1.1)=⇒(10.1.5) show the truth of (10.1.5), thus ∀k ∈ N, φσ

k  (α)

(μ , k) = μ ,

(10.2.3)

wherefrom ∀k ≥ k  , φ α (μ, k) = φ σ

k  (α)

(φ α (μ, k  ), k − k  )

(10.2.2)

=

φσ

k  (α)

(μ , k − k  )

(10.2.3)

=

μ .

Remark 92. As resulting from Theorem 90, the accessible fixed points are final values of the states φ α (μ, ·). Definition 75. Let α ∈ n , μ, μ ∈ Bn and we presume that (μ ) = μ . If μ ∈ O α (μ), then μ is called a rest position (or an equilibrium point, a critical point, a singular point, a stationary point, a steady state, a final value) of φ or of Eq . Corollary 7. For arbitrary α ∈ n , μ ∈ Bn , the following statements are true: (a) lim φ α (μ, k) exists ⇐⇒ card(ωα (μ)) = 1; k→∞

(b) ∀μ ∈ Bn , if ωα (μ) = {μ }, then lim φ α (μ, k) = μ and (μ ) = μ ; k→∞

(c) ∀μ ∈ Bn , if (μ ) = μ and μ ∈ O α (μ), then ωα (μ) = {μ }. Proof. (a) This results from Theorem 40, page 45. (b) We apply Theorem 40 for showing that lim φ α (μ, k) = μ , while (μ ) = μ results k→∞

from Theorem 89. (c) This is a consequence of Theorem 90.

10.3 Morphisms vs. fixed points Theorem 91. The functions  : Bn −→ Bn ,  : Bm −→ Bm are given, together with μ ∈ Bn and (h, h ) ∈ H om(φ, ψ). If (μ) = μ, then (h(μ)) = h(μ). Proof. We take α ∈ n arbitrary. The hypothesis (μ) = μ shows from (10.1.1) and Theorem 88, implication (10.1.1)=⇒(10.1.5), the truth of (10.1.5): ∀k ∈ N,

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Boolean Systems

φ α (μ, k) = μ, therefore we have ∀k ∈ N, 

ψ h (α) (h(μ), k) = h(φ α (μ, k)) = h(μ). This acts as (10.1.7) and we use Theorem 88, implication (10.1.7)=⇒(10.1.1); we conclude that (10.1.1) is true, written as (h(μ)) = h(μ). Remark 93. To be compared Theorem 91 with Theorem 31, page 30.

11 Sources, isolated fixed points, transient points, sinks For any  : Bn −→ Bn , the points μ ∈ Bn are four kinds: sources, isolated fixed points, transient points, and sinks. These possibilities are defined in Section 11.1. In Section 11.2, we show that the isomorphisms I so(φ, ψ) bring the sources of  in the sources of , ..., the sinks of  in the sinks of . Other simple properties are studied in Section 11.3, where we refer to the Hamiltonian paths of the state portraits and to the points of the Cartesian product  × .

11.1 Definition Definition 76. The function  : Bn −→ Bn is given. A point μ ∈ Bn is called1 : (a) source, if μ− = {μ}, μ+ = {μ}, (b) isolated fixed point, if μ− = {μ}, μ+ = {μ}, (c) transient point, if μ− = {μ}, μ+ = {μ}, (d) sink, if μ− = {μ}, μ+ = {μ}. Example 68. We consider the function with the following state portrait, (0, 0, 0)  Q  Q  Q  Q  Q  + s Q (0, 1, 0) (0, 0, 1)      ?+  (0, 1, 1)

(1, 0, 1)

(1, 0, 0)

? - (1, 1, 1)

(1, 1, 0)

where (0, 0, 0), (1, 0, 1) are sources, (1, 0, 0), (1, 1, 0) are isolated fixed points, (0, 0, 1), (0, 1, 1) are transient points and (0, 1, 0), (1, 1, 1) are sinks. For example we have: (0, 0, 0)− = {(0, 0, 0)}, (0, 0, 0)+ = {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)}; 1

In [22] the garden-of-eden states are defined as ‘states that can arise only as initial states of the system and can never be dynamically generated during the course of the subsequent evolution’ and this is equivalent here with μ− = {μ}. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00017-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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Boolean Systems

(1, 0, 0)− = {(1, 0, 0)}, (1, 0, 0)+ = {(1, 0, 0)}; (0, 0, 1)− = {(0, 0, 0), (0, 0, 1)}, (0, 0, 1)+ = {(0, 0, 1), (0, 1, 1)}; (0, 1, 0)− = {(0, 0, 0), (0, 1, 0)}, (0, 1, 0)+ = {(0, 1, 0)}. Example 69. We suppose that  is the constant function ∀μ ∈ Bn , (μ) = μ , see Theorem 11, page 13, and that n ≥ 2. Then μ is a sink (μ+ = [μ , μ ] = {μ }, μ− = [μ , μ ] = Bn ), +

−

μ is a source (μ = [μ , μ ] = Bn , μ = [μ , μ ] = {μ }), and any μ ∈ (μ , μ ) is transient (μ+ = [μ, μ ], μ− = [μ , μ]). Remark 94. If a function  : Bn −→ Bn is given, any point μ ∈ Bn is in exactly one of the situations (a)–(d) from Definition 76. We see that: – the sources and the isolated fixed points, where μ− = {μ}, fulfill ∀λ ∈ Bn , (λ )−1 (μ) ∈ {∅, {μ}}, – the isolated fixed points and the sinks, where μ+ = {μ}, satisfy either of ∀λ ∈ Bn , λ  (μ) = μ and (μ) = μ, in particular the isolated fixed points are fixed points indeed. Remark 95. As far as μ− = {μ} ⇐⇒ O − (μ) = {μ}, μ+ = {μ} ⇐⇒ O + (μ) = {μ}, see Theorem 10, page 11, we get that Definition 76 may be expressed equivalently by replacing μ− , μ+ with O − (μ), O + (μ).

11.2 Morphisms Theorem 92. Let  : Bn → Bn ,  : Bn → Bn and μ ∈ Bn . If (h, h ) ∈ I so(φ, ψ), then (a) μ is a source of  iff h(μ) is a source of , (b) μ is an isolated fixed point of  iff h(μ) is an isolated fixed point of , (c) μ is a transient point of  iff h(μ) is a transient point of , (d) μ is a sink of  iff h(μ) is a sink of . Proof. Theorem 80, page 92 shows that + card(h(μ+  )) = card(h(μ) ), − card(h(μ−  )) = card(h(μ) ).

Remark 96. A special case of isomorphism is represented by duality; from this point of view, (a) of the previous theorem reads as μ is a source for  if and only if μ is a source for ∗ , (b) reads as μ is an isolated fixed point for  if and only if μ is an isolated fixed point for ∗ , ...

Chapter 11 • Sources, isolated fixed points, transient points, sinks

109

11.3 Other properties Theorem 93. If G = (Bn , {(μ, λ (μ))|μ, λ ∈ Bn , μ = λ (μ)}) has a Hamiltonian path (see Definition 38, page 40), then Bn has at most one source, at most one sink, and all the points that are neither source nor sink are transient. n −1

Proof. We put Bn under the form Bn = {μ0 , μ1 , ..., μ2 a Hamiltonian path. We infer

n −1

} and let μ0 → μ1 → ... → μ2

be

{μ0 , μ1 } ⊂ (μ0 )+ ∧ (μ1 )− , {μ1 , μ2 } ⊂ (μ1 )+ ∧ (μ2 )− , ... n −2

{μ2

n −1

, μ2

n −2

} ⊂ (μ2

)+ ∧ (μ2

n −1

)− ,

wherefrom μ1 , ..., μ2 −2 are transient points. For an arbitrary μ ∈ Bn the possibility μ− = n {μ} exists, when μ = μ0 is a source, and the possibility μ+ = {μ} exists also, when μ = μ2 −1 is a sink. n

Theorem 94. We consider the affine space [μ, (μ)] = {μ} and λ ∈ [μ, (μ)]. (a) If λ = μ, λ is either a source, or a transient point, (b) if λ ∈ (μ, (μ)], λ is either a sink, or a transient point. Proof. (a) The hypothesis states that λ+ = [λ, (λ)] = {λ}. If λ is a sink or an isolated fixed point, then λ+ = {λ}, contradiction. (b) We have μ = λ and λ ∈ μ+ , thus μ ∈ λ− , therefore λ− = {λ}. Theorem 95. The functions  : Bn → Bn ,  : Bm → Bm are given and we consider the Cartesian product  ×  : Bn × Bm → Bn × Bm . We have the situation from Table 3 that is to + m be read like this (the first row): if μ ∈ Bn is a source μ−  = {μ}, μ = {μ} and ν ∈ B is a − + − + source ν = {ν}, ν = {ν}, then (μ, ν) is a source (μ, ν)× = {(μ, ν)}, (μ, ν)× = {(μ, ν)}; + − + m if μ ∈ Bn is a source μ−  = {μ}, μ = {μ} and ν ∈ B is a sink ν = {ν}, ν = {ν}, then − + (μ, ν) is a transient point (μ, ν)× = {(μ, ν)}, (μ, ν)× = {(μ, ν)}; if μ ∈ Bn is a source + − + m μ−  = {μ}, μ = {μ} and ν ∈ B is an isolated fixed point ν = {ν}, ν = {ν}, then (μ, ν) is − + a source (μ, ν)× = {(μ, ν)}, (μ, ν)× = {(μ, ν)}, ... Table 3

The Cartesian product.

×

source

sink

is.f.p.

tr.p.

source

source

tr.p.

source

tr.p.

sink

tr.p.

sink

sink

tr.p.

is.f.p.

source

sink

is.f.p.

tr.p.

tr.p.

tr.p.

tr.p.

tr.p.

tr.p.

110

Boolean Systems

Proof. We take μ ∈ Bn , ν ∈ Bm arbitrary, fixed and we apply Theorem 15, page 15. For ex+ − + ample if μ−  = {μ}, μ = {μ} and ν = {ν}, ν = {ν}, then − − (μ, ν)− × = μ × ν = {μ} × {ν} = {(μ, ν)}, + + (μ, ν)+ × = μ × ν = {μ} × {ν} = {(μ, ν)}.

12 Sets of reachable states Section 12.1 of the chapter is dedicated to mathematical preliminaries concerning the convergence of the monotonic sequences of sets Xk ⊂ Bn , k ∈ N. In Section 12.2, we introduce the sequences Ak (μ) ⊂ Bn , k ∈ N, called the sets of states which are reached, from μ, in k time units, and Bk (μ) ⊂ Bn , k ∈ N, called the sets of states that reach μ in k time units. Several properties of these sets that characterize the successors and the predecessors of the points   μ are pointed out, of which the most important are + − O (μ) = Ak (μ) and O (μ) = Bk (μ). An example is given in Section 12.3. k∈N

k∈N

In Section 12.4 we show how the isomorphisms of flows act on the sets of reachable states.

12.1 Convergent sequences of sets Definition 77. The sequence of sets Xk ⊂ Bn , k ∈ N is called convergent, if a rank k  ∈ N exists such that ∀k ≥ k  , Xk = Xk  .

(12.1.1)

In (12.1.1), the set Xk  is called the limit of the sequence and its notation is Xk  = lim Xk . k→∞

Definition 78. The sequence Xk ⊂ Bn , k ∈ N is called (monotonically) increasing if X0 ⊂ X1 ⊂ X2 ⊂ ...

(12.1.2)

X0 ⊃ X1 ⊃ X2 ⊃ ...

(12.1.3)

and (monotonically) decreasing if

If the sequence fulfills one of (12.1.2), (12.1.3) is it called monotonic. Theorem 96. The monotonic sequences of sets are convergent.1 In case that (12.1.2) is true, then  Xk , (12.1.4) lim Xk = k→∞

k∈N

1

This is similar with the convergence of the coordinate-wise monotonic signals, and the convergence of the bounded monotonic sequences of real numbers. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00018-0 Copyright © 2023 Elsevier Inc. All rights reserved.

111

112

Boolean Systems

and if (12.1.3) is true, then lim Xk =

k→∞



(12.1.5)

Xk .

k∈N

Proof. We suppose that Xk , k ∈ N is increasing. We also suppose, against all reason, that it is not convergent, i.e. ∀k  ∈ N, ∃k ≥ k  , Xk = Xk  . We infer the existence of a subsequence Xnk , k ∈ N with Xn0  Xn1  ...  Xnk  ... contradiction with the fact that Bn has finitely many subsets. The sequence Xk , k ∈ N is convergent and we denote with k  , see (12.1.1), the rank that satisfies Xk  = lim Xk . From k→∞

X0 ⊂ X1 ⊂ ... ⊂ Xk  = Xk  +1 = ... we get the truth of (12.1.4). The dual statement is proved similarly.

12.2 Sets of reachable states Definition 79. The function  : Bn −→ Bn and μ ∈ Bn are given. The sets Ak (μ) ⊂ Bn , k ∈ N defined by A0 (μ) = {μ}, Ak+1 (μ) =



(12.2.1) δ+

(12.2.2)

δ∈Ak (μ)

are called the sets of states which are reached, from μ, in k time units. Definition 80. For any μ ∈ Bn , the sets Bk (μ) ⊂ Bn , k ∈ N which are defined in the following way B0 (μ) = {μ}, Bk+1 (μ) =



(12.2.3) δ−

(12.2.4)

δ∈Bk (μ)

are called the sets of states that reach μ in k time units. Theorem 97. The sequences Ak (μ) ⊂ Bn , k ∈ N and Bk (μ) ⊂ Bn , k ∈ N are monotonically increasing.

Chapter 12 • Sets of reachable states

113

Proof. We prove the statement for the first sequence. For k = 0 we get  δ + = A1 (μ). A0 (μ) = {μ} ⊂ μ+ = δ∈A0 (μ)

In the general case we take an arbitrary δ ∈ Ak (μ). We have from the way that Ak+1 (μ) was defined that δ + ⊂ Ak+1 (μ), therefore δ ∈ Ak+1 (μ). The proof for Bk (μ), k ∈ N is similar. Theorem 98. The sequences Ak (μ), k ∈ N and Bk (μ), k ∈ N fulfill Ak+1 (μ) = {δ ⊕  ε i |H ⊂ δ  (δ), δ ∈ Ak (μ)},

(12.2.5)

Bk+1 (μ) = {δ|δ ∈ Bn , [δ, (δ)] ∧ Bk (μ) = ∅}.

(12.2.6)

i∈H

Proof. We infer Ak+1 (μ) =





δ+ =

δ∈Ak (μ)



[δ, (δ)] =

δ∈Ak (μ)

{δ ⊕  ε i |H ⊂ δ  (δ)} i∈H

δ∈Ak (μ)

= {δ ⊕  ε i |H ⊂ δ  (δ), δ ∈ Ak (μ)}, i∈H

Bk+1 (μ) =





ρ− =

ρ∈Bk (μ)

{δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) = ρ}

ρ∈Bk (μ)

= {δ|δ ∈ B , ∃λ ∈ Bn , λ (δ) ∈ Bk (μ)} n

= {δ|δ ∈ Bn , [δ, (δ)] ∧ Bk (μ) = ∅}. Theorem 99. The following properties are fulfilled for arbitrary k ∈ N: ∀δ ∈ Ak (μ), ∀λ ∈ Bn , λ (δ) ∈ Ak+1 (μ),

(12.2.7)

∀δ ∈ Bk+1 (μ), ∃λ ∈ Bn , λ (δ) ∈ Bk (μ).

(12.2.8)

Proof. (12.2.7): We take δ ∈ Ak (μ), λ ∈ Bn arbitrary. We infer: λ (δ) = δ ⊕



i∈δλ (δ)

εi = δ ⊕



εi ,

i∈supp λ∧(δ(δ))

and since supp λ ∧ (δ  (δ)) ⊂ δ  (δ), we conclude from (12.2.5) that λ (δ) ∈ Ak+1 (μ). (12.2.8): We take δ ∈ Bk+1 (μ) arbitrary, and (12.2.6) implies [δ, (δ)] ∧ Bk (μ) = ∅. We obtain the existence of λ ∈ Bn with λ (δ) ∈ [δ, (δ)] ∧ Bk (μ). Theorem 100. The sequences Ak (μ) ⊂ Bn , k ∈ N and Bk (μ) ⊂ Bn , k ∈ N fulfill Ak (μ) = {φ α (μ, k)|α ∈ n },

(12.2.9)

114

Boolean Systems

Bk (μ) = {δ|δ ∈ Bn , ∃α ∈ n , φ α (δ, k) = μ}.

(12.2.10)

Proof. We note first that in (12.2.9), (12.2.10) Ak (μ), Bk (μ) are the result of finitely many iterations of , when asking α : N → Bn or asking in addition that α is progressive produces the same effect. We have adopted the request α ∈ n because the flow φ was defined this way, but a generalization of the flow is possible, and will be made at Definition 179, page 412. (12.2.9): We use the induction on k ∈ N. For k = 0, A0 (μ) = {μ} = {φ α (μ, 0)|α ∈ n }. We suppose that (12.2.9) is true, i.e. k−1

Ak (μ) = {(λ

0

◦ ... ◦ λ )(μ)|λk−1 ∈ Bn , ..., λ0 ∈ Bn }.

Then Ak+1 (μ) =



δ+ =

δ∈Ak (μ)



= δ∈{(λ

k−1

λk−1

= {(λ ◦ 



[δ, (δ)]

δ∈Ak (μ)

{λ (δ)|λ ∈ Bn }

0

◦...◦λ )(μ)|λk−1 ∈Bn ,...,λ0 ∈Bn } 0

◦ ... ◦ λ )(μ)|λ ∈ Bn , λk−1 ∈ Bn , ..., λ0 ∈ Bn } = {φ α (μ, k + 1)|α ∈ n }.

(12.2.10): We reason by induction on k ∈ N and we see first of all that B0 (μ) = {μ} = {δ|δ ∈ Bn , ∃α ∈ n , φ α (δ, 0) = μ}. We suppose that Bk (μ) = {δ|δ ∈ Bn , ∃α ∈ n , φ α (δ, k) = μ} k−1

= {δ|δ ∈ Bn , ∃λk−1 ∈ Bn , ..., ∃λ0 ∈ Bn , (λ

0

◦ ... ◦ φ λ )(δ) = μ},

and we infer Bk+1 (μ)

(12.2.6)

=

{δ|δ ∈ Bn , [δ, (δ)] ∧ Bk (μ) = ∅}

= {δ|δ ∈ Bn , ∃λ ∈ Bn , λ (δ) ∈ {ρ|ρ ∈ Bn , ∃λk−1 ∈ Bn , k−1

..., ∃λ0 ∈ Bn , (λ

0

◦ ... ◦ λ )(ρ) = μ}} k−1

= {δ|δ ∈ Bn , ∃λ ∈ Bn , ∃λk−1 ∈ Bn , ..., ∃λ0 ∈ Bn , (λ

= {δ|δ ∈ Bn , ∃α ∈ n , φ α (δ, k + 1) = μ}. Notation 32. For α ∈ n , μ ∈ Bn , we denote −

0

◦ ... ◦ λ ◦ λ )(δ) = μ}

O α (μ) = {δ|δ ∈ Bn , ∃k ∈ N, φ α (δ, k) = μ}.

Chapter 12 • Sets of reachable states

Theorem 101. We have O + (μ) = O − (μ) =



Ak (μ) =



O α (μ),

k∈N

α∈ n





Bk (μ) =

(12.2.11)

−

O α (μ).

(12.2.12)

α∈ n

k∈N

Proof. (12.2.11): We infer O + (μ) = {(λ ◦ ... ◦ ν )(μ)|λ ∈ Bn , ..., ν ∈ Bn } = {φ α (μ, k)|α ∈ n , k ∈ N}  (12.2.9)  = {φ α (μ, k)|α ∈ n } = Ak (μ) k∈N

=

k∈N



{φ (μ, k)|k ∈ N} = α

α∈ n



O α (μ).

α∈ n

(12.2.11): O − (μ) = {δ|δ ∈ Bn , ∃λ ∈ Bn , ..., ∃ν ∈ Bn , (λ ◦ ... ◦ ν )(δ) = μ} =



= {δ|δ ∈ Bn , ∃α ∈ n , ∃k ∈ N, φ α (δ, k) = μ} (12.2.10)  {δ|δ ∈ Bn , ∃α ∈ n , φ α (δ, k) = μ} = Bk (μ)

k∈N

=



{δ|δ ∈ B , ∃k ∈ N, φ (δ, k) = μ} = n

α

α∈ n



k∈N −

O α (μ).

α∈ n

12.3 Example Example 70. We consider the system - (0, 1, 0) (0, 0, 0) Q Q Q Q Q ? s ? Q (0, 0, 1) (0, 1, 1)

- (1, 0, 0) 3       - (1, 1, 1) - (1, 0, 1) (1, 1, 0) 6

where A0 (1, 1, 1) = {(1, 1, 1)}, A1 (1, 1, 1) = (1, 1, 1)+ = {(1, 1, 1), (1, 1, 0), (1, 0, 1), (1, 0, 0)}, A2 (1, 1, 1) = (1, 1, 1)+ ∨ (1, 1, 0)+ ∨ (1, 0, 1)+ ∨ (1, 0, 0)+ = A1 (1, 1, 1), ...

115

116

Boolean Systems

B0 (1, 1, 1) = {(1, 1, 1)}, B1 (1, 1, 1) = (1, 1, 1)− = {(0, 1, 1), (1, 1, 1)}, B2 (1, 1, 1) = (0, 1, 1)− ∨ (1, 1, 1)− = {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)} ∨ {(0, 1, 1), (1, 1, 1)} = {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)}, B3 (1, 1, 1) = (0, 0, 0)− ∨ (0, 1, 0)− ∨ (0, 0, 1)− ∨ (0, 1, 1)− ∨ (1, 1, 1)− = B2 (1, 1, 1), ...

12.4 Isomorphisms Theorem 102. The functions , : Bn → Bn are given, and the isomorphism (h, h ) ∈ I so(φ, ψ). Then ∀μ ∈ Bn , ∀k ∈ N, h(A k (μ)) = Ak (h(μ)),

(12.4.1)

h(Bk (μ)) = Bk (h(μ)).

(12.4.2)

Proof. We prove (12.4.1) by induction on k ∈ N. For k = 0 we have h(A 0 (μ)) = h({μ}) = {h(μ)} = A0 (h(μ)).

We suppose now that (12.4.1) is true, and 1 p A k (μ) = {μ , ..., μ }.

We infer: 

h(A k+1 (μ)) = h(

+ 1+ δ ) = h(μ1+  ∨ ... ∨ μ ) = h(μ ) ∨ ... ∨ h(μ ) p+

p+

δ∈A k (μ) Theorem 80, page 92

=

=



+ δ =

δ∈h({μ1 ,...,μp })

The proof of (12.4.2) is similar.



p + h(μ1 )+ ∨ ... ∨ h(μ ) =

 δ∈h(A k (μ))

+ δ

δ∈{h(μ1 ),...,h(μp )}

+ (12.4.1) δ =



δ∈A k (h(μ))

+ δ = A k+1 (h(μ)).

Chapter 12 • Sets of reachable states

Remark 97. The previous result gives for the dual functions that ∀μ ∈ Bn , ∀k ∈ N, ∗

 A k (μ) = Ak (μ), ∗

Bk (μ) = Bk (μ).

117

13 Dependence on the initial conditions Roughly speaking, dependence on the initial conditions is the property of a system of having distinct states for distinct initial values of the states, and this is defined in Section 13.1. In Section 13.2 we give several examples. Section 13.3 shows that if a subsystem has dependence on the initial conditions, then the system has the same property. In Section 13.4 we apply the previous idea to point out that if one of ,  is dependent on the initial conditions, then  ×  has the same property. In Section 13.5 we remark that the isomorphic systems posses the same properties of dependence, or independence, on the initial conditions. Several other versions of this property are suggested in Section 13.6.

13.1 Definition Definition 81. Let  : Bn → Bn and α ∈ n . We say that μ, μ ∈ Bn are asymptotically equivalent if ∃k  ∈ N, ∀k ≥ k  , φ α (μ, k) = φ α (μ , k). Remark 98. Saying that μ, μ are not asymptotically equivalent ∀k  ∈ N, ∃k ≥ k  , φ α (μ, k) = φ α (μ , k) means in fact that ∀k ∈ N, φ α (μ, k) = φ α (μ , k). This remark gives, by running through several possibilities of quantifying μ, μ and α, the next definition. Definition 82. The following properties of a system  ∃α ∈ n , ∃μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k),

(13.1.1)

∀α ∈ n , ∃μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k),

(13.1.2)

∀μ ∈ Bn , ∃μ ∈ Bn , ∃α ∈ n , ∀k ∈ N, φ α (μ, k) = φ α (μ , k),

(13.1.3)

∃α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k),

(13.1.4)

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00019-2 Copyright © 2023 Elsevier Inc. All rights reserved.

119

120

Boolean Systems

∀α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k),

(13.1.5)

∀μ ∈ Bn , ∃μ ∈ Bn , ∀α ∈ n , ∀k ∈ N, φ α (μ, k) = φ α (μ , k)

(13.1.6)

are called of dependence on the initial conditions (or on the initial values). Theorem 103. The implications are: (13.1.2) (13.1.6) =⇒ (13.1.5)

=⇒ (13.1.4)

⇐⇒

(13.1.3) ⇐⇒

(13.1.1).

Proof. (13.1.6) =⇒ (13.1.5): Property (13.1.5) is equivalent with ∀μ ∈ Bn , ∀α ∈ n , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k) and let μ ∈ Bn arbitrary, fixed. If, from the hypothesis, we have the existence of μ ∈ Bn such that ∀α ∈ n , ∀k ∈ N, φ α (μ, k) = φ α (μ , k), then obviously for any α ∈ n some μ ∈ Bn exists, which does not depend on α in this case, such that ∀k ∈ N, φ α (μ, k) = φ α (μ , k). (13.1.2) =⇒ (13.1.5): We suppose against all reason that (13.1.5) is false, and we denote with β ∈ n , ν ∈ Bn the computation function and the point that fulfill ∀μ ∈ Bn , ∃k ∈ N, φ β (ν, k) = φ β (μ , k).

(13.1.7)

Let us denote with ν  ∈ Bn , ν  ∈ Bn the points that satisfy, from (13.1.2): ∀k ∈ N, φ β (ν  , k) = φ β (ν  , k).

(13.1.8)

We replace in (13.1.7) μ with ν  , resulting ∃k  ∈ N, φ β (ν, k  ) = φ β (ν  , k  ),

(13.1.9)

then we replace in (13.1.7) μ with ν  , and we get ∃k  ∈ N, φ β (ν, k  ) = φ β (ν  , k  ).

(13.1.10)

We have obtained that for any k ≥ max{k  , k  }, φ β (ν  , k)

(13.1.9)

=

φ β (ν, k)

(13.1.10)

=

φ β (ν  , k)

(13.1.11)

is true. The statements (13.1.8) and (13.1.11) are contradictory. (13.1.5) =⇒ (13.1.2) and (13.1.5) =⇒ (13.1.4) are obvious. (13.1.4) =⇒ (13.1.3): If α ∈ n exists such that for any μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k)

(13.1.12)

Chapter 13 • Dependence on the initial conditions

121

is true, then obviously for any μ ∈ Bn , some α ∈ n exists, which does not depend on μ, such that (13.1.12) is true. We have just proved the truth of ∀μ ∈ Bn , ∃α ∈ n , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k), which is equivalent with (13.1.3). (13.1.3) =⇒ (13.1.4): Let in (13.1.3) μ ∈ Bn arbitrary, fixed. Some ν ∈ Bn and β ∈ n exist such that ∀k ∈ N, φ β (μ, k) = φ β (ν, k).

(13.1.13)

We suppose against all reason that (13.1.4) is false. Then for any α ∈ n , in particular for α = β, some ν  ∈ Bn exists with ∀μ ∈ Bn , ∃k ∈ N, φ β (ν  , k) = φ β (μ , k).

(13.1.14)

Property (13.1.14) is true in the special case when μ = μ, then k  ∈ N exists such that φ β (ν  , k  ) = φ β (μ, k  ),

(13.1.15)

and also in the special case when μ = ν, resulting the existence of k  ∈ N such that φ β (ν  , k  ) = φ β (ν, k  ).

(13.1.16)

For any k ≥ max{k  , k  } we can write φ β (μ, k)

(13.1.15)

=

φ β (ν  , k)

(13.1.16)

=

φ β (ν, k).

(13.1.17)

The statements (13.1.13) and (13.1.17) are contradictory. (13.1.3) =⇒ (13.1.1): We replace (13.1.3), (13.1.1) in this order with the equivalent statements ∀μ ∈ Bn , ∃α ∈ n , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k), ∃μ ∈ Bn , ∃α ∈ n , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k) and at this moment the implication is obvious. (13.1.1) =⇒ (13.1.3): The hypothesis (13.1.1) states the existence of a computation function, denoted β ∈ n , and of some points, that we denote by ν ∈ Bn , ν  ∈ Bn , such that ∀k ∈ N, φ β (ν, k) = φ β (ν  , k).

(13.1.18)

We suppose against all reason that (13.1.3) is false, thus μ ∈ Bn exists with ∀μ ∈ Bn , ∀α ∈ n , ∃k ∈ N, φ α (μ, k) = φ α (μ , k).

(13.1.19)

122

Boolean Systems

We replace μ , α with ν, β in (13.1.19), then we replace μ , α with ν  , β in (13.1.19), and we infer the existence of k  ∈ N, k  ∈ N such that φ β (μ, k  ) = φ β (ν, k  ),

(13.1.20)

φ β (μ, k  ) = φ β (ν  , k  ).

(13.1.21)

We infer that ∀k ≥ max{k  , k  } we have φ β (ν, k)

(13.1.20)

=

φ β (μ, k)

(13.1.21)

=

φ β (ν  , k).

(13.1.22)

Eqs. (13.1.18) and (13.1.22) are contradictory. Remark 99. From a strictly technical point of view, more such properties like (13.1.1)– (13.1.6) can be written, with different quantifications of μ, μ , α. We have intentionally omitted the always false properties, for example ∃α ∈ n , ∃μ ∈ Bn , ∀μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k), etc. Remark 100. The property of dependence on the initial conditions, referring to the existence of points which are not asymptotically equivalent, is a discrete analogue of the sensitive dependence on the initial conditions from the dynamical systems theory.1 In its most general sense, dependence on the initial conditions means the weakest property from (13.1.1)–(13.1.6), which is (13.1.1). Saying that  is independent on the initial conditions refers, in its most general sense, to the negation of the strongest property from (13.1.1)–(13.1.6), which is (13.1.6): ∃μ ∈ Bn , ∀μ ∈ Bn , ∃α ∈ n , ∃k ∈ N, φ α (μ, k) = φ α (μ , k).

(13.1.23)

Remark 101. If  : Bn → Bn is bijective, then the (13.1.4) property is true, with α ∈ n given by ∀k ∈ N, α k = (1, ..., 1). For this, it is enough to see that ∀μ ∈ Bn , ∀μ ∈ Bn , if μ = μ , then ∀k ∈ N, φ α (μ, k) = φ α (μ , k).

13.2 Examples Example 71. In the following state portrait (0, 0) 1

- (0, 1)

- (1, 1)

- (1, 0)

f : X → X has sensitive dependence on initial conditions if ∃ε > 0, ∀x0 ∈ X, ∀U open with x0 ∈ U , ∃y0 ∈ U, ∃k ∈ Z + , d(f k (x0 ), f k (y0 )) > ε.

Chapter 13 • Dependence on the initial conditions

123

we have the case of a system which is independent on the initial conditions, i.e. (13.1.23) is true under the stronger form ∀μ ∈ B2 , ∀μ ∈ B2 , ∀α ∈ 2 , ∃k ∈ N, φ α (μ, k) = φ α (μ , k). This happens because ∀μ ∈ B2 , ∀α ∈ 2 , lim φ α (μ, k) = (1, 0). k→∞

Example 72. The fact that the system (0, 1)



- (0, 0) 

- (1, 0)

? (1, 1) fulfills (13.1.4) with α = (0, 1), (1, 0), (0, 1), (1, 0), ... is seen from the next table: μ/α (0, 0) (0, 1) (1, 0) (1, 1)

(0, 1) (0, 1) (0, 0) (1, 0) (1, 1)

(1, 0) (0, 1) (1, 0) (0, 0) (1, 1)

(0, 1) (0, 0) (1, 0) (0, 1) (1, 1)

(1, 0) (1, 0) (0, 0) (0, 1) (1, 1)

(0, 1) (1, 0) (0, 1) (0, 0) (1, 1)

(1, 0) (0, 0) (0, 1) (1, 0) (1, 1)

(0, 1) (0, 1) (0, 0) (1, 0) (1, 1)

... ... ... ... ...

Property (13.1.2) is false, and in order to note that ∃α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , ∃k ∈ N, φ α (μ, k) = φ α (μ , k), we can take α = (1, 1), (1, 1), (1, 1), ... for which ∀μ ∈ B2 , lim φ α (μ, k) = (1, 1). k→∞

Example 73. By taking a look at the next system - (0, 1) (0, 0) Q Q Q Q Q ? s Q (1, 0) (1, 1) we see that (13.1.5) is true, and for this we must consider all the possibilities. For example, we can suppose that the first non-null α k is α 0 = (1, 0). Then (0, 0), (1, 0) are asymptotically equivalent, but they are not asymptotically equivalent with any of (0, 1), (1, 1). However (13.1.6) is false. This fact results by taking the point (0, 0), for which the existence of μ with ∀α ∈ n , ∀k ∈ N, φ α (μ, k) = φ α (μ , k) is false.

124

Boolean Systems

Example 74. The system characterized by the state portrait (0, 0) 6

(0, 1) 6

? (1, 0)

? (1, 1)

fulfills the (13.1.6) property of dependence on the initial conditions. For this it is enough, if μ ∈ {(0, 0), (1, 0)} to choose μ ∈ {(0, 1), (1, 1)}, and if μ ∈ {(0, 1), (1, 1)} to choose μ ∈ {(0, 0), (1, 0)}; then any α ∈ n fulfills ∀k ∈ N, φ α (μ, k) = φ α (μ , k).

13.3 Subsystem Theorem 104. Let the functions : Bn+m → Bn+m ,  : Bn → Bn , and we suppose that for arbitrary μ ∈ Bn , ν ∈ Bm we have ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ),

(13.3.1)

i.e.  ⊂ . If φ depends on the initial conditions, then γ depends on the initial conditions, in the sense that if φ fulfills (13.1.1) then γ fulfills (13.1.1),..., if φ fulfills (13.1.6) then γ fulfills (13.1.6). Proof. We know from Theorem 69, page 75, that ∀α ∈ n , ∀β ∈ m , ∀μ ∈ Bn , ∀ν ∈ Bm , ∀k ∈ N, ∀i ∈ {1, ..., n}, (α,β)

γi

((μ, ν), k) = φiα (μ, k)

(13.3.2)

takes place. We prove the statement of the theorem in the case of (13.1.4). The hypothesis asks the existence of δ ∈ n such that ∀μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ δ (μ, k) = φ δ (μ , k).

(13.3.3)

For μ ∈ Bn , β ∈ m , ν ∈ Bm arbitrary, μ ∈ Bn exists from (13.3.3) such that ∀k ∈ N, ∃i ∈ {1, ..., n} with (δ,β)

γi

((μ, ν), k)

(13.3.2)

=

φiδ (μ, k)

(13.3.3)

=

φiδ (μ , k)

(13.3.2)

=

(δ,β)

γi

((μ , ν), k).

We infer that γ fulfills (13.1.4) under a slightly stronger form.

13.4 Cartesian product Theorem 105. We consider the systems  : Bn → Bn ,  : Bm → Bm and : Bn+m → Bn+m ,

=  × . If one of φ, ψ depends on the initial conditions, then γ depends on the initial conditions; i.e. φ or ψ fulfills (13.1.1) implies that γ fulfills (13.1.1),..., φ or ψ fulfills (13.1.6) implies that γ fulfills (13.1.6).

Chapter 13 • Dependence on the initial conditions

125

Proof. We have  ⊂  × ,  ⊂  ×  and we apply Theorem 104.

13.5 Isomorphisms Theorem 106. The functions ,  : Bn → Bn are given with the property that an isomorphism (h, h ) ∈ I so(φ, ψ) exists. If φ depends on the initial conditions, then ψ depends on the initial conditions, in the manner φ fulfills (13.1.1) implies that ψ fulfills (13.1.1),..., φ fulfills (13.1.6) implies that ψ fulfills (13.1.6). Proof. We prove the theorem in the case of (13.1.3), when the truth of ∀ν ∈ Bn , ∃ν  ∈ Bn , ∃β ∈ n , ∀k ∈ N, ψ β (ν, k) = ψ β (ν  , k) must be shown. For this, let ν ∈ Bn arbitrary. As h is bijection, μ ∈ Bn exists with h(μ) = ν. From (13.1.3), μ ∈ Bn and α ∈ n exist such that ∀k ∈ N, φ α (μ, k) = φ α (μ , k). But h is bijection, therefore ∀k ∈ N, h(φ α (μ, k)) = h(φ α (μ , k)). This means that ∀k ∈ N, ψ β (ν, k) = ψ β (ν  , k), and we have put ν  = h(μ ), β = h (α). ψ fulfills (13.1.3). Corollary 8. If φ fulfills (13.1.1), then φ ∗ fulfills (13.1.1),..., if φ fulfills (13.1.6), then φ ∗ fulfills (13.1.6). We have denoted as usual with φ ∗ the flow of ∗ , the dual of . ∗ Proof. (θ (1,...,1) , 1 Bn ) ∈ I so(φ, φ ) and we apply Theorem 106.

13.6 Versions of dependence on the initial conditions Remark 102. We can construct other versions of Definition 82 by (a) different quantifications of μ and μ : ∃μ ∈ Bn , ∀μ ∈ Bn  {μ}, respectively ∀μ ∈ n B , ∀μ ∈ Bn  {μ}, (b) interpreting the asymptotic equivalence, see Definition 81, as omega-limit equivalence: φ α (μ, ·) ≈ φ α (μ , ·), (c) replacing  : Bn → Bn in Definitions 81, 82 with  : X → X, X ⊂ Bn , X = ∅, where the truth of ∀λ ∈ Bn , ∀μ ∈ X, λ (μ) ∈ X is asked. Property (13.6.1) is one of invariance of X.

(13.6.1)

126

Boolean Systems

Example 75. The system from Example 74, page 124 fulfills the property of dependence on the initial conditions ∀μ ∈ Bn , ∃μ ∈ Bn , ∀α ∈ n , φ α (μ, ·) ≈ φ α (μ , ·) which is similar to (13.1.6), and uses the idea from Remark 102 (b). Example 76. The situation of the system (0, 0)

(1, 0)

- (0, 1) 6

(1, 1)

suggests the possibility of referring to Remark 102 (c) and the behavior of a function  : X → X, where X ⊂ Bn is nonempty. For X = {(0, 0), (0, 1), (1, 1)}, (13.6.1) is true and all the points of X are asymptotically equivalent (no dependence on the initial conditions), but if we take X = B2 , then (13.1.6) holds.

14 Periodicity The eventually periodic states φ α (μ, ·) may be analyzed like any other eventually periodic signals. A special case here is the one when φ α (μ, ·) and α are both eventually periodic, and this is called in Section 14.1 double eventual periodicity. The main theorems concerning the eventually periodic states are given in Section 14.2. We prove for example that if α is eventually periodic, then φ α (μ, ·) is eventually periodic too. In Section 14.3 we show that the morphisms of flows bring eventually periodic states in eventually periodic states. Some ways of defining new periodicity properties are indicated in Section 14.4.

14.1 Eventual periodicity and double eventual periodicity Remark 103. We consider  : Bn → Bn , α ∈ n and the initial value μ ∈ Bn . From Definition 52, page 50, we know that the state φ α (μ, ·) is called eventually periodic if p ≥ 1 and k  ∈ N exist such that ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p),

(14.1.1)

and the periodicity of φ α (μ, ·) is that special case of eventual periodicity when (14.1.1) is true for k  = 0. The other definitions and results from Chapter 4, Section 4.9 are true also in the special case when the signals x(k) = φ α (μ, k) are the states of a system. Definition 83. If the computation function α is eventually periodic (with the period p), and the state φ α (μ, ·) is eventually periodic itself (with the period p  ):

∀k

≥ k, αk

∃p ≥ 1, ∃p  ≥ 1, ∃k  ∈ N, and ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p  ),

= α k+p

(14.1.2)

then φ α (μ, ·) is said to be double eventually periodic. If (14.1.2) holds with α, φ α (μ, ·) periodic,1 then φ α (μ, ·) is called double periodic. 1

That is if (14.1.2) is true for k  = 0.

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00020-9 Copyright © 2023 Elsevier Inc. All rights reserved.

127

128

Boolean Systems

14.2 Main theorems Theorem 107. The function  : Bn → Bn is given, together with α ∈ n and μ ∈ Bn . We suppose that ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p), and p ≥ 1 is the least number with this property. Then (a) p = 1 if and only if μ is either a sink, or an isolated fixed point, (b) p > 1 if and only if all the points of O α (μ) are transient. Proof. (b) Only if. The hypothesis states the periodicity of φ α (μ, ·) with p > 1 and let μ ∈ O α (μ) arbitrary. Then k  ∈ N, k  ∈ N exist with φ α (μ, k  ) = φ α (μ, k  + 1) = μ ,

(14.2.1)

μ = φ α (μ, k  ) = φ α (μ, k  + 1)

(14.2.2)

true. (14.2.1) implies φ α (μ, k  ) ∈ μ− , μ− = {μ } and (14.2.2) implies φ α (μ, k  + 1) ∈ μ+ , μ+ = {μ }. If. We suppose against all reason that p = 1, i.e. from (a) the point μ ∈ O α (μ) is a sink or an isolated fixed point; μ is not transient, contradiction. Theorem 108. If α ∈ n is periodic: ∃p ≥ 1, ∀k ∈ N, α k = α k+p

(14.2.3)

then, ∀μ ∈ Bn , the state φ α (μ, ·) is double eventually periodic: ∃p  ≥ 1, ∃k  ∈ N, ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p  ).

(14.2.4)

Proof. We denote with μk ∈ Bn , k ∈ N the following values μk = φ α (μ, kp).

(14.2.5)

In the sequence μ0 , μ1 , μ2 , ... let k1 be a rank with the property that k2 > k1 exists such that μk1 = μk2 .

(14.2.6)

We infer: φ α (μ, k1 p + 1) = α (14.2.6)

=

0

α (μk2 )

(14.2.3)

=

k1 p

α

(φ α (μ, k1 p))

k2 p

(μk2 )

φ α (μ, k1 p + 2) = α (14.2.7)

=

1

k1 p+1

α (φ α (μ, k2 p + 1))

(14.2.5)

=

(14.2.5)

=

α

k2 p

α

k1 p

=

α

k2 p+1

(14.2.3)

=

0

α (μk1 )

(φ α (μ, k2 p)) = φ α (μ, k2 p + 1),

(φ α (μ, k1 p + 1))

(14.2.3)

(μk1 )

(14.2.3)

=

1

α (φ α (μ, k1 p + 1))

(φ α (μ, k2 p + 1)) = φ α (μ, k2 p + 2),

(14.2.7)

Chapter 14 • Periodicity

129

... in other words ∀k ≥ k1 p, φ α (μ, k) = φ α (μ, k + (k2 − k1 )p).

(14.2.8)

Theorem 109. Let μ ∈ Bn . If α ∈ n is eventually periodic: ∃p ≥ 1, ∃k  ∈ N, ∀k ≥ k  , α k = α k+p ,

(14.2.9)

then φ α (μ, ·) is double eventually periodic. 

Proof. If p ≥ 1 and k  ∈ N exist such that (14.2.9) is true, then σ k (α) is periodic with the k

period p, from Theorem 51, page 50. We infer from Theorem 108 that φ σ (α) (φ α (μ, k  ), ·) is double eventually periodic, and it satisfies an equation of the form (14.2.8): ∀k ≥ k1 p, φσ

k  (α)

(φ α (μ, k  ), k) = φ σ

k  (α)

(φ α (μ, k  ), k + (k2 − k1 )p),

where 0 ≤ k1 < k2 . We have ∀k ≥ k1 p, 

σ k (φ α (μ, ·))(k)

(6.4.2)page 64

=

φσ

k  (α)

(6.4.2)page 64

=

(φ α (μ, k  ), k) = φ σ

k  (α)

(φ α (μ, k  ), k + (k2 − k1 )p)



σ k (φ α (μ, ·))(k + (k2 − k1 )p),

i.e. ∀k ≥ k1 p, φ α (μ, k + k  ) = φ α (μ, k + k  + (k2 − k1 )p). With the substitution l = k + k  , we infer ∀l ≥ k  + k1 p, φ α (μ, l) = φ α (μ, l + (k2 − k1 )p). Remark 104. The reasoning represented by Theorems 108 and 109, with Theorem 109 presented as a consequence of Theorem 108, could be run in reverse, in the sense that α eventually periodic implies φ α (μ, ·) double eventually periodic (Theorem 109), in particular α periodic implies φ α (μ, ·) double eventually periodic (Theorem 108). This fact is obvious, but the false intuition may be created that strengthening Theorem 108 to: α periodic implies φ α (μ, ·) double periodic is possible. Theorem 110. α ∈ n and μ ∈ Bn are given. If φ α (μ, ·) is periodic with the period p: ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p), then β ∈ n periodic with the period p exists such that ∀k ∈ N, φ β (μ, k) = φ α (μ, k).

(14.2.10)

130

Boolean Systems

p−1

Proof. We denote with k10 , ..., kn0 ∈ {0, p, 2p, ...}, ..., k1 the numbers that fulfill k

j

j +p

j

αi i = αi ∪ αi

p−1

, ..., kn

j +2p

∪ αi

∈ {p − 1, 2p − 1, 3p − 1, ...}

∪ ...

(14.2.11)

where i ∈ {1, ..., n} and j ∈ {0, ..., p − 1}. Obviously ∀i ∈ {1, ..., n}, k0

k

p−1

αi i ∪ ... ∪ αi i



=

j +p

j

(αi ∪ αi

j +2p

∪ αi

∪ ...) =

j ∈{0,...,p−1}



αik = 1.

(14.2.12)

k∈N

We define: k0

k0

β 0 = (α11 , ..., αnn ),

(14.2.13)

... k

p−1

p−1

β p−1 = (α11 , ..., αnkn ).

(14.2.14)

β = β 0 , ..., β p−1 , β 0 , ..., β p−1 , β 0 , ...

(14.2.15)

∀k ∈ N, β k = β k+p ,

(14.2.16)

ω(β) = {β 0 , ..., β p−1 },

(14.2.17)

β 0 ∪ ... ∪ β p−1 = (1, ..., 1),

(14.2.18)

We claim that the sequence2

fulfills

and (14.2.10), i.e. it is the sequence whose existence is stated by the theorem. It is clear that β satisfies the periodicity (14.2.16) thus, from Theorem 52, page 51, (14.2.17) takes place. We have k0

k

p−1

k0

p−1

β 0 ∪ ... ∪ β p−1 = (α11 ∪ ... ∪ α11 , ..., αnn ∪ ... ∪ αnkn )

(14.2.12)

=

(1, ..., 1)

i.e. (14.2.18) is true, showing that β ∈ n . Eq. (14.2.10) is true for k = 0, when both terms are equal with μ. Let now i ∈ {1, ..., n} arbitrary, fixed, for which we have β β0 φi (μ, 1) = i (μ) =

2



i (μ), if βi0 = 1, = μi , otherwise



k0

i (μ), if αi i = 1, μi , otherwise.

/ n . It is not possible to define β = α 0 , ..., α p−1 , α 0 , ..., α p−1 , α 0 , ... because this might give β ∈

Chapter 14 • Periodicity

131

Since ki0 is from (14.2.11) a multiple of p : ki0 = k1 p, we get β

φi (μ, 1) = αi

k1 p

(μ) = αi

k1 p

(φ α (μ, k1 p)) = φiα (μ, k1 p + 1) = φiα (μ, 1).

We can prove similarly that ... φ β (μ, p − 1) = φ α (μ, p − 1). We fix again i ∈ {1, ..., n} arbitrary and we infer  β β p−1 β (φ (μ, p − 1)) = φi (μ, p) = i



p−1

i (φ β (μ, p − 1)), if βi = 1, β φi (μ, p − 1), otherwise k

We have from (14.2.11) the existence of k1 with ki

p−1

β

φi (μ, p) = αi

p−1

i (φ α (μ, p − 1)), if αi i = 1, φiα (μ, p − 1), otherwise.

=

p−1+k1 p

(φ α (μ, p − 1)) = αi

= p − 1 + k1 p, thus p−1+k1 p

(φ α (μ, p − 1 + k1 p))

= φiα (μ, (k1 + 1)p) = μi = φi (μ, 0). β

(14.2.10) is proved. Theorem 111. Let α ∈ n , μ ∈ Bn and we suppose that φ α (μ, ·) is eventually periodic: ∃p ≥ 1, ∃k  ∈ N, ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p);

(14.2.19)

then β ∈ n eventually periodic exists, with the period p and the limit of periodicity k  , such that ∀k ∈ N, φ β (μ, k) = φ α (μ, k).

(14.2.20)

Proof. In the equation ∀k ∈ N, φσ

k  (α)



(φ α (μ, k  ), k) = σ k (φ α (μ, ·))(k),

(14.2.21)



which coincides with (6.4.2)page 64 , the function σ k (φ α (μ, ·))(k) is periodic with the period p, from Theorem 51, page 50. In such circumstances, Theorem 110 states the existence of γ ∈ n periodic with the period p such that ∀k ∈ N, φ γ (φ α (μ, k  ), k) = φ σ

k  (α)

(φ α (μ, k  ), k).

(14.2.22)

132

Boolean Systems

The sequence  βk =

α k , if k ∈ {0, ..., k  − 1},  γ k−k , if k ≥ k 

belongs to n due to γ , and is eventually periodic with the period p and the limit of periodicity k  , as far as ∀k ≥ k  , 



β k = γ k−k = γ k−k +p = β k+p . We have ∀k ∈ N, β

φ (μ, k)

Theorem 64, page 65

= 

(14.2.22)

=

(14.2.21)

=





φ α (μ, k), if k ∈ {0, ..., k  − 1}, φ γ (φ α (μ, k  ), k − k  ), if k ≥ k 

φ α (μ, k), if k ∈ {0, ..., k  − 1},

φσ

k  (α)

(φ α (μ, k  ), k − k  ), if k ≥ k 

φ α (μ, k), if k ∈ {0, ..., k  − 1}, = φ α (μ, k). φ α (μ, (k − k  ) + k  ), if k ≥ k 

Remark 105. Here Theorem 111 generalizes Theorem 110 and the reasoning may be done in reverse, by proving Theorem 111 first and stating the special case represented by Theorem 110 afterwards. The situation is slightly different from the Theorems 108 and 109.

14.3 Morphisms vs. periodicity Theorem 112. The functions  : Bn −→ Bn , : Bm −→ Bm are given. (a) If (h, h ) : φ → ψ is a morphism, then for arbitrary α ∈ n , μ ∈ Bn , we have: φ α (μ, ·)  is (eventually) periodic with the period p =⇒ ψ h (α) (h(μ), ·) is (eventually) periodic with the period p; (b) if (h, h ) : φ → ψ is an isomorphism, then for any α ∈ n , μ ∈ Bn , we infer: φ α (μ, ·) is  (eventually) periodic with the period p ⇐⇒ ψ h (α) (h(μ), ·) is (eventually) periodic with the period p. Proof. (a) Let α ∈ n , μ ∈ Bn , p ≥ 1, k  ∈ N arbitrary with ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p); then 



ψ h (α) (h(μ), k) = h(φ α (μ, k)) = h(φ α (μ, k + p)) = ψ h (α) (h(μ), k + p). Remark 106. A way of interpreting Theorem 112 (a) is: if φ α (μ, ·) is (eventually) periodic  with the period p, then ψ h (α) (h(μ), ·) is (eventually) periodic with the period p  , where p  is p or a divisor of p.

Chapter 14 • Periodicity

133

14.4 Other definitions of periodicity Remark 107.  : Bn → Bn is given. The property of eventual periodicity of φ ∃α ∈ n , ∃μ ∈ Bn , ∃p ≥ 1, ∃k  ∈ N, ∀k ≥ k  , φ α (μ, k) = φ α (μ, k + p) may be nuanced (a) by quantifying α, μ, p, k  differently, (b) by replacing  with  : X → X, where X ⊂ Bn is nonempty and ∀λ ∈ Bn , ∀μ ∈ X, λ (μ) ∈ X, (c) by restricting the values of α and μ to smaller sets than n and Bn . A definition of the (eventual) periodicity of the points μ ∈ Bn exists also. Definition 84. We consider  : Bn → Bn and we fix α ∈ n , μ ∈ Bn , as well as μ ∈ O α (μ). If ∃p ≥ 1, ∃k  ∈ N, φ α (μ, k  ) = μ , ∀k ∈ N, φ α (μ, k  ) = φ α (μ, k  + kp), we say that the point μ is eventually periodic, with the limit of periodicity k  and the period p. In case that k  ∈ {0, ..., p − 1}, we say that μ is periodic (with the period p). Example 77. We consider the system (0, 0) 6

(1, 0) 6

? (0, 1)

? (1, 1)

and the computation functions α, β, γ ∈ 2 defined by 



α1 = 1, 1, 1, ... α2 = 0, 1, 0, 0, 1, 0, 0, 0, 1, ..., β1 = 0, 1, 0, 0, 1, 0, 0, 0, 1, ... β2 = 1, 1, 1, ..., 

γ1 = 1, 1, 1, ... γ2 = 1, 1, 1, ...

We denote with X one of the sets {(0, 0), (0, 1)}, {(1, 0), (1, 1)}. We get that ∀μ ∈ X, O α (μ) = O β (μ) = O γ (μ) = X. Let now μ ∈ X, μ ∈ X arbitrary both. We conclude that:

134

Boolean Systems

– α is not periodic, φ α (μ, ·) is not periodic, μ is not periodic; – β is not periodic, φ β (μ, ·) has the period 2 and μ has the period 2; – γ has the period 1, φ γ (μ, ·) has the period 2 and μ has the period 2.

15 Path-connectedness and topological transitivity A path in X ⊂ Bn from μ ∈ X to μ ∈ X is a state φ α (μ, ·), with α ∈ n , having the property that k  exists such that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ . We say that μ and μ are path-connected. Several path-connectedness properties are introduced in Section 15.1. Topological transitivity, defined in Section 15.2, is the property that ∀μ ∈ X, O α (μ) = X. Examples of systems and sets that fulfill path-connectedness and topological transitivity requests are given in Section 15.3. Some properties concerning path-connectedness and topological transitivity are analyzed in Section 15.4. In Section 15.5 we show that the morphisms of flows bring path-connected sets in pathconnected sets, and topologically transitive sets in topologically transitive sets. In Section 15.6 we show that the Cartesian product of path-connected sets is pathconnected, but the similar property for the topologically transitive sets is false in general. The path-connected components of a set X are defined in Section 15.7 and several examples are given.

15.1 Path-connectedness Definition 85. Let  : Bn → Bn , X ⊂ Bn , α ∈ n , and μ, μ ∈ X. The state φ α (μ, ·) is called a path in X from μ to μ if k  ∈ N exists such that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X,

(15.1.1)

φ α (μ, k  ) = μ .

(15.1.2)

We say then that μ and μ are path-connected (by , in X). Definition 86. The set X is path-connected (by ) if one of ∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ ,

(15.1.3)

∀μ ∈ X, ∃α ∈ n , ∀μ ∈ X, ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ ,

(15.1.4)

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00021-0 Copyright © 2023 Elsevier Inc. All rights reserved.

135

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∃α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ ,

(15.1.5)

∀α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ

(15.1.6)

holds. Remark 108. Definitions 85 and 86 include the possibilities μ = μ and X = {μ}. The requests of non-triviality μ = μ and card(X) > 1 may be mentioned explicitly. Theorem 113.  : Bn → Bn , X ⊂ Bn , X = ∅, and μ, μ ∈ X are given, μ = μ . The following statements are equivalent: (a) α ∈ n exists such that the state φ α (μ, ·) is a path in X from μ to μ in the sense of Definition 85, (b) G |X has a path μ → ... → μ starting in μ, and ending in μ , see Definition 34, page 39 and Notation 13, page 39. Proof. Statement (a) shows the existence of α ∈ n and k  ≥ 1 so that (15.1.1), (15.1.2) become 0

μ, α (μ), ..., (α (α

k  −1

k  −1

0

◦ ... ◦ α )(μ) ∈ X,

(15.1.7)

◦ ... ◦ α )(μ) = μ . 0

0

Statement (b) is essentially the same, but in the list μ, α (μ), ..., (α (15.1.7) only the distinct successive values occur.

(15.1.8) k  −1

0

◦ ... ◦ α )(μ) from

Remark 109. By comparing the paths from the previous theorem, item (a) and item (b), the time instant k  that satisfies φ α (μ, k  ) = μ and the length k  of μ → ... → μ from G |X fulfill k  ≥ k  ; this is true because the arrows (ν, λ (ν)) from G |X satisfy ν = λ (ν). Theorem 114. If card(X) > 1, the following statements are equivalent: (a) X is path-connected in the sense of Definition 86, (15.1.3), (b) G |X is a path-connected component of G , see Definition 37, page 39. Proof. Let μ, μ ∈ X arbitrary, such that μ = μ . We apply Theorem 113.

15.2 Topological transitivity Definition 87. Let  : Bn → Bn and X ⊂ Bn , X = ∅. The properties ∀μ ∈ X, ∃α ∈ n , O α (μ) = X,

(15.2.1)

∃α ∈ n , ∀μ ∈ X, O α (μ) = X,

(15.2.2)

Chapter 15 • Path-connectedness and topological transitivity

∀α ∈ n , ∀μ ∈ X, O α (μ) = X

137

(15.2.3)

are called of topological transitivity (of X, relative to ). Remark 110. In Definition 87 we may have card(X) = 1; then (15.2.1)–(15.2.3) are true if and only if (μ) = μ. Theorem 115. If card(X) > 1, the properties (15.1.3)–(15.1.6), (15.2.1)–(15.2.3) fulfill (15.1.6)

(15.2.3)

=⇒ (15.1.5) ⇑ =⇒ (15.2.2)

=⇒ (15.1.4) ⇑ =⇒ (15.2.1)

⇐⇒ (15.1.3)

Proof. (15.2.1)=⇒(15.1.4) Let μ ∈ X arbitrary, fixed. The hypothesis shows the existence of α ∈ n such that O α (μ) = X. Let μ ∈ X arbitrary. Some k  ∈ N exists then with ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ . (15.1.3)=⇒(15.1.4) We put X under the form X = {μ0 , μ1 , ..., μp }, p ≥ 1. Let in (15.1.4) μ ∈ X arbitrary and, without losing the generality, we take μ = μ0 . We apply (15.1.3) for  ∈ N with ∀k ∈ {0, ..., k  }, – μ = μ0 , μ = μ1 and we get the existence of α 01 ∈ n , k01 01 01 01  α 0 α 0 1 φ (μ , k) ∈ X and φ (μ , k01 ) = μ ,  ∈ N with ∀k ∈ {0, ..., k  }, – μ = μ1 , μ = μ2 and we get the existence of α 12 ∈ n , k12 12 12 12  α 1 α 1 2 φ (μ , k) ∈ X and φ (μ , k12 ) = μ , ...  – μ = μp−1 , μ = μp and we get the existence of α p−1p ∈ n , kp−1p ∈ N with ∀k ∈   }, φ α (μp−1 , k) ∈ X and φ α (μp−1 , kp−1p ) = μp . {0, ..., kp−1p We define now α ∈ n in the following way: ⎧  − 1}, α 01,k , if k ∈ {0, ..., k01 ⎪ ⎪ ⎪  ⎨  , ..., k  + k  − 1}, α 12,k−k01 , if k ∈ {k01 01 12 αk = ... ⎪ ⎪ ⎪ p−1p,k−k  −k  −...−k  ⎩ 01 12 p−2p−1 , if k ≥ k  + k  + ... + k  α 01 12 p−2p−1 p−1p

p−1p

and we have from Theorem 64, page 65, that   }, φ α (μ0 , k) = φ α 01 (μ0 , k) ∈ X, ∀k ∈ {0, ..., k01  ) = φ α 01 (μ0 , k  ) = μ1 , φ α (μ0 , k01 01 

 , ..., k  + k  }, φ α (μ0 , k) = φ α (μ1 , k − k  ) ∈ X, ∀k ∈ {k01 01 12 01  + k  ) = φ α 12 (μ1 , k  ) = μ2 , φ α (μ0 , k01 12 12 12

...

138

Boolean Systems ⎧   + ... + k     ⎪ p−2p−1 , ..., k01 + k12 + ... + kp−1p }, ⎨ ∀k ∈ {k01 + k12p−1p  − k  − ... − k  φ α (μ0 , k) = φ α (μp−1 , k − k01 12 p−2p−1 ) ∈ X, ⎪ p−1p ⎩ α 0     (μp−1 , kp−1p ) = μp . φ (μ , k01 + k12 + ... + kp−1p ) = φ α

 + k  + ... + k  (15.1.4) is true, with k  ≤ k01 12 p−1p . (15.1.6)=⇒(15.2.3) We fix in (15.2.3) α ∈ n , μ ∈ X arbitrary. We prove

O α (μ) ⊂ X. We suppose against all reason that O α (μ)  X = ∅, thus k  ∈ N, μ ∈ O α (μ)  X exist with the property that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  + 1) = μ ∈ / X. We denote μ =  α  0 k φ (μ, k ) and we choose β ∈ n arbitrary, such that β = α . We take μ ∈ X arbitrary, with μ = μ . In this situation, the statement (15.1.6) written for β, μ , μ as α, μ, μ takes the form: ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ β (μ , k) ∈ X, φ β (μ , k  ) = μ . (15.2.4) is false, because μ = μ implies k  ≥ 1 and k

/ X, φ β (μ , 1) = β (μ ) = α (μ ) = μ ∈ 0

contradiction. We prove X ⊂ O α (μ) and we take μ ∈ X arbitrary. We obtain from (15.1.6) that μ ∈ O α (μ). The rest of the implications are obvious.

15.3 Examples Example 78. We give the example of the system (1, 1) - (0, 1)

(0, 0) 

-

 (1, 0)

(15.2.4)

Chapter 15 • Path-connectedness and topological transitivity

139

For X = {(0, 0), (0, 1)}, properties (15.1.3)–(15.1.5) are true and properties (15.1.6), (15.2.1)– (15.2.3) are false (in the case of this system, for any μ ∈ X and any α ∈ 2 , the progressiveness of α makes O α (μ) = X be false). Example 79. For the following system (1, 1, 1)

(0, 0, 1)

-

? (0, 1, 1)

(1, 1, 0)  6

? (1, 0, 1)

-

(0, 0, 0) 

(1, 0, 0) 6

(0, 1, 0)

the set X = {(1, 1, 1), (1, 0, 0), (0, 1, 0), (1, 0, 1)} is (15.1.3)-path-connected, as indicated by the arrows. We see that – the transfer condition of (1, 0, 0) → (1, 0, 1) at k + 1 is that α2k = 0, α3k = 1, and (1, 0, 1) is not accessible from another point of X, – the transfer condition of (1, 0, 0) → (1, 1, 1) at k + 1 is that α2k = 1, α3k = 1, and (1, 1, 1) is not accessible from another point of X, therefore the choice of α2k depends on μ = (1, 0, 0), μ ∈ {(1, 0, 1), (1, 1, 1)}, see (15.1.3). The just given arguments show in fact the validity of the stronger property here, X is (15.2.1)topologically transitive relative to . At the same time (15.1.5), (15.1.6), (15.2.2), (15.2.3) are false. Example 80. We look now at the system (0, 0) 6 ? (1, 0)



- (0, 1)

(1, 1)

The set X = {(1, 0), (0, 0), (0, 1)} is (15.1.5)-path-connected, and also (15.2.2)-topologically transitive relative to , as the computation function α = (0, 1), (1, 0), (0, 1), (1, 0), ... fulfills ∀μ ∈ X, O α (μ) = X. On the other hand the (15.2.3) and the (15.1.6)-path-connectedness are false, for example if α = (1, 1), (1, 1), (1, 1), ... we get φ α ((0, 0), 1) = (1, 1) ∈ / X.

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Boolean Systems

Example 81. In the next case (0, 0) 6 (1, 0) 

- (0, 1)

? (1, 1)

the set X = B2 is (15.1.6)-path-connected and also (15.2.3)-topologically transitive relative to . All the other properties are true also. Remark 111. Taking into account the possibilities which are expressed by Theorem 115 and which are not found in Examples 78–81, it might be interesting finding examples such that (a) (15.1.3), (15.1.4), (15.1.5), (15.2.1) be fulfilled and (15.1.6), (15.2.2), (15.2.3) be not fulfilled, (b) (15.1.3), (15.1.4) be fulfilled and the rest of the properties be not fulfilled.

15.4 Some properties Theorem 116. Let  : Bn → Bn , the set X ⊂ Bn and μ ∈ X. For any k ∈ {(15.1.3), ..., (15.1.6)}, the following statements hold: (a) if card(X) > 1 and μ is a source, or a sink, or an isolated fixed point, then X is not k-path-connected; (b) if card(X) > 1 and X is k-path-connected, then μ is a transient point; (c) if μ is an isolated fixed point and X is k-path-connected, then X = {μ}. Proof. (a) We suppose that card(X) > 1 and μ is a source or an isolated fixed point, O − (μ) = {μ}. We take μ ∈ X arbitrary, μ = μ. Then ∀α ∈ n , ∀k  ∈ N, φ α (μ , k  ) = μ, thus (15.1.3) is false. Similarly if μ is a sink or an isolated fixed point, O + (μ) = {μ}, then for arbitrary μ ∈ X, μ = μ, we have ∀α ∈ n , ∀k  ∈ N, φ α (μ, k  ) = μ , thus (15.1.3) is false again. X is not (15.1.3)-path-connected. Theorem 117. We suppose that the nonempty sets X1 ⊂ Bn , X2 ⊂ Bn are (15.1.6)-pathconnected, and X1 ∧ X2 = ∅. Then X1 = X2 . Proof. Let α ∈ n and μ ∈ X1 ∧ X2 arbitrary. (15.2.3) is true and we get X1 = O α (μ) = X2 .

Chapter 15 • Path-connectedness and topological transitivity

141

Example 82. We have the example of the system (1, 0, 0) 

? (0, 0, 0) 6 (0, 1, 0) 

(1, 0, 1) 6

(1, 1, 0)

- (0, 0, 1)

- (1, 1, 1)

? (0, 1, 1)

The sets X1 = {(0, 0, 0), (0, 0, 1), (1, 0, 1), (1, 0, 0)} and X2 = {(0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0)} are (15.1.3)-path-connected and X1 ∧ X2 = {(0, 0, 0), (0, 0, 1)} fulfills no pathconnectedness property. Theorem 118. If X1 , X2 ⊂ Bn are k-path-connected, k ∈ {(15.1.3), ..., (15.1.5)} and X1 ∧ X2 = ∅, then X1 ∨ X2 is (15.1.3)-path-connected. Proof. We can suppose without loss that X1 , X2 are (15.1.3)-path-connected and we take μ ∈ X1 ∨ X2 , μ ∈ X1 ∨ X2 arbitrary, fixed. We must show the existence of α ∈ n and k  ∈ N such that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X1 ∨ X2 ,

(15.4.1)

φ α (μ, k  ) = μ .

(15.4.2)

If μ, μ ∈ X1 or μ, μ ∈ X2 the statement is true, thus we can suppose, without losing the generality, that μ ∈ X1 and μ ∈ X2 . We take an arbitrary μ ∈ X1 ∧ X2 now. As X1 is (15.1.3)path-connected, we infer the existence of β ∈ n , k  ∈ N such that ∀k ∈ {0, ..., k  }, φ β (μ, k) ∈ X1 ,

(15.4.3)

φ β (μ, k  ) = μ

(15.4.4)

and as X2 is (15.1.3)-path-connected, we get the existence of γ ∈ n , k  ∈ N such that ∀k ∈ {0, ..., k  }, φ γ (μ , k) ∈ X2 ,

(15.4.5)

φ γ (μ , k  ) = μ .

(15.4.6)

We define α by  α = k

β k , if k ∈ {0, ..., k  − 1},  γ k−k , if k ≥ k 

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Boolean Systems

and we have



∀k ∈ {0, ..., k  }, φ α (μ, k) = φ β (μ, k)

(15.4.3)

∈

X1 ,

(15.4.4) φ α (μ, k  ) = φ β (μ, k  ) = μ ,



∀k ∈ {k  , ..., k  + k  }, φ α (μ, k) = φ γ (μ , k − k  ) φ α (μ, k 

(15.4.5)

∈

X2 ,

(15.4.6) + k  ) = φ γ (μ , k  ) = μ ,

thus (15.4.1), (15.4.2) take place with k  = k  + k  . Example 83. In Example 82, the (15.1.3)-path-connected sets X1 , X2 satisfy X1 ∧ X2 = ∅, thus X1 ∨ X2 is (15.1.3)-path-connected.

15.5 Morphisms Theorem 119. We consider the functions  : Bn → Bn , : Bm → Bm and the nonempty set X ⊂ Bn . (a) If (h, h ) ∈ H om(φ, ψ) and X is k-path-connected (by ), then h(X) is k-pathconnected (by ), k ∈ {(15.1.3), ..., (15.1.5)}, (b) if (h, h ) ∈ I so(φ, ψ) and X is (15.1.6)-path-connected (by ), then h(X) is (15.1.6)path-connected (by ). Proof. (a) (15.1.3): We take ν ∈ h(X), ν  ∈ h(X) arbitrary, thus μ ∈ X, μ ∈ X exist such that h(μ) = ν, h(μ ) = ν  . The (15.1.3)-path-connectedness of X gives the existence of α ∈ n and k  ∈ N such that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ . We infer that ∀k ∈ {0, ..., k  }, h(φ α (μ, k)) ∈ h(X) and h(φ α (μ, k  )) = h(μ ), therefore ∀k ∈   {0, ..., k  }, ψ h (α) (ν, k) ∈ h(X) and ψ h (α) (ν, k  ) = ν  . Theorem 120. (a) For arbitrary (h, h ) ∈ H om(φ, ψ), the k-topological transitivity of X relative to  implies the k-topological transitivity of h(X) relative to , where k ∈ {(15.2.1), (15.2.2)}, (b) if (h, h ) ∈ I so(φ, ψ), the (15.2.3)-topological transitivity of X relative to  implies the (15.2.3)-topological transitivity of h(X) relative to . Proof. (a) (15.2.2): The hypothesis states the existence of α ∈ n with the property that  α (μ) = X. We want to prove that h (α) satisfies ∀ν ∈ h(X), O h (α) (ν) = h(X). For ∀μ ∈ X, O

this, let ν ∈ h(X) arbitrary. Then μ ∈ X exists with h(μ) = ν and we have: α h(X) = h(O (μ)) = h({φ α (μ, k)|k ∈ N}) = {h(φ α (μ, k))|k ∈ N} 





h (α) = {ψ h (α) (h(μ), k)|k ∈ N} = {ψ h (α) (ν, k)|k ∈ N} = O

(ν).

Chapter 15 • Path-connectedness and topological transitivity

143

15.6 Cartesian products Theorem 121. Let  : Bn → Bn , X ⊂ Bn , X = ∅, : Bm → Bm , Y ⊂ Bm , Y = ∅ and k ∈ {(15.1.3), ..., (15.1.5)}. If X is k-path-connected by  and Y is k-path-connected by , then X × Y is k-path-connected by  × . Proof. We consider the case k = (15.1.3), when the hypothesis states ∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ , ∀ν ∈ Y, ∀ν  ∈ Y, ∃β ∈ m , ∃k  ∈ N, ∀k ∈ {0, ..., k  }, ψ β (ν, k) ∈ Y and ψ β (ν, k  ) = ν  and we must prove that ∀(μ, ν) ∈ X × Y, ∀(μ , ν  ) ∈ X × Y, ∃(α, β) ∈ n × m , ∃k1 ∈ N, ∀k ∈ {0, ..., k1 }, (φ α (μ, k), ψ β (ν, k)) ∈ X × Y and (φ α (μ, k1 ), ψ β (ν, k1 )) = (μ , ν  ). We take μ ∈ X, μ ∈ X, ν ∈ Y, ν  ∈ Y arbitrary. We get the existence of α ∈ n , k  ∈ N and β ∈ m , k  ∈ N such that ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ , ∀k ∈ {0, ..., k  }, ψ β (ν, k) ∈ Y and ψ β (ν, k  ) = ν  . In case that k  = k  , the conclusion ∀k ∈ {0, ..., k1 }, (φ α (μ, k), ψ β (ν, k)) ∈ X × Y and (φ α (μ, k1 ), ψ β (ν, k1 )) = (μ , ν  ) 



is true with k1 = k  = k  ; if k  < k  , we can consider that α satisfies α k = α k +1 = ... = α k (0, ..., 0) ∈ Bn such that the conclusion holds with k1 = k  etc.

 −1

=

Remark 112. Stating that the Cartesian product of topologically transitive sets is topologically transitive is false in general. This happens because instead of (α,β)

β

(α,β)

β

α O× (μ, ν) = O (μ) × O (ν)

we have the weaker property α O× (μ, ν) ⊂ O (μ) × O (ν),

see Theorem 71, page 77.

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Boolean Systems

15.7 Path-connected components Definition 88. We consider the nonempty set X ⊂ Bn and the partition X1 , ..., Xp ⊂ X of k-path-connected subsets of X, p ≥ 1, k ∈ {(15.1.3), ..., (15.1.6)}. Then X1 , ..., Xp are called the k-path-connected components of X. Remark 113. X1 , ..., Xp with card(X1 ) > 1, ..., card(Xp ) > 1 are the (15.1.3)-path connected components of X if and only if G |X1 , ..., G |Xp are the path-connected components of G , see Definition 37, page 39 and Theorem 114, page 136. Remark 114. Let  : Bn → Bn , : Bm → Bm . If X1 , ..., Xp are the k-path-connected (by ) components of X, k ∈ {(15.1.3), ..., (15.1.5)}, and (h, h ) ∈ H om(φ, ψ), then h(X1 ), ..., h(Xp ) are the k-path-connected (by ) components of h(X), see Theorem 119, page 142. The same is true for k = (15.1.6) and (h, h ) ∈ I so(φ, ψ). Example 84. In Example 79, page 139, the sets {(1, 1, 1), (0, 1, 0), (1, 0, 0), (1, 0, 1)}, {(0, 1, 1)}, {(0, 0, 0)}, {(1, 1, 0)}, {(0, 0, 1)} are the (15.1.3)-path-connected components of B3 . Example 85. The set B2 has in Example 80 two (15.1.5)-path-connected components, {(1, 0), (0, 0), (0, 1)} and {(1, 1)}. Example 86. In Example 81, B2 has one (15.1.6)-path-connected component. Example 87. The space B2 has for the following system (0, 0) 6

(1, 0) 6

? (0, 1)

? (1, 1)

two (15.1.6)-path-connected components, {(0, 0), (0, 1)} and {(1, 0), (1, 1)}.

16 Chaos In literature there are several definitions of chaotic behavior of a system on a set X. In general, authors ask for topological transitivity and sensitive dependence on the initial conditions. Invariance, a property that will be introduced later, is a consequence of the topological transitivity in this context. Authors exist that ask, in addition to the previous requests, the density of the set of periodic points μ ∈ X in X, i.e. here the periodicity of the state. Section 16.1 defines the chaos and several examples are given in Section 16.2. Section 16.3 shows that the morphisms of flows bring the chaotic behavior of a system to the chaotic behavior of another system.

16.1 Definition Theorem 122. Let  : Bn → Bn and X ⊂ Bn , X = ∅. We consider the properties ∀μ ∈ X, ∃α ∈ n , (O α (μ) = X and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)),

(16.1.1)

∃α ∈ n , ∀μ ∈ X, (O α (μ) = X and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)),

(16.1.2)

∀α ∈ n , ∀μ ∈ X, (O α (μ) = X and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)),

(16.1.3)

∀μ ∈ X, ∃α ∈ n , (O α (μ) = X and ∃p ≥ 1, ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p) and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)),

(16.1.4)

∃α ∈ n , ∀μ ∈ X, (O α (μ) = X and ∃p ≥ 1, ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p) and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)),

(16.1.5)

∀α ∈ n , ∀μ ∈ X, (O α (μ) = X and ∃p ≥ 1, ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p) and ∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k)).

(16.1.6)

(a) The truth of (16.1.1) implies card(X) ≥ 2, Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00022-2 Copyright © 2023 Elsevier Inc. All rights reserved.

145

146

Boolean Systems

(b) statement (16.1.6) is false, (c) we have the implications: =⇒ (16.1.5) =⇒ ⇓ =⇒ (16.1.2) =⇒

(16.1.6) ⇓ (16.1.3)

(16.1.4) ⇓ (16.1.1).

Proof. (a) Let μ ∈ X arbitrary. The hypothesis states the existence of α ∈ n and μ ∈ X with the property that μ = φ α (μ, 0) = φ α (μ , 0) = μ , thus card(X) ≥ 2. (b) Let α = (0, ..., 0), (1, ..., 1), (0, ..., 0), (0, ..., 0), (1, ..., 1), (0, ..., 0), (0, ..., 0), (0, ..., 0), ...          1

2

3

and μ ∈ X arbitrary. The existence of μ ∈ X with ∀k ∈ N, φ α (μ, k) = φ α (μ , k) shows (like at (a)) that card(X) ≥ 2, thus the equality O α (μ) = X implies the non constancy of φ α (μ, ·). We infer that in the equation ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p)

(16.1.7)

we have p ≥ 2. From the way that α was chosen, we have the existence of k  ∈ N with the property that 





α k = α k +1 = ... = α k +p−1 = (0, ..., 0), therefore φ α (μ, k  + 1) = φ α (μ, k  + 2) = ... = φ α (μ, k  + p).

(16.1.8)

But Theorem 52, page 51 implies, from (16.1.7) and (16.1.8), that O α (μ) = {φ α (μ, k  + 1)}, card(O α (μ)) = card(X) = 1, contradiction. Definition 89. The properties (16.1.1)–(16.1.5) are called of chaos. If one of them is fulfilled, we say that function  exhibits a chaotic behavior on X. Remark 115. Several possibilities of adapting to asynchronicity the general concept of chaos exist, for example instead of (16.1.1) we can try and ∃μ etc.

∀μ ∈ X, ∃α ∈ n , (O α (μ) = X ∈ X, O α (μ ) ⊂ X and ∀k ∈ N, φ α (μ, k) = φ α (μ , k))

Chapter 16 • Chaos

147

16.2 Examples Example 88. The next system - (0, 1) (0, 0) Q 6 Q Q Q Q ? s Q (1, 0) (1, 1) fulfills properties (16.1.1) and (16.1.4) for X = {(0, 0), (1, 0)}. Indeed, for μ = (0, 0) we can take α = (1, 0), (1, 1), (1, 0), (1, 1), ... and for μ = (1, 0) we can take α = (1, 1), (1, 0), (1, 1), (1, 0), ... In both situations O α (μ) = X, p = 2 is a period of φ α (μ, ·) and μ ∈ X  {μ} satisfies the dependence on the initial conditions property ∀k ∈ N, φ α (μ, k) = φ α (μ , k). Let us suppose now against all reason that α in (16.1.1) is independent on μ and (16.1.2) holds. We can get from (0, 0) to (1, 0) with α 0 = (1, 0) only, otherwise (0,1) (0, 0) = (0, 1) ∈ / X, (1,1) (0, 0) = (1, 1) ∈ / X; and in addition α 0 = (1, 0) brings (1, 0) in (0, 0). Therefore, a possibility to switch between the elements of X by choosing x(0) be any of (0, 0), (1, 0) is to choose the computation function α = (1, 0), (1, 0), (1, 0), ... but such an α is not progressive. If we take α progressive such that x(k) = φ α (μ, k) switches between the elements of X, we have that α depends on μ, as we have seen. The falsity of (16.1.2) is proved. Example 89. We have in the following state portrait an example of satisfaction of (16.1.2) and (16.1.5) - (0, 1) (0, 0) Q 6 Q Q Q Q ? s Q (1, 1) (1, 0) by taking X = {(0, 0), (1, 1)}. For this, it is enough to choose α = (1, 1), (1, 1), (1, 1), ... On the other hand (16.1.3) is false. To be compared this example with the previous one. Example 90. The system (0, 0) 

? (1, 0)

(0, 1) 6 - (1, 1)

gives for X = B2 an example when (16.1.3) holds. Note that in (16.1.3), for any α ∈ 2 and any μ ∈ B2 , we must choose μ ∈ B2 equal with μ.

148

Boolean Systems

16.3 Morphisms Theorem 123. We suppose that ,  : Bn → Bn , X ⊂ Bn , X = ∅ are given and (h, h ) : φ → ψ is an isomorphism. Then (a) (16.1.1) implies α (μ) = h(X) ∀μ ∈ h(X), ∃α ∈ n , (O and ∃μ ∈ h(X), ∀k ∈ N, ψ α (μ, k) = ψ α (μ , k)),

(b) (16.1.2) implies α (μ) = h(X) ∃α ∈ n , ∀μ ∈ h(X), (O  α and ∃μ ∈ h(X), ∀k ∈ N, ψ (μ, k) = ψ α (μ , k)),

(c) (16.1.3) implies α (μ) = h(X) ∀α ∈ n , ∀μ ∈ h(X), (O and ∃μ ∈ h(X), ∀k ∈ N, ψ α (μ, k) = ψ α (μ , k)),

(d) (16.1.4) implies α (μ) = h(X) ∀μ ∈ h(X), ∃α ∈ n , (O α and ∃p ≥ 1, ∀k ∈ N, ψ (μ, k) = ψ α (μ, k + p) and ∃μ ∈ h(X), ∀k ∈ N, ψ α (μ, k) = ψ α (μ , k)),

(e) (16.1.5) implies α (μ) = h(X) ∃α ∈ n , ∀μ ∈ h(X), (O and ∃p ≥ 1, ∀k ∈ N, ψ α (μ, k) = ψ α (μ, k + p) and ∃μ ∈ h(X), ∀k ∈ N, ψ α (μ, k) = ψ α (μ , k)).

Proof. (d) Let μ ∈ h(X) arbitrary, thus ν ∈ X exists with h(ν) = μ. From (16.1.4) we have the existence of α ∈ n with α O (ν) = X,

(16.3.1)

∃p ≥ 1, ∀k ∈ N, φ α (ν, k) = φ α (ν, k + p),

(16.3.2)

∃ν  ∈ X, ∀k ∈ N, φ α (ν, k) = φ α (ν  , k).

(16.3.3)

Theorem 120, page 142 and (16.3.1) show that h (α)

O

(μ) = h(X),

Theorem 112, page 132 shows that for p whose existence is stated at (16.3.2), we obtain 



∀k ∈ N, ψ h (α) (μ, k) = ψ h (α) (μ, k + p),

Chapter 16 • Chaos

149

and from Theorem 106, page 125 we infer that for μ ∈ h(X) defined by μ = h(ν  ), where ν  is the one whose existence is given by (16.3.3), we infer 



∀k ∈ N, ψ h (α) (μ, k) = ψ h (α) (μ , k).

17 Nonwandering points and Poisson stability A point μ is nonwandering if μ ∈ ωα (μ) and it is Poisson stable if O α (μ) = ωα (μ). The two concepts are defined in Sections 17.1, 17.2, where several examples are also given. In Section 17.3 we prove that the omega-limit points are nonwandering and Poisson stable. In Section 17.4 we show that the morphisms of flows bring nonwandering points in nonwandering points and Poisson stable points in Poisson stable points.

17.1 Nonwandering points Definition 90. Let the function  : Bn → Bn . A point μ ∈ Bn is nonwandering (or recurrent) if one of ∃α ∈ n , μ ∈ ωα (μ),

(17.1.1)

∀α ∈ n , μ ∈ ωα (μ)

(17.1.2)

holds. Remark 116. The point μ is nonwandering if the state φ α (μ, ·) gets back to its initial value μ infinitely many times. Remark 117. The periodic points – which were defined at Definition 84, page 133 – and, more general, the omega-limit points fulfill (17.1.1). The last statement will be proved at Theorem 125. The fixed points fulfill (17.1.2). Example 91. For the following system, the point (0, 0) is not nonwandering, it ‘wanders’ to one of (0, 1), (1, 0), (1, 1), (0, 1) 

(0, 0)

- (1, 0)

? (1, 1) and (0, 1), (1, 0), (1, 1) fulfill (17.1.2). Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00023-4 Copyright © 2023 Elsevier Inc. All rights reserved.

151

152

Boolean Systems

Example 92. The system (0, 1) 

(0, 0)



- (1, 0)

? (1, 1) fulfills the property that all the points are nonwandering: (0, 0), (1, 0) fulfill (17.1.1), while the fixed points (0, 1), (1, 1) fulfill (17.1.2). Example 93. In the case of the system (0, 0) 6

(1, 0) 6

? (0, 1)

? (1, 1)

all the points μ ∈ B2 fulfill (17.1.2).

17.2 Poisson stability Definition 91. Let  : Bn → Bn and μ ∈ Bn . The point μ is called Poisson stable if one of ∃α ∈ n , O α (μ) = ωα (μ),

(17.2.1)

∀α ∈ n , O α (μ) = ωα (μ)

(17.2.2)

holds. Remark 118. The Poisson stability property (17.2.1) holds for example in case of the periodicity of the state φ α (μ, ·). Example 94. In the state portrait (0, 0) 6

(0, 1) 6 3     ? ?  - (1, 1) (1, 0) the points (0, 0), (1, 0) are (17.2.1)-Poisson stable and (0, 1), (1, 1) are (17.2.2)-Poisson stable. Example 95. The identity 1Bn satisfies that any μ ∈ Bn is (17.2.2)-Poisson stable.

Chapter 17 • Nonwandering points and Poisson stability

153

Example 96. For the system - (0, 1)

(0, 0) 6 (1, 0) 

? (1, 1)

any μ ∈ B2 is (17.2.2)-Poisson stable. Remark 119. In the case of the system (0, 1) 

- (0, 0) 

- (1, 0)

? (1, 1) we take α = (0, 1), (1, 1), (1, 0), (1, 1), (1, 0), (1, 1), (1, 0), ... The point (0, 0) is (17.1.1)-nonwandering and it is not Poisson stable, because O α (0, 0) = {(0, 1), (0, 0), (1, 0)} and ωα (0, 0) = {(0, 0), (1, 0)}.

17.3 Properties Theorem 124. The (17.2.1)-Poisson stable points are (17.1.1)-nonwandering and the (17.2.2)-Poisson stable points are (17.1.2)-nonwandering. Proof. In order to prove the first statement, we take an arbitrary μ ∈ Bn , and we suppose that α ∈ n exists such that O α (μ) = ωα (μ). Then μ ∈ O α (μ) = ωα (μ). Theorem 125. The following property is true: ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ ωα (μ), ∃β ∈ n such that ωβ (μ ) = O β (μ ) = ωα (μ), μ ∈ ωβ (μ ). Proof. We take α ∈ n , μ ∈ Bn , and μ ∈ ωα (μ) arbitrary. Since k1 ∈ N exists with ωα (μ) = {φ α (μ, k)|k ≥ k1 }, 

we have the existence of k  ≥ k1 such that μ = φ α (μ, k  ). We define β by β = σ k (α), and we get O β (μ ) = {φ β (μ , k)|k ∈ N} = {φ α (μ, k)|k ≥ k  } = ωα (μ).

(17.3.1)

154

Boolean Systems

In order to prove that the (17.2.1)-Poisson stability of μ O β (μ ) = ωβ (μ )

(17.3.2)

holds, we want to show first the truth of O β (μ ) ⊂ ωβ (μ ).

(17.3.3)

Let us take ν ∈ O β (μ ) arbitrary. From (17.3.1) we have ν ∈ ωα (μ), hence the sets {k|k ∈ N, φ β (μ , k) = ν} = {k − k  |k ≥ k  , φ α (μ, k) = ν} are infinite, therefore ν ∈ ωβ (μ ). The inclusion (17.3.3) takes place, therefore (17.3.2) is true. Like in Theorem 124, μ ∈ O β (μ ) = ωβ (μ ).

17.4 Morphisms Theorem 126. The functions  : Bn → Bn , : Bm → Bm and μ ∈ Bn are given. (a) If (h, h ) ∈ H om(φ, ψ), then β

α ∃α ∈ n , μ ∈ ω (μ) =⇒ ∃β ∈ m , h(μ) ∈ ω (h(μ)), β

β

α α (μ) = ω (μ) =⇒ ∃β ∈ m , O (h(μ)) = ω (h(μ)), ∃α ∈ n , O

(17.4.1) (17.4.2)

(b) if (h, h ) ∈ I so(φ, ψ), then m = n and β

α ∀α ∈ n , μ ∈ ω (μ) ⇐⇒ ∀β ∈ m , h(μ) ∈ ω (h(μ)), β

β

α α (μ) = ω (μ) ⇐⇒ ∀β ∈ m , O (h(μ)) = ω (h(μ)). ∀α ∈ n , O

(17.4.3) (17.4.4)

α (μ), thus h(μ) ∈ h(ωα (μ)) Proof. (17.4.1): We suppose that α ∈ n exists such that μ ∈ ω  Theorem 78, page 91

h (α)

= ω (h(μ)). We have obtained the existence of β ∈ m , namely β = h (α), β with the property h(μ) ∈ ω (h(μ)). (17.4.4): =⇒ We have m = n and let β ∈ n arbitrary. Then α ∈ n with the property α (μ) = ωα (μ). We get h (α) = β satisfies O  β

h (α)

O (h(μ)) = O

α = h(ω (μ))

(h(μ))

Theorem 78

=

Theorem 78

h (α)

ω

=

α h(O (μ)) β

(h(μ)) = ω (h(μ)).

18 Invariance The invariance of a set A ⊂ Bn , A = ∅ relative to a system  : Bn −→ Bn is the property that all the orbits of  with the initial value in A, are included in A. Five definitions of invariance (four of which are non-equivalent) are given in Section 18.1, and Section 18.2 has several examples of invariant sets. The invariant subsets are the topic of Section 18.3. A first group of properties of invariance (such as: the orbits and the omega-limit sets of the states of  are invariant, the intersection and the union of the invariant sets are invariant) is given in Section 18.4. The way that morphisms bring invariant sets in invariant sets is shown in Section 18.5, and Section 18.6 contains the special case of the symmetries relative to translations. If a set A ⊂ Bn is invariant relative to a subsystem  : Bn −→ Bn of  : Bn+m −→ Bn+m , then the invariance of the set A × Bm relative to  follows. This implication, together with the inverse one, is studied in Section 18.7, while Section 18.8 treats similar implications on Cartesian products of systems. We note in Section 18.9 that topological transitivity implies invariance and pathconnectedness. In Section 18.10 we give a Lyapunov-Lagrange type invariance theorem, that is the Boolean asynchronous analogue of Theorem 3.3.1 from page 93, and Theorem 4.1.3 from page 151, in [33]. Section 18.11 shows that other possibilities of defining invariance exist.

18.1 Definition Remark 120. The invariance of the nonempty sets A ⊂ Bn has been present several times in this book. Thus – in the definition of a system, in opposition with the attribute ‘universal’, we have mentioned at page 72 that in certain circumstances of invariance of a set A, the possibility of restricting the state space to A exists, – at Theorem 86, page 101 referring to nullclines, we have showed that N Ci = Bn partitions Bn in two sets, (Bn )0i and (Bn )1i that contain, together with a point μ, all the orbits starting from μ, and this is a property of invariance, – at Remark 102, page 125, when suggesting several versions of the property of dependence on the initial conditions, we have noted that  : Bn → Bn may be replaced by  : A → A, where the truth of the invariance ∀λ ∈ Bn , ∀μ ∈ A, λ (μ) ∈ A should hold, – at page 145, when writing about chaos in literature, we mentioned invariance, which is here a consequence of the topological transitivity. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00024-6 Copyright © 2023 Elsevier Inc. All rights reserved.

155

156

Boolean Systems

And we recall also the directed graph G |A = (X, E |A ), A ⊂ Bn see Notation 13, page 39, where G = (Bn , E ) is the state portrait of  : Bn → Bn and E |A = {(μ, λ (μ))|μ ∈ A, λ ∈ Bn , λ (μ) ∈ A, μ = λ (μ)}. In this case all the arrows (μ, λ (μ)) of G starting from A are in E |A if A is invariant. All the previous examples that anticipated implicitly or explicitly this chapter dedicated to invariance show its importance. Lemma 1. If  : A → A and A is a finite set, then the following statements are equivalent: (a)  is injective, (b)  is surjective, (c)  is bijective. Proof. (a)=⇒(b) We suppose against all reason that  is not surjective, i.e. ν ∈ A exists with ∀μ ∈ A, (μ) = ν. As card(A) > card((A)), we infer that μ , μ ∈ A exist, μ = μ with the property that (μ ) = (μ ), contradiction. (b)=⇒(a) We suppose against all reason that  is not injective, therefore μ , μ ∈ A exist,  μ = μ , such that (μ ) = (μ ). We obtain card(A) > card((A)), thus ν ∈ A exists with ∀μ ∈ A, (μ) = ν, contradiction. (b)=⇒(c) As  is surjective implies that  is injective, we infer that  is bijective. (c)=⇒(b) Obvious. Theorem 127. Let the function  : Bn −→ Bn and the set A ⊂ Bn , A = ∅. The relations ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A,

(18.1.1)

∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A,

(18.1.2)

∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A,

(18.1.3)

∀λ ∈ Bn , λ (A) ⊂ A,

(18.1.4)

∀λ ∈ Bn , λ (A) = A

(18.1.5)

fulfill (18.1.5) =⇒ (18.1.4) ⇐⇒ (18.1.3) =⇒ (18.1.2) =⇒ (18.1.1). Proof. The implications are obvious in general, we prove (18.1.3)⇐⇒(18.1.4), with (18.1.3) written under the form ∀α ∈ n , ∀μ ∈ A, ∀k ∈ N, φ α (μ, k) ∈ A.

(18.1.6)

(18.1.3)=⇒(18.1.4) Let λ ∈ Bn , μ ∈ A arbitrary, fixed. We take α ∈ n arbitrary, with α 0 = λ. Then:

Chapter 18 • Invariance

λ (μ) = φ α (μ, 1)

(18.1.6)

∈

157

A.

(18.1.4)=⇒(18.1.3) We take α ∈ n , μ ∈ A arbitrary, fixed and we prove (18.1.6) by induction on k. For k = 0, μ = φ α (μ, 0) ∈ A and we suppose now that φ α (μ, k) ∈ A. Then: k

φ α (μ, k + 1) = α (φ α (μ, k))

(18.1.4)

∈

A.

Definition 92. Relations (18.1.1)–(18.1.5) are called of invariance of A. We say that A is a k-invariant set, k ∈ {1, ..., 5} (relative to ). Remark 121. Using several non-equivalent definitions of the same concept, as resulted by the translations, in many situations, of a real numbers dynamical systems concept, to a Boolean dynamical systems concept, is treated in our work something like: (18.1.1)invariance, ..., (18.1.5)-invariance. We stress on the fact that the proposed terminology is in this case that of 1-invariance, ..., 5-invariance as an exception, simply because invariance will be present frequently from now and this seems to be the most natural way to proceed. Remark 122. The properties (18.1.1)–(18.1.3) refer to flows and the properties (18.1.4), (18.1.5) refer to generator functions. Note that φ,  represent the same system, and note also that the compatibility between these two points of view on invariance expressed by Theorem 127 includes the equivalence (18.1.3)⇐⇒(18.1.4). Remark 123. The invariance of a set A is the property of a system that, if it starts from an initial value of the state μ ∈ A, all the subsequent values of the state are in A. The different behaviors of the system in this situation make us have several definitions of invariance. Remark 124. Lemma 1 interprets property (18.1.5) in the following way: ∀λ ∈ Bn , the restriction of λ : Bn → Bn to A has the values in A, and the resulted λ : A → A function (abusive notation!) is bijective. Remark 125. Invariance as defined by Definition 92 may be called also stability in the sense of Lyapunov. Another type of stability, i.e. asymptotic stability, which is characterized by lim φ α (μ, k) ∈ A instead of ∀k ∈ N, φ α (μ, k) ∈ A (i.e. O α (μ) ⊂ A), will be introduced in k→∞

Chapter 29.

18.2 Examples Example 97. The identity 1B2 (0, 0)

(0, 1)

(1, 0)

(1, 1)

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Boolean Systems

has the property that any nonempty subset of B2 is 5-invariant. The same is true in the general case of 1Bn : Bn → Bn . Example 98. For any  : Bn → Bn , the space Bn is k-invariant, k ∈ {1, ..., 4}. Example 99. We have seen at Theorem 86, page 101, the way that the property N Ci = Bn , i ∈ {1, ..., n} partitions the space Bn in two k-invariant sets (Bn )0i and (Bn )1i , k ∈ {1, ..., 4}. Example 100. For a fixed point μ ∈ Bn , the set {μ} is 5-invariant. The sets A ⊂ Bn of fixed points ∀μ ∈ A, (μ) = μ are 5-invariant. This will be proved in Theorem 131, but see again Example 97 also. Example 101. We consider the state portrait - (1, 1, 1) (0, 1, 0)  (1, 1, 0) 3  Q 6  Q  Q  Q  Q  s Q (0, 0, 0) (1, 0, 0) (0, 1, 1)

(0, 0, 1)

(1, 0, 1)

where the set A = {(0, 0, 0), (1, 1, 0), (1, 1, 1)} is 1-invariant. In order to see this, we notice that for α ∈ 3 with α 0 = (1, 1, 0), α 1 = (0, 0, 1) we have φ α ((0, 0, 0), ·) = (0, 0, 0), (1, 1, 0), (1, 1, 1), (1, 1, 1), ... ∈ A, for β ∈ 3 with β 0 = (0, 0, 1) we have φ β ((1, 1, 0), ·) = (1, 1, 0), (1, 1, 1), (1, 1, 1), ... ∈ A, and for arbitrary γ ∈ 3 we have φ γ ((1, 1, 1), ·) = (1, 1, 1), (1, 1, 1), ... ∈ A. We prove now that (18.1.2) is false. Indeed, the requests 

λ (0, 0, 0) ∈ {(0, 0, 0), (1, 1, 0)}, λ (1, 1, 0) ∈ {(1, 1, 0), (1, 1, 1)}

are satisfied by λ ∈ {(0, 0, 0), (0, 0, 1), (1, 1, 0), (1, 1, 1)} ∧ {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)} = {(0, 0, 0), (0, 0, 1)}. As ∀λ ∈ {(0, 0, 0), (0, 0, 1)}, λ (0, 0, 0) = (0, 0, 0), in order that (18.1.2) be true, the inclusion O α (0, 0, 0) ⊂ A should take place under the form ∀k ∈ N, φ α ((0, 0, 0), k) = (0, 0, 0), O α ((0, 0, 0)) = {(0, 0, 0)}, where α ∈ 3 has ∀k ∈ N, α k ∈ {(0, 0, 0), (0, 0, 1)}, contradiction with its requirement of progressiveness. We have obtained that the choice of α ∈ 3 such that (18.1.1) holds depends on μ ∈ A.

Chapter 18 • Invariance

159

Example 102. Let the function (μ1 , μ2 ) = (μ1 ∪ μ2 , 1) and the set A = {(0, 0), (0, 1)}. - (0, 1) (0, 0) Q Q Q Q Q ? s Q - (1, 1) (1, 0) A is 2-invariant: for this, it is enough to choose α ∈ 2 with α 0 = (0, 1) and see that φ α ((0, 0), ·) = (0, 0), (0, 1), (0, 1), ... ∈ A, φ α ((0, 1), ·) = (0, 1), (0, 1), (0, 1), ... ∈ A. A is not 3-invariant since if, for α ∈ 2 , we take α 0 = (1, 1), we get ∀k ≥ 1, φ α ((0, 0), k) = (1, 1) ∈ / A. A = {(0, 1)} is 5-invariant. For this system, B = {(0, 0), (1, 0), (1, 1)} is 2-invariant, B  = {(1, 0), (1, 1)} is 3-invariant, and B  = {(1, 1)} is 5-invariant. Example 103. From the state portrait of the function - (0, 1, 0) (0, 0, 0) Q Q Q Q Q ? s ? Q (1, 0, 0) (1, 1, 0) 

(1, 0, 1) 

(0, 0, 1)      ?+ ?  (1, 1, 1)  (0, 1, 1)

we notice that the set A = {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0)} is 3-invariant. The subsets A = {(1, 0, 0), (1, 1, 0)}, A = {(0, 1, 0), (1, 1, 0)} are 3-invariant also, A = {(0, 0, 0), (1, 1, 0)} is 2invariant and A = {(1, 1, 0)} is 5-invariant. We see that B = {(0, 0, 1), (1, 0, 1), (0, 1, 1), (1, 1, 1), (1, 1, 0)} is 3-invariant, together with some of its subsets. Example 104. We have the example of (μ1 , μ2 ) = (μ1 , μ2 ) (0, 0) 6

(0, 1) 6

? (1, 0)

? (1, 1)

where the whole space B2 is 5-invariant. Indeed, for different values of λ ∈ B2 , from λ (μ1 , μ2 ) = (μ1 ⊕ λ1 , μ2 ) we get two functions, 1B2 (μ1 , μ2 ) = (μ1 , μ2 ) and (μ1 , μ2 ) = (μ1 ⊕ 1, μ2 ). The functions 1B2 and  are both bijective. In this example the sets {(0, 0), (1, 0)}, {(0, 1), (1, 1)} are 5-invariant too.

160

Boolean Systems

18.3 Invariant subset Definition 93. Let  : Bn −→ Bn and the sets ∅ = A ⊂ X ⊂ Bn . If A is k-invariant, k ∈ {1, ..., 5}, then it is called a k-invariant subset of X. Remark 126. The invariant subsets occur often, for example A = Bn is trivially a 3invariant subset of X = Bn . Remark 127. (a) In Definition 93, normally, the set X is not invariant. It is of interest, in such situations, to think of A as the maximal/minimal invariant subset of X, in the sense of the inclusion. For example, we can take in Example 104 X = {(0, 0), (1, 0), (0, 1)} which is not invariant, while A = {(0, 0), (1, 0)} is the only invariant subset (5-invariance). (b) Another possibility is the one when A, X are both invariant and they have different types of invariance. A situation like ∃α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A, not (∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A) is non-contradictory (the invariance of A is weaker than the invariance of X). We can give here the case of the function  from Example 101, page 158, where X = {(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)} is 2-invariant, and in order to see this, it is sufficient to take α ∈ 3 with α 0 = α 1 = (1, 1, 0). The subset A = {(0, 0, 0), (1, 1, 0), (1, 1, 1)} is 1-invariant, as we have already remarked there. At the same time, ∀λ ∈ Bn , λ (X) = X, ∀λ ∈ Bn , λ (A) ⊂ A, not (∀λ ∈ Bn , λ (A) = A) is contradictory (the invariance of A is weaker than the invariance of X again). For this, see Theorem 135 to follow, page 165, and its proof. (c) It is convenient, as a third possibility, to consider that A and X have the same type of invariance. An example for this is given by the function  from Example 104, page 159, where A = {(0, 0), (1, 0)} and X = B2 are 5-invariant.

18.4 Properties Theorem 128. Let  : Bn → Bn and μ ∈ Bn . We have the 4-invariance ∀λ ∈ Bn , λ (O + (μ)) ⊂ O + (μ).

Chapter 18 • Invariance

161

Proof. We take λ ∈ Bn and δ ∈ O + (μ) arbitrary, therefore ν ∈ Bn , ..., ξ ∈ Bn exist such that δ = (ν ◦ ... ◦ ξ )(μ). We can write λ (δ) = λ ((ν ◦ ... ◦ ξ )(μ)) = (λ ◦ ν ◦ ... ◦ ξ )(μ) ∈ O + (μ). Theorem 129. For τ ∈ Bn , we have that Bn satisfies the 5-invariance property ∀λ ∈ Bn , (θ τ )λ (Bn ) = Bn .

(18.4.1)

Proof. For any λ ∈ Bn , (θ τ )λ = θ τ λ is bijection, thus (18.4.1) is true. Example 105. We have drawn below the state portrait of function  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 , μ1 ⊕ μ2 ). (0, 0)

(1, 0) 

(0, 1)

-

(1, 1)

This function is not a translation, but its restriction to {(0, 0), (0, 1)} behaves like 1B2 = θ (0,0) , and its restriction to {(1, 0), (1, 1)} behaves like θ (0,1) . {(0, 0), (0, 1)}, {(1, 0), (1, 1)}, B2 and some other sets are 5-invariant. Theorem 130. If  : Bn → Bn satisfies that μ ∈ Bn is not a fixed point, then {μ} is not invariant in the sense of Definition 92, page 157. Proof. We suppose that (μ) = μ ⊕ ε i1 ⊕ ... ⊕ ε ip , where i1 , ..., ip ∈ {1, ..., n}, and let α ∈ n arbitrary. From the progressiveness of α we get the existence of the least k ∈ N with / {μ}. supp α k ∧ {i1 , ..., ip } = ∅. We have φ α (μ, k + 1) ∈ Theorem 131. If A ⊂ Bn fulfills ∀μ ∈ A, (μ) = μ, then it is 5-invariant. Proof. We suppose against all reason that the property is false, i.e. ∃λ ∈ Bn , λ (A) = A and two non-exclusive possibilities exist. Case λ (A)  A = ∅ / A. But λ (μ) = μ ∈ A, contradiction. when μ ∈ A exists such that λ (μ) ∈ λ Case A   (A) = ∅ Then ν ∈ A exists such that ∀μ ∈ A, λ (μ) = ν. As λ (ν) = ν, we have obtained a contradiction again. Theorem 132. Function  : Bn → Bn is given. For any α ∈ n and any μ ∈ Bn , the sets O α (μ), ωα (μ) are 1-invariant.

162

Boolean Systems

Proof. Let α ∈ n , μ ∈ Bn and μ ∈ O α (μ) arbitrary, fixed. We have the existence of k  ∈ N with μ = φ α (μ, k  ) and therefore ∀k ≥ k  , φ α (μ, k) = φ σ

k  (α)

(μ , k − k  ).

We infer Oσ

k  (α)

(μ ) = {φ σ

k  (α)

(μ , k)|k ∈ N} = {φ σ

k  (α)

(φ α (μ, k  ), k)|k ∈ N}

= {φ α (μ, k + k  )|k ∈ N} = {φ α (μ, k)|k ≥ k  } ⊂ {φ α (μ, k)|k ∈ N} = O α (μ). We have just shown that ∀μ ∈ O α (μ), ∃β ∈ n , O β (μ ) ⊂ O α (μ), 

with β = σ k (α). Similarly, we suppose that ωα (μ) = {φ α (μ, k)|k ≥ k1 } for k1 ∈ N, and we take μ ∈ ωα (μ) arbitrary, thus k  ≥ k1 exists with μ = φ α (μ, k  ). We infer Oσ

k  (α)

(μ ) = {φ σ

k  (α)

(μ , k)|k ∈ N} = {φ α (μ, k)|k ≥ k  } = ωα (μ).

We have just proved that ∀μ ∈ ωα (μ), ∃β ∈ n , O β (μ ) ⊂ ωα (μ), 

where β = σ k (α). Remark 128. A comparison is now possible between the last statement of Theorem 132, relative to the invariance of ωα (μ), together with its proof, and Theorem 125, page 153, together with its proof. Theorem 133. Let A = {μ1 , ..., μp } ⊂ Bn , p ≥ 1. (a) If p = 1, then ∀k ∈ {1, ..., 5}, the k-invariance of A is equivalent with (μ1 ) = μ1 ; (b) if p ≥ 2, then (b.1) the 1-invariance of A is equivalent with 1

p

∃α 1 ∈ n , ..., ∃α p ∈ n , {μ1 , ..., μp } = O α (μ1 ) ∨ ... ∨ O α (μp ), (b.2) the 2-invariance of A is equivalent with ∃α ∈ n , {μ1 , ..., μp } = O α (μ1 ) ∨ ... ∨ O α (μp ), (b.3) the 3-invariance of A is equivalent with 1

p

∀α 1 ∈ n , ..., ∀α p ∈ n , {μ1 , ..., μp } = O α (μ1 ) ∨ ... ∨ O α (μp ),

Chapter 18 • Invariance

163

(b.4) the 4-invariance of A is equivalent with {μ1 , ..., μp } = μ1+ ∨ ... ∨ μp+ . Proof. (b) The hypothesis asks that p ≥ 2. 1 (b.1) =⇒ The computation functions α 1 ∈ n , ..., α p ∈ n exist such that O α (μ1 ) ⊂ p 1 p 1 p A, ..., O α (μp ) ⊂ A, thus O α (μ1 ) ∨ ... ∨ O α (μp ) ⊂ A. As μ1 ∈ O α (μ1 ), ..., μp ∈ O α (μp ) im1 p 1 p ply A ⊂ O α (μ1 ) ∨ ... ∨ O α (μp ), we infer A = O α (μ1 ) ∨ ... ∨ O α (μp ). ⇐= The computation functions α 1 ∈ n , ..., α p ∈ n exist with the property A = 1 p i 1 α O (μ1 ) ∨ ... ∨ O α (μp ) and let i ∈ {1, ..., p} arbitrary. We get O α (μi ) ⊂ O α (μ1 ) ∨ ... ∨ p O α (μp ). (b.4) =⇒ ∀λ ∈ Bn , λ (μ1 ) ∈ A implies μ1+ ⊂ A, ..., ∀λ ∈ Bn , λ (μp ) ∈ A implies μp+ ⊂ A, therefore μ1+ ∨ ... ∨ μp+ ⊂ A. On the other hand μ1 ∈ μ1+ , ..., μp ∈ μp+ shows that A ⊂ μ1+ ∨ ... ∨ μp+ . We infer A = μ1+ ∨ ... ∨ μp+ . ⇐= The hypothesis states that A = μ1+ ∨ ... ∨ μp+ and let λ ∈ Bn , i ∈ {1, ..., p} arbitrary. We get λ (μi ) ∈ μi+ , thus λ (μi ) ∈ A. Problem 10. To be written Theorem 133 (b.5), which is missing. Theorem 134. For the nonempty sets A, B ⊂ Bn , we have ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A and ∀μ ∈ B, ∃α ∈ n , O α (μ) ⊂ B =⇒ ∀μ ∈ A ∨ B, ∃α ∈ n , O α (μ) ⊂ A ∨ B,

(18.4.2)

∀λ ∈ Bn , λ (A) ⊂ A and ∀λ ∈ Bn , λ (B) ⊂ B and A ∧ B = ∅ =⇒ ∀λ ∈ Bn , λ (A ∧ B) ⊂ A ∧ B,

(18.4.3)

∀λ ∈ Bn , λ (A) ⊂ A and ∀λ ∈ Bn , λ (B) ⊂ B =⇒ ∀λ ∈ Bn , λ (A ∨ B) ⊂ A ∨ B,

(18.4.4)

∀λ ∈ Bn , λ (A) = A and ∀λ ∈ Bn , λ (B) = B and A ∧ B = ∅ =⇒ ∀λ ∈ Bn , λ (A ∧ B) = A ∧ B,

(18.4.5)

∀λ ∈ Bn , λ (A) = A and ∀λ ∈ Bn , λ (B) = B =⇒ ∀λ ∈ Bn , λ (A ∨ B) = A ∨ B,

(18.4.6)

∀λ ∈ Bn , λ (A) = A and ∀λ ∈ Bn , λ (B) = B and A  B = ∅ =⇒ ∀λ ∈ Bn , λ (A  B) = A  B.

(18.4.7)

Proof. We prove (18.4.5). Let λ ∈ Bn and μ ∈ A ∧ B arbitrary. The hypothesis states that λ (μ) ∈ A and λ (μ) ∈ B are both true, thus λ (μ) ∈ A ∧ B. This proves that λ (A ∧ B) ⊂ A ∧ B.

164

Boolean Systems

In order to show that λ (A ∧ B) = A ∧ B, we suppose against all reason that μ ∈ A ∧ B exists, μ = μ , such that λ (μ) = λ (μ ). We infer that λ (A)  A, representing a contradiction with the hypothesis. It has resulted that such a μ does not exist, therefore λ (A ∧ B) = A ∧ B holds. We prove (18.4.7). If, against all reason, the set A  B is not 5-invariant, we have the existence of λ ∈ Bn with λ (A  B) = A  B. This means the existence of μ ∈ A  B such that λ (μ) ∈ A ∧ B (since λ (A) = A). On the other hand, from (18.4.5), μ ∈ A ∧ B exists with λ (μ ) = λ (μ).

(18.4.8)

We have μ ∈ / B, μ ∈ B, therefore μ = μ . Eq. (18.4.8) contradicts the bijectivity of λ : A → A. Remark 129. If A, B are 1-invariant, then A ∨ B is 1-invariant, see (18.4.2), but A ∧ B is not. Such properties do not hold also for the 2-invariance of A ∧ B and A ∨ B. The intersection and the union of k-invariant sets are k-invariant sets however for k ∈ {3, 4, 5}. Remark 130. In the proof of the 5-invariance of A  B from (18.4.7) we have used the bijectivity of λ : A → A. The proof cannot be made for k-invariances of A  B with k ∈ {1, ..., 4} and these properties are false. Example 106. In the case of the system (1, 1) (1, 0) Q k 6 Q Q Q Q Q  (0, 1) (0, 0) the sets A = {(0, 0), (0, 1)}, B = {(0, 0), (1, 1)} fulfill both (18.1.2)page 156 , as for α = (0, 1), (1, 1), (1, 1), (1, 1), ... we have O α (0, 0) = {(0, 0), (0, 1)}, O α (0, 1) = {(0, 1)} and on the other hand for β = (1, 1), (1, 1), (1, 1), ... we have O β (0, 0) = {(0, 0), (1, 1)}, O β (1, 1) = {(1, 1)}. But A ∧ B = {(0, 0)} is not 2-invariant and in fact {(0, 0)} makes the 1-invariance, ..., 5-invariance be all false. Example 107. We give the example of a system - (0, 0, 1) (0, 0, 0) Q Q Q Q Q ? s Q (1, 0, 0) (1, 0, 1)

(0, 1, 0) 

(0, 1, 1)      ?  + (1, 1, 0) (1, 1, 1)

Chapter 18 • Invariance

165

for which the sets A = {(0, 0, 0), (1, 0, 0)} and B = {(0, 1, 1), (0, 1, 0)} are 2-invariant. In order to see this, we can take α ∈ 3 with α 0 = (1, 0, 0) in the case of A and with α 0 = (0, 0, 1) in the case of B. We prove now that A ∨ B is 1-invariant only, and we note that the requests of 2invariance (referring to (0, 0, 0), (0, 1, 1) since (1, 0, 0), (0, 1, 0) are fixed points)  λ  (0, 0, 0) ∈ {(0, 0, 0), (1, 0, 0)}, λ (0, 1, 1) ∈ {(0, 1, 1), (0, 1, 0)} give λ ∈ {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0)} ∧ {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)} = {(0, 0, 0), (0, 1, 0)}. There is no α ∈ 3 such that ∀μ ∈ A ∨ B, O α (μ) ⊂ A ∨ B and ∀k ∈ N, α k ∈ {(0, 0, 0), (0, 1, 0)}. Theorem 135. If  : Bn → Bn , the set X ⊂ Bn is 5-invariant and Y ⊂ X is 4-invariant, then Y is 5-invariant. Proof. The hypothesis states that ∀λ ∈ Bn , λ (X) = X,

(18.4.9)

∀λ ∈ Bn , λ (Y ) ⊂ Y

(18.4.10)

are true and we fix an arbitrary λ ∈ invariant:

Bn .

We suppose against all reason that Y is not 5-

λ (Y )  Y, i.e. μ, μ ∈ Y exist, μ = μ such that λ (μ) = λ (μ ). This is in contradiction with λ (X) = X. Y is 5-invariant. Theorem 136. The following statements are equivalent: (a) X is 5-invariant, (b) ∀λ ∈ Bn , a partition X1 , ..., Xk of X exists, k ≥ 1, with the property that ∀i ∈ {1, ..., k}, ∃pi ≥ 1, ∀μ ∈ Xi , Xi = {μ, (λ )(μ), ..., (λ )(pi −1) (μ)} and (λ )(pi ) (μ) = μ.

(18.4.11)

Proof. If card(X) = 1 then (a), (b) are both equivalent with (μ) = μ, so that we can suppose in the rest of the proof that card(X) > 1. (a)=⇒(b) We take λ ∈ Bn and μ ∈ X arbitrary, fixed both of them. If λ (μ) = μ, then we denote X1 = {μ} and we have p1 = 1. We suppose now that μ, (λ )(μ), ..., (λ )(p1 −1) (μ) are distinct, p1 ≥ 2 and (λ )(p1 ) (μ) ∈ {μ, (λ )(μ), ..., (λ )(p1 −1) (μ)}. We claim that (λ )(p1 ) (μ) = μ.

(18.4.12)

166

Boolean Systems

Indeed, if this would not be true, then q ≥ 1 would exist such that (λ )(p1 −1) (μ) = (λ )(q−1) (μ), (λ )(p1 ) (μ) = (λ )(q) (μ), representing a contradiction with the request of bijectivity of λ : X → X. (18.4.12) is true and we denote X1 = {μ, (λ )(μ), ..., (λ )(p1 −1) (μ)}. If X = X1 statement (b) is true, thus we can suppose that X = X1 . From λ (X) = X,

(18.4.13)

λ (X1 ) = λ ({μ, (λ )(μ), ..., (λ )(p1 −1) (μ)}) = {(λ )(μ), (λ )(2) (μ), ..., μ} = X1 we infer λ (X  X1 ) = X  X1 , and we take μ ∈ X  X1 arbitrary, fixed. Then p2 ≥ 1 exists, similarly with the previous reasoning, such that μ, (λ )(μ), ..., (λ )(p2 −1) (μ) are distinct and (λ )(p2 ) (μ) = μ. We denote X2 = {μ, (λ )(μ), ..., (λ )(p2 −1) (μ)}. As X2 ⊂ X  X1 , X1 and X2 are disjoint. If X = X1 ∨ X2 statement (b) is true, thus we can suppose that X = X1 ∨ X2 . From (18.4.13) and λ (X1 ∨ X2 ) = λ (X1 ) ∨ λ (X2 ) = X1 ∨ X2 we infer λ (X  (X1 ∨ X2 )) = X  (X1 ∨ X2 ) and we take some μ ∈ X  (X1 ∨ X2 ) arbitrary, fixed... We finally obtain X1 , ..., Xk that fulfill (b). (b)=⇒(a) We take λ ∈ Bn arbitrary. The hypothesis shows the existence of the partition X1 , ..., Xk of X such that ∀i ∈ {1, ..., k}, ∃pi ≥ 1 with ∀μ ∈ Xi , (18.4.11) is true. We get for any i: (λ )(Xi ) = (λ )({μ, (λ )(μ), ..., (λ )(pi −1) (μ)}) = {(λ )(μ), (λ )(2) (μ), ..., μ} = Xi . In addition we can write λ (X) = λ (X1 ∨ ... ∨ Xk ) = λ (X1 ) ∨ ... ∨ λ (Xk ) = X1 ∨ ... ∨ Xk = X. Theorem 137. Let  : Bn −→ Bn and the set H ⊂ Bn , H = ∅. The sequence of sets defined by: X0 = H, ∀k ∈ N, Xk+1 =

 λ∈Bn

λ (Xk )

Chapter 18 • Invariance

167

is descending and convergent. If the limit X = lim Xk is nonempty, then X is 5-invariant. k→∞

Proof. For any k ∈ N we obtain  Xk+1 = λ (Xk ) = (0,...,0) (Xk ) ∧ λ∈Bn

= Xk ∧





λ (Xk )

λ∈Bn {(0,...,0)}

λ (Xk ) ⊂ Xk ,

λ∈Bn {(0,...,0)}

thus the sequence (Xk ) is descending and has a limit: ∃k1 ∈ N, ∀k ≥ k1 , Xk = Xk1 that we denote by X. We suppose that X = ∅ and let λ ∈ Bn arbitrary, for which we prove the truth of λ (X) ⊂ X.

(18.4.14)

Indeed, for any ν ∈ X = Xk1 , we get λ (ν) ∈ Xk1 +1 = X, thus (18.4.14) holds. We conclude: X

(18.4.14)

⊃

λ (X) = λ (Xk1 ) ⊃



k1 +1

λ

(Xk1 ) = Xk1 +1 = X,

λk1 +1 ∈Bn

i.e. X is 5-invariant. Remark 131. The origins of the previous theorem are given by a possibility of introducing the attractors. We do not reproduce such reasonings, since the attractors will be treated later.

18.5 Morphisms Theorem 138. The functions  : Bn −→ Bn ,  : Bm −→ Bm and the set A ⊂ Bn , A = ∅ are given. (a) If (h, h ) ∈ H om(φ, ψ), then  α (μ) ⊂ A ∀μ ∈ A, ∃α ∈ n , O (18.5.1) β =⇒ ∀ν ∈ h(A), ∃β ∈ m , O (ν) ⊂ h(A), (b) if (h, h ) ∈ H om(φ, ψ), then  α (μ) ⊂ A ∃α ∈ n , ∀μ ∈ A, O β =⇒ ∃β ∈ m , ∀ν ∈ h(A), O (ν) ⊂ h(A), (c) if (h, h ) ∈ H om(φ, ψ) and h is surjective, we have  α (μ) ⊂ A ∀α ∈ n , ∀μ ∈ A, O β =⇒ ∀β ∈ m , ∀ν ∈ h(A), O (ν) ⊂ h(A),

(18.5.2)

(18.5.3)

168

Boolean Systems

(d) if (h, h ) ∈ H om(φ, ψ) and h is surjective, then ∀λ ∈ Bn , λ (A) = A =⇒ ∀ξ ∈ Bm ,  ξ (h(A)) = h(A).

(18.5.4)

Proof. (a) Let ν ∈ h(A) arbitrary, thus μ ∈ A exists with h(μ) = ν. The hypothesis shows α (μ) ⊂ A, thus h(O α (μ)) ⊂ h(A). We have the existence of the existence of α ∈ n with O   β = h (α) that fulfills β

h (α)

O (ν) = O

(h(μ))

Theorem 78, page 91

=

α h(O (μ)) ⊂ h(A).

(18.5.5)

(b) We show that β = h (α) satisfies the property ∀ν ∈ h(A), O (ν) ⊂ h(A) and we take α (μ) ⊂ A we get ν ∈ h(A) arbitrary. We have the existence of μ ∈ A with h(μ) = ν. From O α h(O (μ)) ⊂ h(A), therefore (18.5.5) is true. (c) Let β ∈ m and ν ∈ h(A) arbitrary. The hypothesis asks the existence of α ∈ n and α (μ) ⊂ A. (18.5.5) takes place. μ ∈ A with h (α) = β, h(μ) = ν and O m (d) We take ξ ∈ B , ν ∈ h(A) arbitrary, and we consider some computation function β ∈ m with β 0 = ξ . The hypothesis implies the existence of α ∈ n with h (α) = β, thus h (α)0 = ξ . We have also the existence of μ ∈ A with h(μ) = ν and, from the bijectivity of 0 0 α : A → A (abusive notation!), the existence of μ ∈ A with α (μ ) = μ. We infer β



 ξ (h(μ )) = ψ β (h(μ ), 1) = ψ h (α) (h(μ ), 1) = h(φ α (μ , 1)) = h(α (μ )) = h(μ) = ν. 0

We have just proved that the function h(A)  δ →  ξ (δ) ∈ h(A) is surjective, wherefrom  ξ (h(A)) = h(A). Remark 132. A special case of the previous theorem is given by h = θ τ , where τ ∈ Bn . This is mentioned in the next section.

18.6 Symmetry relative to translations Corollary 9. Let ,  : Bn −→ Bn , τ ∈ Bn , and A ⊂ Bn , A = ∅. (a) If (θ τ , h ) ∈ H om(φ, ψ), then  α (μ) ⊂ A ∀μ ∈ A, ∃α ∈ n , O β =⇒ ∀ν ∈ A ⊕ τ, ∃β ∈ n , O (ν) ⊂ A ⊕ τ, (b) if (θ τ , h ) ∈ H om(φ, ψ), we have  α (μ) ⊂ A ∃α ∈ n , ∀μ ∈ A, O β =⇒ ∃β ∈ n , ∀ν ∈ A ⊕ τ, O (ν) ⊂ A ⊕ τ, (c) in case that (θ τ , h ) ∈ H om(φ, ψ) with h surjective, we can write  α (μ) ⊂ A ∀α ∈ n , ∀μ ∈ A, O β =⇒ ∀β ∈ n , ∀ν ∈ A ⊕ τ, O (ν) ⊂ A ⊕ τ,

(18.6.1)

(18.6.2)

(18.6.3)

Chapter 18 • Invariance

169

(d) if (θ τ , h ) ∈ H om(φ, ψ) with h surjective, ∀λ ∈ Bn , λ (A) = A =⇒ ∀ξ ∈ Bn ,  ξ (A ⊕ τ ) = A ⊕ τ,

(18.6.4)

where we have denoted A ⊕ τ = θ τ (A). Example 108. We take  : B2 → B2 defined by ∀μ ∈ B2 , (μ1 , μ2 ) = (μ1 , μ2 ). (0, 0)



(1, 0)



- (0, 1)

- (1, 1)

We can prove that (θ (1,0) , 1B2 ) ∈ Aut (), therefore (θ (1,0) , 1 B2 ) ∈ Aut (φ). For A = {(0, 0), (0, 1)} we have A ⊕ (1, 0) = {(1, 0), (1, 1)} and the 5-invariance of A implies the 5-invariance of A ⊕ (1, 0), from the statement (d) of Corollary 9 written in the special case when  = . We can take in this example A = {(1, 0), (1, 1)} and A ⊕ (1, 0) = {(0, 0), (0, 1)}, too.

18.7 Subsystems Theorem 139. We consider the systems  : Bn+m → Bn+m ,  : Bn → Bn with ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ). (a) The set A ⊂ Bn , A = ∅ is given, and we define A ⊂ Bn+m by A = A × Bm . The following implications hold:  α (μ) ⊂ A ∀μ ∈ A, ∃α ∈ n , O β =⇒ ∀ξ ∈ A , ∃β ∈ n+m , O (ξ ) ⊂ A , 



(18.7.1)

α (μ) ⊂ A ∃α ∈ n , ∀μ ∈ A, O β  =⇒ ∃β ∈ n+m , ∀ξ ∈ A , O (ξ ) ⊂ A ,

(18.7.2)

α (μ) ⊂ A ∀α ∈ n , ∀μ ∈ A, O β =⇒ ∀β ∈ n+m , ∀ξ ∈ A , O (ξ ) ⊂ A .

(18.7.3)

(b) For A ⊂ Bn+m , we define the set A ⊂ Bn in the following way: A = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ A }.

170

Boolean Systems

We infer







∀ξ ∈ A , ∃β ∈ n+m , O (ξ ) ⊂ A α (μ) ⊂ A, =⇒ ∀μ ∈ A, ∃α ∈ n , O

(18.7.4)

∃β ∈ n+m , ∀ξ ∈ A , O (ξ ) ⊂ A α (μ) ⊂ A, =⇒ ∃α ∈ n , ∀μ ∈ A, O

(18.7.5)

∀β ∈ n+m , ∀ξ ∈ A , O (ξ ) ⊂ A α (μ) ⊂ A, =⇒ ∀α ∈ n , ∀μ ∈ A, O

(18.7.6)

β

β

β

∀υ ∈ Bn+m ,  υ (A ) = A =⇒ ∀λ ∈ Bn , λ (A) = A.

(18.7.7)

Proof. (a) (18.7.2) The hypothesis states the existence of α ∈ n such that α (μ) ⊂ A. ∀μ ∈ A, O

(18.7.8)

Let δ ∈ m arbitrary. We aim to prove that (α,δ)

∀μ ∈ A, ∀ν ∈ Bm , O

(μ, ν) ⊂ A × Bm

(18.7.9)

i.e. the conclusion of (18.7.2) is true for β = (α, δ). Let for this μ ∈ A and ν ∈ Bm arbitrary, fixed. We get from (18.7.8) the fact that ∀k ∈ N, φ α (μ, k) ∈ A. Theorem 69, page 75 shows that ∀i ∈ {1, ..., n}, ∀k ∈ N, (α,δ)

ψi

((μ, ν), k) = φiα (μ, k),

therefore (18.7.9) is true. (b) If we denote with π : Bn+m → Bn the function ∀μ ∈ Bn , ∀ν ∈ Bm , π(μ, ν) = μ, we get, similarly with Theorem 28, page 28 that (π, π) ∈ H om(, ). Furthermore, π ∈ n+m,n implies that (π,  π ) ∈ H om(ψ, φ), and π,  π are both surjective. We apply Theorem 138, page 167.

18.8 Cartesian products Theorem 140. The systems  : Bn → Bn ,  : Bm → Bm ,  =  ×  and the sets A ⊂ Bn , B ⊂ Bm , C ⊂ Bn+m are given. We denote with A ⊂ Bn , B  ⊂ Bm the sets A = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ C}, B  = {ν|ν ∈ Bm , ∃μ ∈ Bn , (μ, ν) ∈ C}.

Chapter 18 • Invariance

171

For any k ∈ {1, ..., 5}, we have: (a) if A is k-invariant relative to  and B is k-invariant relative to  then A × B is kinvariant relative to , (b) if C is k-invariant relative to , then A is k-invariant relative to  and B  is kinvariant relative to . Proof. ,  are subsystems of  and the statements of the theorem result in general from Theorem 139, page 169. (a) If k = 5, we see that the conjunction of ∀λ ∈ Bn , λ (A) = A, ∀δ ∈ Bm ,  δ (B) = B implies for arbitrary λ ∈ Bn , δ ∈ Bm that  (λ,δ) (A × B) = ( × )(λ,δ) (A × B)

Theorem 8, page 10

=

(λ ×  δ )(A × B)

= λ (A) ×  δ (B) = A × B.

18.9 Invariance and path-connectedness vs. topological transitivity Theorem 141. For ∅  X ⊂ Bn and γ ∈ n , the following implications hold: ∀μ ∈ X, ∃α ∈ n , O α (μ) = X ∀μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X and =⇒ ∀μ ∈ X, ∃α ∈ n , ∀μ ∈ X, ∃k  ∈ N, ⎩ ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ ,

(18.9.1)

∃α ∈ n , ∀μ ∈ X, O α (μ) = X ∃α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X and =⇒ ∃α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k  ∈ N, ⎩ ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ ,

(18.9.2)

∀α ∈ n , ∀μ ∈ X, O α (μ) = X ∀α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X and ∀α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k  ∈ N, ∀k ∈ {0, ..., k  }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ .

(18.9.3)

⎧ ⎨

⎧ ⎨

⇐⇒

⎧ ⎨ ⎩

Proof. If card(X) = 1, the statements are trivially fulfilled since ∀μ ∈ X, (μ) = μ. We suppose that card(X) > 1. Then topological transitivity obviously implies invariance, and we use Theorem 115, page 137.

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Boolean Systems

Remark 133. We notice that the implications were written under that form in order to point out as explicitly as possible the three properties which are involved: topological transitivity, invariance and path-connectedness. We surely have

=⇒

⎧ ⎨ ⎩

∀μ ∈ X, ∃α ∈ n , O α (μ) = X ∀μ ∈ X, ∃α ∈ n , (O α (μ) ⊂ X and ∀μ ∈ X, ∃k  ∈ N,  ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k  ) = μ )

instead of (18.9.1), for example, which is more compact, as invariance and path-connectedness have common quantified variables.

18.10 A Lyapunov-Lagrange type invariance theorem Definition 94. We use to say about a function ϕ : N → N that it is strictly increasing if ∀k ∈ N, ∀k  ∈ N, k < k  =⇒ ϕ(k) < ϕ(k  ) and decreasing if ∀k ∈ N, ∀k  ∈ N, k < k  =⇒ ϕ(k) ≥ ϕ(k  ). Definition 95. We define the Hamming distance (see Definition 8, page 4) between μ ∈ Bn and the set A ⊂ Bn , A = ∅ by d(μ, A) = min card(μ  ν). ν∈A

Theorem 142. Let  : Bn → Bn and the nonempty set A ⊂ Bn . We consider the following statements involving the functions V : Bn → N and ϕ1 , ϕ2 : N → N: ϕ1 (0) = ϕ2 (0) = 0,

(18.10.1)

ϕ1 , ϕ2 are strictly increasing,

(18.10.2)

∀μ ∈ Bn , ϕ1 (d(μ, A)) ≤ V (μ) ≤ ϕ2 (d(μ, A)),

(18.10.3)

∀μ ∈ A, ∃α ∈ n , V (φ α (μ, ·)) is decreasing,

(18.10.4)

∃α ∈ n , ∀μ ∈ A, V (φ α (μ, ·)) is decreasing,

(18.10.5)

∀μ ∈ A, ∀α ∈ n , V (φ α (μ, ·)) is decreasing,

(18.10.6)

∀μ ∈ A, ∀μ ∈ A, ∀λ ∈ Bn , μ = μ =⇒ λ (μ) = λ (μ ).

(18.10.7)

(a) The functions V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.4) hold if and only if A is 1-invariant; (b) the functions V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.5) hold if and only if A is 2-invariant;

Chapter 18 • Invariance

173

(c) the functions V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.6) hold if and only if A is 3-invariant; (d) the functions V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.6), (18.10.7) hold if and only if A is 5-invariant. Proof. (a) If. The hypothesis states that A is 1-invariant. We prove the truth of the implication for V (μ) = d(μ, A) and ϕ1 = ϕ2 = 1N . Indeed, (18.10.1), (18.10.2), (18.10.3) are true. For (18.10.4), we take an arbitrary μ ∈ A. The hypothesis states the existence of α ∈ n with the property that ∀k ∈ N, φ α (μ, k) ∈ A, in other words ∀k ∈ N, d(φ α (μ, k), A) = 0. This implies V (φ α (μ, 0)) ≥ V (φ α (μ, 1)) ≥ V (φ α (μ, 2)) ≥ ... Only if. V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.4) hold and we suppose against all reason that the 1-invariance property of A is false, i.e. μ ∈ A exists such that 

/ A. ∀α  ∈ n , ∃k  ∈ N, φ α (μ , k  ) ∈

(18.10.8)

We can write 0

(18.10.1)

=

ϕ1 (0) = ϕ1 (d(μ , A))

(18.10.3)

≤

V (μ )

(18.10.3)

≤

ϕ2 (d(μ , A)) = ϕ2 (0)

(18.10.1)

=

0,

therefore V (μ ) = 0. We get from (18.10.4) the existence of α ∈ n with 0 = V (μ ) = V (φ α (μ , 0)) ≥ V (φ α (μ , 1)) ≥ V (φ α (μ , 2)) ≥ ... ≥ 0,

(18.10.9)

wherefrom ∀k ∈ N, V (φ α (μ , k)) = 0.

(18.10.10)

The truth ∀k ∈ N of 0 ≤ ϕ1 (d(φ α (μ , k), A))

(18.10.3)

≤

V (φ α (μ , k))

(18.10.10)

=

(18.10.11)

0

proves that ∀k ∈ N, ϕ1 (d(φ α (μ , k), A)) = 0

(18.10.12)

i.e., from (18.10.1), (18.10.2), ∀k ∈ N, d(φ α (μ , k), A) = 0, in other words ∀k ∈ N, φ α (μ , k) ∈ A. We have obtained a contradiction with (18.10.8). A is 1-invariant. (c) If. We suppose that A is 3-invariant. We prove similarly with the If part of the proof of (a) that V (μ) = d(μ, A) and ϕ1 = ϕ2 = 1N fulfill (18.10.1), (18.10.2), (18.10.3), (18.10.6). Only if. The proof is similar with the Only if part of the proof of (a), with slight differences concerning the use of the quantifiers. We suppose the existence of the functions V , ϕ1 , ϕ2 such that (18.10.1), (18.10.2), (18.10.3), (18.10.6) hold and we take α ∈ n , μ ∈ A arbitrary, fixed. We have 0

(18.10.1)

=

ϕ1 (0) = ϕ1 (d(μ, A))

(18.10.3)

≤

V (μ)

(18.10.3)

≤

ϕ2 (d(μ, A)) = ϕ2 (0)

(18.10.1)

=

0,

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Boolean Systems

thus V (μ) = 0. From (18.10.6), function V satisfies 0 = V (μ) = V (φ α (μ, 0)) ≥ V (φ α (μ, 1)) ≥ V (φ α (μ, 2)) ≥ ... ≥ 0, therefore ∀k ∈ N, V (φ α (μ, k)) = 0.

(18.10.13)

But the truth ∀k ∈ N of 0 ≤ ϕ1 (d(φ α (μ, k), A))

(18.10.3)

≤

V (φ α (μ, k))

(18.10.13)

=

0

shows us that ∀k ∈ N, ϕ1 (d(φ α (μ, k), A)) = 0.

(18.10.14)

As α, μ were arbitrary, (18.10.14) is true for any α ∈ n and any μ ∈ A. We suppose now against all reason that the 3-invariance property of A is false, thus α ∈ n , μ ∈ A, k ∈ N / A, in other words d(φ α (μ, k), A) > 0. We infer from (18.10.2) that exist such that φ α (μ, k) ∈ ϕ1 (d(φ α (μ, k), A)) > 0, representing a contradiction with (18.10.14). We have obtained that A is 3-invariant. (d) If. The 5-invariance of A implies from Theorem 127, page 156 that it is 3-invariant thus, taking into account item (c), V , ϕ1 , ϕ2 exist such that (18.10.1), (18.10.2), (18.10.3), (18.10.6) take place. Let now λ ∈ Bn arbitrary. The bijectivity of the function λ : A → A means, for arbitrary μ ∈ A, μ ∈ A with μ = μ , that λ (μ) = λ (μ ), i.e. (18.10.7) is true. Only if. The existence of V , ϕ1 , ϕ2 such that (18.10.1), (18.10.2), (18.10.3), (18.10.6) be true shows due to (c) that A is 3-invariant, thus it is 4-invariant, see Theorem 127. From the 4-invariance of A, from the injectivity (18.10.7) of λ with arbitrary λ ∈ Bn , and from Lemma 1, page 156 we have that λ : A → A is surjective, therefore A is 5-invariant. Example 109. (Example 101, page 158 revisited) The functions ϕ1 , ϕ2 : N → N, ϕ1 = ϕ2 = 1N and V : B3 → N, ∀μ ∈ B3 , V (μ) = d(μ, A) satisfy (18.10.1), (18.10.2), (18.10.3), (18.10.4) for A = {(0, 0, 0), (1, 1, 0), (1, 1, 1)}. In (18.10.4) for any μ ∈ A, we choose α ∈ 3 such that ∀k ∈ N, d(φ α (μ, k), A) = 0, i.e. V (φ α (μ, k)) = 0.

18.11 Other possibilities of defining invariance Remark 134. We consider  : Bn → Bn , γ ∈ n and the set A  Bn . The requests ∀μ ∈ Bn  A, ∃α ∈ n , O α (μ) ⊂ Bn  A,

(18.11.1)

∀μ ∈ Bn  A, O γ (μ) ⊂ Bn  A,

(18.11.2)

∀μ ∈ Bn  A, ∀α ∈ n , O α (μ) ⊂ Bn  A,

(18.11.3)

Chapter 18 • Invariance

175

∀λ ∈ Bn , λ (Bn  A) ⊂ Bn  A,

(18.11.4)

∀λ ∈ Bn , λ (Bn  A) = Bn  A,

(18.11.5)

that obviously fulfill (18.11.5) =⇒ (18.11.4) ⇐⇒ (18.11.3) =⇒ (18.11.2) =⇒ (18.11.1), represent new possibilities of understanding the invariance of A. They state, instead of ‘nothing gets out’, the usual 1-invariance, ..., 5-invariance of A, that we can also think of ‘nothing gets in’. We shall not use these definitions, but the idea is present under some form in the next chapter.

19 Relatively isolated sets, isolated set The invariant, disjoint sets are called relatively isolated. They are defined in Section 19.1, and in Section 19.2 we give two examples. Some properties are mentioned in Section 19.3, for example if A1 , ..., Ap are relatively isolated, then  is dependent on the initial conditions. Section 19.4 characterizes this concept in the situation when the orbits included in invariant sets are nullclines. The isomorphisms transform the relatively isolated sets in relatively isolated sets, as shown in Section 19.5. If a family A1 , ..., Ap ⊂ Bn of sets is relatively isolated by a subsystem  : Bn → Bn of  : Bn+m → Bn+m , then the family A1 × Bm , ..., Ap × Bm is relatively isolated by , and this is discussed in Section 19.6.

19.1 Definition Remark 135. We use the properties of 1-invariance, ..., 5-invariance from (18.1.1)page 156 – (18.1.5)page 156 in the following Definitions 96–98 and we keep in mind that (18.1.3)page 156 is equivalent with (18.1.4)page 156 . Definition 96. Let  : Bn → Bn . The nonempty sets A1 , ..., Ap ⊂ Bn , p ≥ 2 are said to be relatively k-isolated (by ), k ∈ {1, ..., 5}, if ∀i ∈ {1, ..., p}, ∀j ∈ {1, ..., p}, i = j =⇒ Ai ∧ Aj = ∅ and in addition one of the following k-invariance properties ∀i ∈ {1, ..., p}, ∀μ ∈ Ai , ∃α ∈ n , O α (μ) ⊂ Ai , ∀i ∈ {1, ..., p}, ∃α ∈ n , ∀μ ∈ Ai , O α (μ) ⊂ Ai , ∀i ∈ {1, ..., p}, ∀α ∈ n , ∀μ ∈ Ai , O α (μ) ⊂ Ai , ∀i ∈ {1, ..., p}, ∀λ ∈ Bn , λ (Ai ) ⊂ Ai , ∀i ∈ {1, ..., p}, ∀λ ∈ Bn , λ (Ai ) = Ai holds. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00025-8 Copyright © 2023 Elsevier Inc. All rights reserved.

177

178

Boolean Systems

Definition 97. The set A is called k-isolated, k ∈ {1, ..., 5} if ∅  A  Bn and A, Bn  A are k-invariant. Definition 98. The k-invariant set A ⊂ Bn , k ∈ {1, ..., 5} is said to be k-indecomposable, if no partition A1 , ..., Ap , p ≥ 2 of relatively k-isolated subsets of A exists.

19.2 Examples Example 110. There is a limit situation of relatively 5-isolated sets given by the identity 1Bn : Bn −→ Bn , when any singleton {μ}, μ ∈ Bn is an isolated fixed point. In this situation the set {μ} is 5-invariant and the state functions are constant, equal with μ. Example 111. We show that in the case of the system  (1, 0) - (0, 0) Q k 6 QQ Q Q Q Q Q Q Q s Q ? -Q (1, 1)  (0, 1) the set A = {(0, 0), (1, 0)} is 1-isolated  ∀μ ∈ A, ∃α ∈ 2 , O α (μ) ⊂ A, ∀μ ∈ (B2  A), ∃α ∈ 2 , O α (μ) ⊂ (B2  A).

(19.2.1)

In this respect, we take for (0, 0), the function α = (1, 0), (1, 1), (1, 0), (1, 1), ... and for (1, 0), the function α = (1, 1), (1, 0), (1, 1), (1, 0), ... Showing that B2  A = {(0, 1), (1, 1)} is 1-invariant is similar, we take for (0, 1), the function α = (1, 1), (1, 0), (1, 1), (1, 0), ... and for (1, 1), the function α = (1, 0), (1, 1), (1, 0), (1, 1), ... We notice also that the stronger 2-isolation property  ∃α ∈ 2 , ∀μ ∈ A, O α (μ) ⊂ A, ∃α ∈ 2 , ∀μ ∈ (B2  A), O α (μ) ⊂ (B2  A) is false.

19.3 Properties Remark 136. If the sets A1 , ..., Ap ⊂ Bn , p ≥ 2 are relatively k-isolated, k ∈ {1, 3, 4, 5}, then A1 ∨ ... ∨ Ap is k-invariant, see Theorem 134, page 163. Remark 137. If A1 , ..., Ap are relatively k-isolated, k ∈ {1, 2}, it is possible that ∃i ∈ {1, ..., p}, ∃j ∈ {1, ..., p}, i = j and ∃μ ∈ Ai , ∃α ∈ n , O α (μ) ∧ Aj = ∅.

Chapter 19 • Relatively isolated sets, isolated set

179

This is not the case for k ∈ {3, ..., 5}. Theorem 143. The relatively k-isolated sets A1 , ..., Ap are given, p ≥ 2 and k ∈ {3, ..., 5}. (a) For any i ∈ {1, ..., p}, j ∈ {1, ..., p}, i = j we have ∀μ ∈ Ai , ∀α ∈ n , O α (μ) ∧ Aj = ∅; (b) if A1 ∨ ... ∨ Ap = Bn , then the dependence on the initial conditions property (13.1.5)page 120 : ∀α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , ∀k ∈ N, φ α (μ, k) = φ α (μ , k) is true. Proof. (a) We suppose without losing the generality that k = 3. For any i, j ∈ {1, ..., p}, the fact that ∀α ∈ n , ∀μ ∈ Ai , O α (μ) ⊂ Ai shows that if j = i, then for arbitrary α ∈ n , μ ∈ Ai we have O α (μ) ∧ Aj ⊂ Ai ∧ Aj = ∅. (b) We suppose that k = 3 and let α ∈ n , μ ∈ Bn arbitrary. Then i ∈ {1, ..., p} exists such that μ ∈ Ai , and we take j ∈ {1, ..., p}, j = i, μ ∈ Aj arbitrary. For any k ∈ N, we infer φ α (μ, k) ∈ Ai , φ α (μ , k) ∈ Aj and φ α (μ, k) = φ α (μ , k).

19.4 When the orbits included in invariant sets are nullclines Remark 138. Recall that for a nonempty set A ⊂ Bn , we have denoted, see Notation 31, page 100, A0i = {μ|μ ∈ A, μi = 0}, A1i = {μ|μ ∈ A, μi = 1} i ∈ {1, ..., n}. We have A0i ∧ A1i = ∅, A0i ∨ A1i = A, and, more general, for i1 , ..., ip ∈ {1, ..., n} and λ, λ ∈ Bp distinct we can write λ

λ

λ

Aλi11 ∧ ... ∧ Aipp ∧ Ai11 ∧ ... ∧ Aipp = ∅, and



λ

Aλi11 ∧ ... ∧ Aipp = A,

(19.4.1)

λ∈Bp λ

i.e. the sets Aλi11 ∧ ... ∧ Aipp , λ ∈ Bp , of which at least one is nonempty, are a partition of A.

180

Boolean Systems

Theorem 144. The function  : Bn −→ Bn and the nonempty set A ⊂ Bn are given, and we ask that A ⊂ N Ci1 ∧ ... ∧ N Cip , where i1 , ..., ip ∈ {1, ..., n} and p ≥ 1. λ

(a) ∀k ∈ {1, ..., 5}, A is k-invariant implies that the nonempty sets Aλi11 ∧ ... ∧ Aipp , λ ∈ Bp are relatively k-isolated; λ (b) ∀k ∈ {1, 3, 4, 5}, if the nonempty sets Aλi11 ∧ ... ∧ Aipp , λ ∈ Bp are relatively k-isolated then A is k-invariant. Proof. (a) We make the proof for k = 2, when ∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A

(19.4.2) λ

takes place. As A = ∅ and (19.4.1) is true, we infer that a nonempty set Aλi11 ∧ ... ∧ Aipp exists, λ

λ

λ ∈ Bp . Let Aλi11 ∧ ... ∧ Aipp = ∅ and μ ∈ Aλi11 ∧ ... ∧ Aipp , both arbitrary, in particular we obtain μi1 = λ1 , ..., μip = λp . From (19.4.2) we infer O α (μ) ⊂ A.

(19.4.3)

But Corollary 6, page 99 states the fact that ∀k ∈ N, φiα1 (μ, k) = μi1 , ..., φiαp (μ, k) = μip , thus λ

O α (μ) ⊂ (Bn )λi11 ∧ ... ∧ (Bn )ipp .

(19.4.4)

From (19.4.3) and (19.4.4) we get λ

λ

λ

O α (μ) ⊂ A ∧ (Bn )λi11 ∧ ... ∧ (Bn )ipp = (A ∧ (Bn )λi11 ) ∧ ... ∧ (A ∧ (Bn )ipp ) = Aλi11 ∧ ... ∧ Aipp . λ

This proves that Aλi11 ∧ ... ∧ Aipp is 2-invariant. The statement of the theorem is true because λ

all the nonempty sets Aλi11 ∧ ... ∧ Aipp are disjoint.  λ λ (b) Let k ∈ {1, 3, 4, 5} arbitrary. The set A = Ai11 ∧ ... ∧ Aipp is k-invariant as union of λ∈Bp

k-invariant sets. Example 112. For the system with the following state portrait,  (1, 0, 0) - (0, 0, 0) Q k 6 QQ AQ Q A QQ Q Q Q A ? s Q Q A (1, 1, 0)  - (0, 1, 0) A A A (1, 1, 1)  - (0, 1, 1) A  3  6   A    A  AU ?  ?+  - (0, 0, 1) (1, 0, 1)

Chapter 19 • Relatively isolated sets, isolated set

181

the set A = {(1, 1, 0), (0, 1, 0), (1, 1, 1), (0, 1, 1)} is 1-isolated. We have N C3 = B3  {(1, 0, 0)} and A ⊂ N C3 , thus A03 = {(1, 1, 0), (0, 1, 0)} and A13 = {(1, 1, 1), (0, 1, 1)} are relatively 1isolated subsets of A. The set B = B3  A = {(1, 0, 0), (0, 0, 0), (1, 0, 1), (0, 0, 1)} is 1-isolated too. Its subsets 0 B3 = {(1, 0, 0), (0, 0, 0)}, B31 = {(1, 0, 1), (0, 0, 1)} are relatively 1-isolated subsets of B, and the inclusion B ⊂ N C3 is false.

19.5 Isomorphisms Theorem 145. Let the functions ,  : Bn −→ Bn , the isomorphism (h, h ) ∈ I so(φ, ψ) and the nonempty sets A1 , ..., Ap ⊂ Bn , p ∈ {2, ..., 2n } which fulfill ∀i ∈ {1, ..., p}, ∀j ∈ {1, ..., p}, i = j =⇒ Ai ∧ Aj = ∅. Then ∀i ∈ {1, ..., p}, ∀j ∈ {1, ..., p}, i = j =⇒ h(Ai ) ∧ h(Aj ) = ∅,

(19.5.1)

and the following implications hold: α (μ) ⊂ A ∀i ∈ {1, ..., p}, ∀μ ∈ Ai , ∃α ∈ n , O i β =⇒ ∀i ∈ {1, ..., p}, ∀ν ∈ h(Ai ), ∃β ∈ n , O (ν) ⊂ h(Ai ), α (μ) ⊂ A ∀i ∈ {1, ..., p}, ∃α ∈ n , ∀μ ∈ Ai , O i β =⇒ ∀i ∈ {1, ..., p}, ∃β ∈ n , ∀ν ∈ h(Ai ), O (ν) ⊂ h(Ai ), α (μ) ⊂ A ∀i ∈ {1, ..., p}, ∀α ∈ n , ∀μ ∈ Ai , O i β =⇒ ∀i ∈ {1, ..., p}, ∀β ∈ n , ∀ν ∈ h(Ai ), O (ν) ⊂ h(Ai ),

∀i ∈ {1, ..., p}, ∀λ ∈ Bn , λ (Ai ) = Ai =⇒ ∀i ∈ {1, ..., p}, ∀ξ ∈ Bn ,  ξ (h(Ai )) = h(Ai ). Proof. We apply Theorem 138, page 167.

19.6 Subsystem Theorem 146. The systems  : Bn+m → Bn+m ,  : Bn → Bn are considered, with  a subsystem of  : ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ). The sets A1 , ..., Ap ⊂ Bn , A 1 , ..., A p ⊂ Bn+m are also given, where ∀i ∈ {1, ..., p}, A i = Ai × Bm .

182

Boolean Systems

Then ∀k ∈ {1, ..., 4}, if A1 , ..., Ap are relatively k-isolated by , we have that A 1 , ..., A p are relatively k-isolated by . Proof. This is a consequence of Theorem 139 (a), page 169, if we take in consideration that ∀i ∈ {1, ..., p}, ∀j ∈ {1, ..., p}, i = j =⇒ Ai ∧ Aj = ∅ implies ∀i ∈ {1, ..., p}, ∀j ∈ {1, ..., p}, i = j =⇒ A i ∧ A j = ∅. Remark 139. A special case of the previous theorem is: if A is k-isolated by a subsystem  of , then the set A = A × Bm is k-isolated by , for k ∈ {1, ..., 4}. Remark 140. The properties expressed by Theorem 146 and Remark 139 may be stated for Cartesian products also, for example: given the systems  : Bn → Bn and ϒ : Bm → Bm , if A ⊂ Bn is k-isolated by , then A × Bm is k-isolated by  × ϒ, for any k ∈ {1, ..., 4}.

20 Maximal invariant subset Let  : Bn → Bn . A set is the maximal invariant subset of X ⊂ Bn if it is invariant and it includes any other invariant subset of X. The maximal invariant subsets of X, denoted 1 , ..., X 5 , are defined in Section 20.1, and several examples are given in Section 20.2. Xmax max 1 3 4 The fact that Xmax = {μ|μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X}, Xmax = Xmax = {μ|μ ∈ X, ∀α ∈ α 5 n λ n , O (μ) ⊂ X}, Xmax = {μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ B ,  (A) = A} and other important properties of these subsets are presented in Section 20.3. 1 , ..., X 5 In Section 20.4 we analyze the situation of Xmax max when the orbit is a nullcline. Section 20.5 points out that the isomorphisms of flows bring maximal invariant subsets in maximal invariant subsets. k k In Section 20.6 we present the relation between the sets Xmax, and Ymax, , k ∈ {1, ..., 4} n+m n+m when  is a subsystem of  : B →B ; in Section 20.7, given the systems  : Bn → k k Bn ,  : Bm → Bm , we present the relation between Xmax, , Ymax, and (X × Y )kmax,× , k ∈ {1, ..., 5}.

20.1 Definition Definition 99. Let  : Bn → Bn and the arbitrary sets A, X ⊂ Bn , with A ⊂ X, A = ∅. The set A is called the maximal invariant subset of X if one of  ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A, (20.1.1) ∀Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y ) =⇒ Y ⊂ A, 



∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, ∀Y, (∅  Y ⊂ X and ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ Y ⊂ A,

(20.1.2)

∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, ∀Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ Y ⊂ A,

(20.1.3)





∀λ ∈ Bn , λ (A) ⊂ A, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) ⊂ Y ) =⇒ Y ⊂ A,

(20.1.4)

∀λ ∈ Bn , λ (A) = A, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ Y ⊂ A

(20.1.5)

holds. We shall refer to the maximal 1-invariant, ..., the maximal 5-invariant subset of X. 1 , ..., X 5 . Notation 33. The maximal invariant subsets of X are denoted with Xmax max Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00026-X Copyright © 2023 Elsevier Inc. All rights reserved.

183

184

Boolean Systems

Remark 141. Definition 99 states nothing about the invariance of X (see Remark 127, page 160). 1 , ..., X 5 Theorem 147. The set X ⊂ Bn is given, X = ∅. If Xmax max exist, then 5 4 3 2 1 ⊂ Xmax = Xmax ⊂ Xmax ⊂ Xmax . Xmax 5 is 5-invariant, it is also 4-invariant from Theorem 127, page 156, thus from Proof. As Xmax 4 5 4 . The inclusions X 4 3 2 the way that Xmax was defined we have Xmax ⊂ Xmax max ⊂ Xmax ⊂ Xmax ⊂ 1 Xmax follow similarly. 3 4 , we Theorem 127 implies also that Xmax is 4-invariant thus, from the definition of Xmax 3 4 have Xmax ⊂ Xmax .

20.2 Examples Example 113. For the following system (0, 0, 1) 

(1, 1, 0) 

(0, 0, 0) Q Q Q Q Q ? s Q (1, 0, 0) (1, 0, 1)

(0, 1, 0)

? (1, 1, 1)

(0, 1, 1)

the set X = {(0, 0, 0), (1, 0, 1), (1, 1, 1)} is 1-invariant, and it is not 2-invariant. This is remarked from the fact that the requirements 

λ (0, 0, 0) ∈ {(0, 0, 0), (1, 0, 1)}, λ (1, 0, 1) ∈ {(1, 0, 1), (1, 1, 1)}

give λ ∈ {(0, 0, 0), (0, 1, 0), (1, 0, 1), (1, 1, 1)} ∧ {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)} = {(0, 0, 0), (0, 1, 0)}. Any α : N → B3 which is candidate to fulfill the 2-invariance ∀μ ∈ X, O α (μ) ⊂ X, fulfills also ∀k ∈ N, α k ∈ {(0, 0, 0), (0, 1, 0)} and O α (0, 0, 0) = {(0, 0, 0)}. But α is not progressive in these conditions, contradiction. We have obtained that the choice of α ∈ 3 such that O α (μ) ⊂ X depends on μ, X is 1 k 1-invariant. In this case Xmax = X, and ∀k ∈ {2, ..., 5}, Xmax = {(1, 1, 1)}.

Chapter 20 • Maximal invariant subset

185

Example 114. Let the system - (1, 0) (0, 1)  (0, 0) Q k 3  6 Q  Q  Q  Q ?  Q  (1, 1) The set X = {(0, 0), (1, 1), (0, 1)} is 2-invariant and its 2-invariant subsets are: {(0, 0), (1, 1)}, {(0, 0), (0, 1)}, {(0, 1), (1, 1)}, {(0, 1)} and {(0, 0), (1, 1), (0, 1)}. We can take α = (1, 1), (1, 1), 2 1 3 4 5 = X. Obviously Xmax = X and, in addition, Xmax = Xmax = Xmax = (1, 1), ... and we have Xmax {(0, 1)}. Example 115. The system (0, 0)  3     ?  - (0, 1) (1, 1) (1, 0) 6

is given and we take X = {(1, 0), (1, 1), (0, 1)}, which is 2-invariant. For α = (0, 1), (1, 0), (0, 1), 2 1 3 4 5 (1, 0), ... we get Xmax = X = Xmax and on the other hand Xmax = Xmax = Xmax = {(0, 1)}. Example 116. In the case of the system (0, 0) 6 (1, 0) 

- (0, 1)

? (1, 1)

the set X = B2 is 3-invariant, in the circumstances that there is exactly one 3-invariant 1 2 3 4 , while = Xmax = Xmax = Xmax subset of B2 , namely B2 itself. And the conclusion is X = Xmax 5 Xmax does not exist. Example 117. The following system (0, 0) 6

(1, 0) 6

? (0, 1)

? (1, 1)

fulfills the property that X = B2 is 5-invariant, and it has three 5-invariant subsets: {(0, 0), (0, 1)}, {(1, 0), (1, 1)} and B2 ; we have that B2 = (B2 )5max . On the other hand X = 1 5 = ... = Xmax = {(0, 0), (0, 1)}. {(0, 0), (0, 1), (1, 0)} is not invariant, and it fulfills Xmax

186

Boolean Systems

20.3 Main properties Theorem 148. For any k ∈ {1, ..., 5}, X is k-invariant if and only if X = the maximal invariant subset of itself. Proof. Let k ∈ {1, ..., 5} arbitrary. =⇒ X is k-invariant and X is the greatest subset of itself, thus x is the greatest kinvariant subset of itself. ⇐= X is the greatest k-invariant subset of itself implies that X is k-invariant. Theorem 149. For any k ∈ {1, ..., 5}, we have k k = (Xmax )kmax . Xmax

(20.3.1)

k is k-invariant, thus (20.3.1) is true from TheoProof. We fix k ∈ {1, ..., 5} arbitrary. Xmax rem 148.

Theorem 150. For any k ∈ {1, 3, 4, 5}, the following property is true: X ⊂ Bn contains a nonempty set A ⊂ X which is k-invariant if and only if the maximal k-invariant subset of X exists. Proof. We fix k ∈ {1, 3, 4, 5} arbitrary. =⇒ The hypothesis states the existence of a k-invariant subset of X and let A1 , ..., Ap be the set of these k-invariant subsets. Then A1 ∨ ... ∨ Ap is a subset of X, which is k-invariant, from Theorem 134, page 163. If A is a k-invariant subset of X, then i ∈ {1, ..., p} exists with k A = Ai , thus A ⊂ A1 ∨ ... ∨ Ap . We have Xmax = A1 ∨ ... ∨ Ap . k , which is k-invariant. ⇐= The hypothesis states the existence of Xmax Remark 142. If A1 , A2 are 2-invariant, we can not establish if A1 ∨ A2 is 2-invariant or not. This missing property is a gap in Theorem 134, page 163, which propagates to Theorem 150. Theorem 151. We consider the nonempty set X ⊂ Bn and we denote X = {μ|μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X}, X = {μ|μ ∈ X, ∀α ∈ n , O α (μ) ⊂ X}, X = {μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) = A}. (a) If X = ∅, then the set X is 1-invariant: ∀μ ∈ X , ∃β ∈ n , O β (μ) ⊂ X .

(20.3.2)

(b) If X = ∅, then the set X is 3-invariant: ∀α ∈ n , ∀μ ∈ X , O α (μ) ⊂ X .

(20.3.3)

Chapter 20 • Maximal invariant subset

187

(c) In case that X = ∅, then X is 5-invariant: ∀ν ∈ Bn , ν (X ) = X .

(20.3.4)

Proof. (a) We suppose that X = ∅ and we must show ∀μ ∈ X , ∃β ∈ n , ∀μ ∈ O β (μ), ∃γ ∈ n , O γ (μ ) ⊂ X.

(20.3.5)

Let an arbitrary μ ∈ X . We have from the definition of X the existence of α ∈ n with O α (μ) ⊂ X.

(20.3.6)

We take in (20.3.5) β = α and an arbitrary μ ∈ O α (μ), meaning the existence of k ∈ N with μ = φ α (μ, k ). We define, in order to show the truth of (20.3.5)

γ = σ k (α), and we have: O γ (μ ) = {φ γ (μ , k)|k ∈ N} = {φ σ

k (α)

(φ α (μ, k ), k)|k ∈ N}

= {φ α (μ, k + k )|k ∈ N} = {φ α (μ, k)|k ≥ k } ⊂ {φ α (μ, k)|k ∈ N} = O α (μ)

(20.3.6)

⊂

X.

(20.3.5) is proved, thus (20.3.2) is also true. (b) We ask that X = ∅ and we must prove ∀α ∈ n , ∀μ ∈ X , ∀μ ∈ O α (μ), ∀γ ∈ n , O γ (μ ) ⊂ X.

(20.3.7)

We take α ∈ n , μ ∈ X , μ ∈ O α (μ), and γ ∈ n arbitrary, fixed. From the definition of X we get ∀β ∈ n , O β (μ) ⊂ X.

(20.3.8)

We have the existence of k ∈ N with μ = φ α (μ, k ) and we define β ∈ n in the following way:  k α , if k ∈ {0, ..., k − 1}, k β = γ k−k , if k ≥ k . We infer: O γ (μ ) = {φ γ (μ , k)|k ∈ N} = {φ γ (φ α (μ, k ), k)|k ∈ N} = {φ σ

k (β)

(φ β (μ, k ), k)|k ∈ N} = {φ β (μ, k + k )|k ∈ N}

= {φ β (μ, k)|k ≥ k } ⊂ {φ β (μ, k)|k ∈ N} = O β (μ) Statement (20.3.7) is proved and (20.3.3) is proved as well.

(20.3.8)

⊂

X.

188

Boolean Systems

(c) We suppose that X = ∅. We prove first that X is 4-invariant, ∀ν ∈ Bn , ν (X ) ⊂ X , and we take ν ∈ Bn , μ ∈ X arbitrary. From the definition of X we get the existence of A ⊂ X such that μ ∈ A and ∀λ ∈ Bn , λ (A) = A. We infer ν (μ) ∈ ν (A) = A and ∀λ ∈ Bn , λ (A) = A, therefore ν (μ) ∈ X . We prove now that X is 5-invariant, i.e. the truth of (20.3.4). We take ν ∈ Bn , μ ∈ X and μ ∈ X arbitrary, with μ = μ and we suppose against all reason that ν (μ) = ν (μ ). We know that ∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) = A,

(20.3.9)

∃B ⊂ X, μ ∈ B and ∀λ ∈ Bn , λ (B) = B.

(20.3.10)

From (20.3.9), (20.3.10) and Theorem 134, page 163, we can write μ, μ ∈ A ∨ B and ∀λ ∈ Bn , λ (A ∨ B) = A ∨ B, contradiction (the restriction of ν to A ∨ B is bijective). (20.3.4) is true. Theorem 152. For X ⊂ Bn , X , X , X like in the hypothesis of Theorem 151, the following statements are true: 1 , (a) if X = ∅, then X is the maximal 1-invariant subset of X, i.e. X = Xmax 3 , (b) if X = ∅, then X is the maximal 3-invariant subset of X, i.e. X = Xmax 5 (c) in case that X = ∅, then X = Xmax . Proof. (a) The first statement of (20.1.1)page 183 , i.e. the invariance of X was proved at Theorem 151. We consider now an arbitrary A ⊂ X, A = ∅ fulfilling the invariance property ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A. For any μ ∈ A, we get μ ∈ X and since α ∈ n exists with O α (μ) ⊂ A ⊂ X, we infer μ ∈ X . We have just proved that ∀A, (∅  A ⊂ X and ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A) =⇒ A ⊂ X , i.e. the second statement of (20.1.1)page 183 takes place too. (b) The first statement of (20.1.3)page 183 , consisting in the invariance of X , was shown at Theorem 151. Let A an arbitrary set that fulfills A = ∅, A ⊂ X and also the invariance ∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A and we fix α ∈ n , μ ∈ A arbitrary themselves. From the fact that μ ∈ X and, on the other hand, that O α (μ) ⊂ A ⊂ X, we get μ ∈ X . We have just inferred the truth of the implication ∀A, (∅  A ⊂ X and ∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A) =⇒ A ⊂ X . The second statement of (20.1.3)page 183 holds.

Chapter 20 • Maximal invariant subset

189

(c) The first property of (20.1.5)page 183 consisting in the 5-invariance of X was proved at Theorem 151. In order to prove the second property, we consider an arbitrary set A with the properties that ∅  A ⊂ X and ∀λ ∈ Bn , λ (A) = A, and let μ ∈ A. We obviously infer that μ ∈ X , thus A ⊂ X and the second statement of (20.1.5)page 183 is true too. Notation 34. We denote for any A ⊂ Bn , A = ∅:  O + (μ). O + (A) = μ∈A k Corollary 10. For any nonempty X ⊂ Bn , if Xmax exist, k ∈ {1, 3, 4, 5}, then 1 (a) Xmax is equal with

{μ|μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X}, 3 (b) Xmax is equal with

{μ|μ ∈ X, ∀α ∈ n , O α (μ) ⊂ X}, 4 is equal with any of (c) Xmax 3 Xmax ,

{μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) ⊂ A}, {μ|∃A ⊂ X, μ ∈ A and O + (A) ⊂ A}, 5 (d) Xmax is equal with

{μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) = A}, and we have 5 ⊂ {μ|∃A ⊂ X, μ ∈ A and O + (A) = A}. Xmax 3 4 Proof. (c) Xmax = Xmax was stated in Theorem 147, page 184. We prove

{μ|μ ∈ X, ∀α ∈ n , O α (μ) ⊂ X} ⊂ {μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) ⊂ A} and let μ ∈ X arbitrary, with the property that ∀α ∈ n , O α (μ) ⊂ X. The set A = O + (μ) sat O α (μ) ⊂ X and μ ∈ O + (μ). The fact that ∀λ ∈ Bn , λ (O + (μ)) ⊂ O + (μ) isfies O + (μ) = α∈n

was proved at Theorem 128, page 160. We prove {μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) ⊂ A} ⊂ {μ|∃A ⊂ X, μ ∈ A and O + (A) ⊂ A}.

190

Boolean Systems

We take μ arbitrary, for which the hypothesis states the existence of A ⊂ X with μ ∈ A and ∀λ ∈ Bn , λ (A) ⊂ A. We infer  λ (A) ⊂ A, λ∈Bn

 



(λ ◦ λ )(A) ⊂

λ∈Bn λ ∈Bn



λ (A) ⊂ A,

λ∈Bn

... wherefrom O + (A) = A ∨



λ (A) ∨

 



(λ ◦ λ )(A) ∨ ... ⊂ A ∨ A ∨ A ∨ ... = A.

λ∈Bn λ ∈Bn

λ∈Bn

We prove now {μ|∃A ⊂ X, μ ∈ A and O + (A) ⊂ A} ⊂ {μ|μ ∈ X, ∀α ∈ n , O α (μ) ⊂ X}. Let μ arbitrary, having the property that the set A ⊂ X exists with μ ∈ A and O + (A) ⊂ A. In this case for any α ∈ n we have O α (μ) ⊂ O + (μ) ⊂ O + (A) ⊂ A ⊂ X. (d) We prove {μ|∃A ⊂ X, μ ∈ A and ∀λ ∈ Bn , λ (A) = A} ⊂ {μ|∃A ⊂ X, μ ∈ A and O + (A) = A}. We take μ arbitrary, with the property that a set A ⊂ X exists such that μ ∈ A and ∀λ ∈ Bn , λ (A) = A. We get:    λ (A) ∨ (λ ◦ λ )(A) ∨ ... = A ∨ A ∨ A ∨ ... = A. O + (A) = A ∨ λ∈Bn λ ∈Bn

λ∈Bn

Remark 143. As far as A ⊂ O + (A), the statements O + (A) ⊂ A and O + (A) = A that have previously occurred are equivalent, thus we rediscover at Corollary 10 (d) the statement 5 4 3 Xmax ⊂ Xmax = Xmax of Theorem 147, page 184. 2 . Problem 11. To be studied Xmax

20.4 Maximality vs. nullclines Theorem 153. Let X ⊂ Bn nonempty and we ask that X ⊂ N Ci1 ∧ ... ∧ N Cip , i1 , ..., ip ∈ k k {1, ..., n}, p ≥ 1. Then ∀k ∈ {1, ..., 5}, if Xmax exists, we infer for any μ ∈ Xmax that μi

μi

μi

μi

k k (Xmax )i1 1 ∧ ... ∧ (Xmax )ip p = (Xi1 1 ∧ ... ∧ Xip p )kmax .

(20.4.1)

Chapter 20 • Maximal invariant subset

191

k exProof. We consider k ∈ {1, ..., 5}, the k-invariant subset A ⊂ X (the possibility A = Xmax ists), and μ ∈ A, arbitrary all of them. We intersect the members of k A ⊂ Xmax ⊂X μi

μi

with (Bn )i1 1 ∧ ... ∧ (Bn )ip p and we get in succession μi

μi

μi

μi

k A ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p ⊂ Xmax ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p μi

μi

⊂ X ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p , μi

μi

μi

μi

k k (A ∧ (Bn )i1 1 ) ∧ ... ∧ (A ∧ (Bn )ip p ) ⊂ (Xmax ∧ (Bn )i1 1 ) ∧ ... ∧ (Xmax ∧ (Bn )ip p ) μi

μi

⊂ (X ∧ (Bn )i1 1 ) ∧ ... ∧ (X ∧ (Bn )ip p ), μi

μi

μi

μi

μi

μi

k k )i1 1 ∧ ... ∧ (Xmax )ip p ⊂ Xi1 1 ∧ ... ∧ Xip p . Ai1 1 ∧ ... ∧ Aip p ⊂ (Xmax k implies, from Theorem 144, page 180, that the We note that the k-invariance of A, Xmax μip μi1 μi1 k k )μip , which are nonempty, are also k-invariant. sets Ai1 ∧ ... ∧ Aip , (Xmax )i1 ∧ ... ∧ (Xmax ip μi

μi

μi

μi

k ) 1 ∧ ... ∧ (X k ) p is the greatest k-invariant subset of X 1 ∧ ... ∧ X p . Moreover, (Xmax max ip i1 i1 ip (20.4.1) is true.

20.5 Isomorphisms Theorem 154. The systems ,  : Bn → Bn are given, together with (h, h ) ∈ I so(φ, ψ). Let X ⊂ Bn nonempty. We have ∀k ∈ {1, ..., 5}, k h(Xmax, ) = h(X)kmax, .

(20.5.1)

2 Proof. We make the proof for k = 2. The hypothesis states for A = Xmax, the truth of 2 (20.1.2)page 183 , in particular Xmax, ⊂ X is 2-invariant, 2 α 2 ∃α ∈ n , ∀μ ∈ Xmax, , O (μ) ⊂ Xmax, . 2 ) ⊂ h(X): By Theorem 138, page 167, we infer the 2-invariance of h(Xmax, β

2 2 ), O (ν) ⊂ h(Xmax, ). ∃β ∈ n , ∀ν ∈ h(Xmax,

We take an arbitrary set Y ⊂ h(X), which is nonempty and 2-invariant: ∃β ∈ n , ∀ν ∈ Y , O (ν) ⊂ Y . β

The set h−1 (Y ) ⊂ X is nonempty then, and 2-invariant: α ∅  h−1 (Y ) ⊂ X and ∃α ∈ n , ∀μ ∈ h−1 (Y ), O (μ) ⊂ h−1 (Y ).

(20.5.2)

192

Boolean Systems

Because the second statement of (20.1.2)page 183 α 2 ∀Y, (∅  Y ⊂ X and ∃α ∈ n , ∀μ ∈ Y, O (μ) ⊂ Y ) =⇒ Y ⊂ Xmax,

has the hypothesis fulfilled as (20.5.2), we infer 2 h−1 (Y ) ⊂ Xmax, .

It has resulted that 2 Y ⊂ h(Xmax, ), 2 i.e. h(Xmax, ) is maximal. (20.5.1) is proved for k = 2.

Corollary 11. We have ∀k ∈ {1, ..., 5}, k

k Xmax, = Xmax,∗ .

Proof. This is a special case of Theorem 154, when  = ∗ and h = θ (1,...,1) (the complement).

20.6 Subsystems Theorem 155. The systems  : Bn → Bn ,  : Bn+m → Bn+m are given, and we suppose that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ). Let X ⊂ Bn nonempty and Y = X × Bm . Then ∀k ∈ {1, ..., 4}, k k Ymax, = Xmax, × Bm .

Proof. For k = 3, the hypothesis states that 3 α 3 ∀α ∈ n , ∀μ ∈ Xmax, , O (μ) ⊂ Xmax, , α 3 ∀Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O (μ) ⊂ Y ) =⇒ Y ⊂ Xmax , ,

(20.6.1) (20.6.2)

and we must show (α,β)

3 ∀α ∈ n , ∀β ∈ m , ∀μ ∈ Xmax, , ∀ν ∈ Bm , O

3 (μ, ν) ⊂ Xmax, × Bm ,

∀Y , (∅  Y ⊂ X × Bm (α,β) and ∀α ∈ n , ∀β ∈ m , ∀(μ, ν) ∈ Y , O (μ, ν) ⊂ Y ) 3 m =⇒ Y ⊂ Xmax , × B .

(20.6.3)

(20.6.4)

Chapter 20 • Maximal invariant subset

193

Statement (20.6.3) results from Theorem 139 (a), page 169. We suppose now, against all reason, that (20.6.4) is false ∃Y , (∅  Y ⊂ X × Bm and ∀α ∈ n , ∀β ∈ m , ∀(μ, ν) ∈ Y , O 3 m and Y ⊂ Xmax , × B ,

(α,β)

(μ, ν) ⊂ Y )

and we define Y = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ Y }. The 3-invariance of Y means, due to Theorem 139 (b), page 169, that Y is 3-invariant, and 3 3 m Y ⊂ Xmax , × B implies that Y ⊂ Xmax , . We have obtained a contradiction with (20.6.2).

20.7 Cartesian products Theorem 156. We consider the systems  : Bn → Bn ,  : Bm → Bm ,  =  ×  and the nonempty sets X ⊂ Bn , Y ⊂ Bm . Then ∀k ∈ {1, ..., 5}, k k (X × Y )kmax, = Xmax, × Ymax, .

Proof. We take k = 2. We have from the hypothesis 2 α 2 , O (μ) ⊂ Xmax, , ∃α ∈ n , ∀μ ∈ Xmax, α 2 ∀A, (∅  A ⊂ X and ∃α ∈ n , ∀μ ∈ A, O (μ) ⊂ A) =⇒ A ⊂ Xmax , , β

2 2 ∃β ∈ m , ∀ν ∈ Ymax, , O (ν) ⊂ Ymax, , β

2 ∀B, (∅  B ⊂ Y and ∃β ∈ m , ∀ν ∈ B, O (ν) ⊂ B) =⇒ B ⊂ Ymax , .

(20.7.1) (20.7.2) (20.7.3) (20.7.4)

2 Theorem 140, page 170 shows that (20.7.1) and (20.7.3) imply the 2-invariance of Xmax, × 2 Ymax, relative to . Let now C ⊂ X × Y nonempty, 2-invariant (α,β)

∃(α, β) ∈ n+m , ∀(μ, ν) ∈ C, O

(μ, ν) ⊂ C

2 2 and we suppose against all reason that C ⊂ Xmax, × Ymax, . This shows the existence of 2 2 2 / Xmax, × Ymax, , for example μ ∈ / Xmax, . From Theorem 140, page (μ , ν ) ∈ C, (μ , ν ) ∈ 170 we infer that the set

A = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ C} 2 is 2-invariant relative to . We have obtained the contradiction A ⊂ Xmax, , μ ∈ A.

21 Minimal invariant superset 1 , ..., Given  : Bn → Bn , the minimal invariant supersets of X ⊂ Bn are denoted with Xmin 5 Xmin . Their definitions and some examples are the topics of Sections 21.1 and 21.2. Several properties of these sets are presented in Section 21.3. In Section 21.4 we study the situation when X ⊂ N Ci1 ∧ ... ∧ N Cip and the orbits are nullclines. The purpose of Section 21.5 is that of proving that the isomorphisms transform minimal invariant supersets in minimal invariant supersets. In Section 21.6 we point out the way that the minimal invariant superset of a subsystem is related to the minimal invariant superset of the whole system. 1 × In Section 21.7 we show that we cannot prove the properties (X × Y )1min,× = Xmin, 5 5 5 1 Ymin, , ..., (X × Y )min,× = Xmin, × Ymin, .

21.1 Definition Definition 100. We consider the function  : Bn → Bn and the sets ∅  X ⊂ A ⊂ Bn . If either of the properties  ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A, (21.1.1) n ∀Y, (X ⊂ Y ⊂ B and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y ) =⇒ A ⊂ Y, 

∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, and ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ A ⊂ Y,

(21.1.2)

∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, ∀Y, (X ⊂ Y ⊂ Bn and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ A ⊂ Y,

(21.1.3)

∀Y, (X ⊂ Y 

⊂ Bn



∀λ ∈ Bn , λ (A) ⊂ A, and ∀λ ∈ Bn , λ (Y ) ⊂ Y ) =⇒ A ⊂ Y,

(21.1.4)

∀λ ∈ Bn , λ (A) = A, and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ A ⊂ Y

(21.1.5)

∀Y, (X ⊂ Y

⊂ Bn

∀Y, (X ⊂ Y

⊂ Bn



is satisfied, then A is called the minimal 1-invariant, ..., minimal 5-invariant superset of X. 1 , ..., X 5 . Notation 35. The minimal invariant supersets of X are denoted with Xmin min Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00027-1 Copyright © 2023 Elsevier Inc. All rights reserved.

195

196

Boolean Systems

Remark 144. Definition 100 states nothing about the invariance of X. 1 , ..., X 5 exist. Then Theorem 157. The set X ⊂ Bn , X = ∅ is given and we suppose that Xmin min 1 2 3 4 5 Xmin ⊂ Xmin ⊂ Xmin = Xmin = Xmin . 2 is 2-invariant, thus it is 1-invariant, from Theorem 127, page 156. From Proof. The set Xmin 1 we infer X 1 ⊂ X 2 . The inclusions X 2 ⊂ X 3 ⊂ X 4 ⊂ X 5 result the definition of Xmin min min min min min min similarly. 3 , which is 3-invariant, is also 4-invariant, from Theorem 127. The definition of But Xmin 4 gives X 4 ⊂ X 3 . We obtained that X 3 = X 4 . Xmin min min min min 4 ⊂ X 5 is 5-invariant from Theorem 135, On the other hand the 4-invariant subset Xmin min 5 gives X 5 ⊂ X 4 . It has resulted that X 4 = X 5 . page 165 thus the definition of Xmin min min min min

21.2 Examples Example 118. We consider the system (0, 0)  Q  Q  Q  Q  Q ?  + s Q (0, 1) (1, 0) (1, 1) k sets, if any, must be different from X. We The set X = {(0, 0)} is not invariant, and the Xmin 2 does see that the sets {(0, 0), (1, 0)}, {(0, 0), (1, 1)}, {(0, 0), (0, 1)} are all 2-invariant, thus Xmin 3 2 not exist. We get however Xmin = B .

Example 119. In the next state portrait (1, 0, 1) 

(0, 1, 1) 

- (1, 0, 0)      ? ?  + - (0, 1, 0) (0, 0, 0) (0, 0, 1) 6

- (1, 1, 0)

(1, 1, 1)

we denote X = {(0, 0, 1), (1, 0, 0)}. X is not invariant, and some invariant sets containing X are: {(0, 0, 1), (1, 0, 0), (1, 1, 0)} is 1-invariant, {(0, 0, 1), (1, 0, 0), (0, 1, 0)} is 2-invariant, as α = (1, 1, 1), (1, 1, 1), (1, 1, 1), ... fulfills this property, {(0, 0, 1), (1, 0, 0), (0, 0, 0)} is 1-invariant. We 1 does not exist and X 2 = {(0, 0, 1), (1, 0, 0), (0, 1, 0)}. We infer also that X 3 = get that Xmin min min 3 B  {(1, 1, 1)}.

Chapter 21 • Minimal invariant superset

197

Example 120. We give now the example of the system (0, 0) 

- (0, 1)

(1, 0)

- (1, 1)

5 = {(0, 0), (0, 1)} = X 3 = The sets X = {(0, 0)}, Y = {(0, 0), (1, 1)} are not invariant and Xmin min 5 3 4 4 Xmin , Ymin = {(0, 0), (0, 1), (1, 1)} = Ymin = Ymin .

21.3 Properties Theorem 158. The system  : Bn → Bn is given, together with X ⊂ Bn nonempty. For any k . k ∈ {1, ..., 5}, X is k-invariant if and only if X = Xmin Proof. We fix an arbitrary k ∈ {1, ..., 5}. =⇒ We suppose that X is k-invariant. For any set Y that fulfills X ⊂ Y ⊂ Bn and kk . invariance we have X ⊂ Y , thus X = Xmin ⇐= Obvious. Theorem 159. For , X like previously and k ∈ {1, ..., 5} arbitrary, we can write k k Xmin = (Xmin )kmin .

(21.3.1)

k Proof. As Xmin is k-invariant, we apply the only if part of the previous theorem. (21.3.1) holds.

Theorem 160. For  : Bn → Bn and X ⊂ Bn , X = ∅, the following equivalence takes place k Y k-invariant exists, X ⊂ Y ⊂ Bn ⇐⇒ Xmin exists

where k ∈ {3, 4, 5}. Proof. We fix k ∈ {3, 4, 5} arbitrary. =⇒ We suppose that a k-invariant Y exists, X ⊂ Y ⊂ Bn , and we denote with Y1 , ..., Yp these sets. The intersection Y1 ∧ ... ∧ Yp is nonempty, as far as X ⊂ Y1 ∧ ... ∧ Yp , and Theorem 134, page 163, implies that Y1 ∧ ... ∧ Yp is k-invariant. If Y is an arbitrary k-invariant set satisfying X ⊂ Y ⊂ Bn , then i ∈ {1, ..., p} exists with Y = Yi . We obtain Yi ⊃ Y1 ∧ ... ∧ Yp , k . therefore Y1 ∧ ... ∧ Yp = Xmin k k ⊂ Bn . ⇐= If Xmin exists, it satisfies the property that it is k-invariant and X ⊂ Xmin Remark 145. We can compare now Theorem 160 with Theorem 150, page 186. Like in the case of that theorem, there is a gap in Theorem 134 consisting in the fact that the implication (A1 , A2 are k-invariant) and (A1 ∧ A2 = ∅) =⇒ A1 ∧ A2 is k-invariant

198

Boolean Systems

is not necessarily true for k ∈ {1, 2}. The gap propagates. k exists, Theorem 161. We consider  : Bn → Bn and X ⊂ Bn , X = ∅. Then ∀k ∈ {3, 4, 5}, if Xmin k + we have Xmin = O (X).

Proof. For any μ ∈ X, the fact that O + (μ) is 4-invariant was proved at Theorem 128, page 160, therefore the set  O + (X) = O + (μ) μ∈X

is 4-invariant from Theorem 134, page 163. We prove the minimality ∀Y, (X ⊂ Y ⊂ Bn and ∀λ ∈ Bn , λ (Y ) ⊂ Y ) =⇒ O + (X) ⊂ Y now. For this, let Y an arbitrary 4-invariant set such that X ⊂ Y ⊂ Bn . We suppose against all reason that O + (X) ⊂ Y , meaning that O + (X)  Y = ∅. Let ν ∈ O + (X)  Y arbitrary, where + O (X) = O α (μ), therefore α ∈ n , μ ∈ X and k ∈ N exist with ν = φ α (μ, k ). But α∈n μ∈X

μ ∈ Y obviously, and this implies that k ≥ 1. In fact, a time instant k ≥ 0 exists, k < k , with the property that ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ Y , and φ α (μ, k + 1) ∈ / Y . The fact that φ α (μ, k + k

/ Y represents a contradiction with the 4-invariance of Y . O + (X) ⊂ Y 1) = α (φ α (μ, k )) ∈ is proved.

21.4 Minimality vs. nullclines Theorem 162. We consider the system  : Bn → Bn , X ⊂ Bn nonempty, and we suppose that k exists, we X ⊂ N Ci1 ∧ ... ∧ N Cip , where i1 , ..., ip ∈ {1, ..., n}, p ≥ 1. For any k ∈ {1, ..., 5}, if Xmin infer ∀μ ∈ X that μi

μi

μi

μi

k k (Xmin )i1 1 ∧ ... ∧ (Xmin )ip p = (Xi1 1 ∧ ... ∧ Xip p )kmin .

(21.4.1)

Proof. The proof follows the same steps as the proof of Theorem 153, page 190. We take k ∈ k exists), and μ ∈ X, all arbitrary. {1, ..., 5}, the k-invariant set A ⊃ X (the possibility A = Xmin In the double inclusion k X ⊂ Xmin ⊂A μi

μi

we intersect the members with (Bn )i1 1 ∧ ... ∧ (Bn )ip p . We infer successively μi

μi

μi

μi

k ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p X ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p ⊂ Xmin μi

μi

⊂ A ∧ (Bn )i1 1 ∧ ... ∧ (Bn )ip p , μi

μi

μi

μi

k k (X ∧ (Bn )i1 1 ) ∧ ... ∧ (X ∧ (Bn )ip p ) ⊂ (Xmin ∧ (Bn )i1 1 ) ∧ ... ∧ (Xmin ∧ (Bn )ip p )

Chapter 21 • Minimal invariant superset

199

μi

μi

⊂ (A ∧ (Bn )i1 1 ) ∧ ... ∧ (A ∧ (Bn )ip p ), μi

μi

μi

μi

μi

μi

k k Xi1 1 ∧ ... ∧ Xip p ⊂ (Xmin )i1 1 ∧ ... ∧ (Xmin )ip p ⊂ Ai1 1 ∧ ... ∧ Aip p . k , A, that the nonempty sets Theorem 144, page 180 shows, from the k-invariance of Xmin μip μip μi1 μi1 k k k )μi1 ∧...∧(X k )μip (Xmin )i1 ∧...∧(Xmin )ip , Ai1 ∧...∧Aip are k-invariant. In addition, (Xmin i1 min ip μi

μi

is the least k-invariant superset of Xi1 1 ∧ ... ∧ Xip p . Eq. (21.4.1) is proved.

21.5 Isomorphisms Theorem 163. Let the systems ,  : Bn → Bn , ∅  X ⊂ Bn and (h, h ) ∈ I so(φ, ψ). We have ∀k ∈ {1, ..., 5}, k ) = h(X)kmin, . h(Xmin,

(21.5.1)

3 Proof. The proof is made for k = 3. We know from the hypothesis (21.1.3) that Xmin, fulfills 3 X ⊂ Xmin, and it is 3-invariant 3 α 3 , O (μ) ⊂ Xmin, . ∀α ∈ n , ∀μ ∈ Xmin, 3 ) and we know also from Theorem 138, page 167, that the 3We get that h(X) ⊂ h(Xmin, invariance β

3 3 ), O (ν) ⊂ h(Xmin, ) ∀β ∈ n , ∀ν ∈ h(Xmin,

holds. We take now an arbitrary 3-invariant set Y , where h(X) ⊂ Y ⊂ Bn : ∀β ∈ n , ∀ν ∈ Y , O (ν) ⊂ Y . β

We know that the set h−1 (Y ) ⊃ X is 3-invariant: α (μ) ⊂ h−1 (Y ). X ⊂ h−1 (Y ) ⊂ Bn and ∀α ∈ n , ∀μ ∈ h−1 (Y ), O

In the second statement of (21.1.3) α 3 (μ) ⊂ Y ) =⇒ Xmin, ⊂ Y, ∀Y, (X ⊂ Y ⊂ Bn and ∀α ∈ n , ∀μ ∈ Y, O

the hypothesis is true under the form (21.5.2), thus we infer 3 ⊂ h−1 (Y ). Xmin,

The conclusion is 3 ) ⊂ Y , h(Xmin, 3 ) is minimal. Statement (21.5.1) is proved. i.e. h(Xmin,

(21.5.2)

200

Boolean Systems

Corollary 12. For any k ∈ {1, ..., 5}, k

k Xmin, = Xmin,∗ .

21.6 Subsystems Theorem 164. Let the systems  : Bn → Bn ,  : Bn+m → Bn+m such that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ), the set X ⊂ Bn , X = ∅ and Y = X × Bm . We have ∀k ∈ {1, ..., 4}, k k = Xmin, × Bm . Ymin,

Proof. For k = 1, we know from the hypothesis that 1 α 1 , ∃α ∈ n , O (μ) ⊂ Xmin, , ∀μ ∈ Xmin, α 1 (μ) ⊂ Y ) =⇒ Xmin ∀Y, (X ⊂ Y ⊂ Bn and ∀μ ∈ Y, ∃α ∈ n , O , ⊂ Y,

(21.6.1) (21.6.2)

and we must prove (α,β)

1 , ∀ν ∈ Bm , ∃α ∈ n , ∃β ∈ m , O ∀μ ∈ Xmin,

1 (μ, ν) ⊂ Xmin, × Bm ,

∀Y , (X × Bm ⊂ Y ⊂ Bn × Bm and (α,β) ∀(μ, ν) ∈ Y , ∃α ∈ n , ∃β ∈ m , O (μ, ν) ⊂ Y ) 1 m =⇒ Xmin , × B ⊂ Y .

(21.6.3)

(21.6.4)

Statement (21.6.3) is a consequence of Theorem 139 (a), page 169. We suppose against all reason that (21.6.4) is false ∃Y , X × Bm ⊂ Y ⊂ Bn × Bm and (α,β) ∀(μ, ν) ∈ Y , ∃α ∈ n , ∃β ∈ m , O (μ, ν) ⊂ Y 1 m and Xmin , × B ⊂ Y . We denote Y = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ Y } and we have, from X × Bm ⊂ Y , that Y ⊃ X. Theorem 139 (b), page 169 implies that Y is 1-invariant α ∀μ ∈ Y, ∃α ∈ n , O (μ) ⊂ Y

and, from (21.6.2), we get 1 Xmin , ⊂ Y.

(21.6.5)

Chapter 21 • Minimal invariant superset

201

1 m Our hypothesis that was made against all reason Xmin , × B ⊂ Y means that 1 m / Y , ∃μ ∈ Xmin , , ∃ν ∈ B , (μ, ν) ∈

(21.6.6)

1 m ∀μ ∈ Xmin , , ∃ν ∈ B , (μ, ν) ∈ Y .

(21.6.7)

and (21.6.5) means

The statements (21.6.6), (21.6.7) are contradictory. (21.6.4) is true.

21.7 Cartesian products Remark 146. A natural attempt comes here, to rewrite Theorem 156, page 193 in the case of the minimal invariant supersets. In this respect, we consider the systems  : Bn → Bn ,  : Bm → Bm ,  =  ×  and the nonempty sets X ⊂ Bn , Y ⊂ Bm . For any k ∈ {1, ..., 5}, k k × Ymin, (X × Y )kmin, = Xmin,

should be true. And here is the intended proof, which is standard, made (without restricting the generality) for k = 2. The hypothesis states that 2 α 2 , O (μ) ⊂ Xmin, , ∃α ∈ n , ∀μ ∈ Xmin, α 2 ∀A, (X ⊂ A ⊂ Bn and ∃α ∈ n , ∀μ ∈ A, O (μ) ⊂ A) =⇒ Xmin , ⊂ A, β

2 2 ∃β ∈ m , ∀ν ∈ Ymin, , O (ν) ⊂ Ymin, , β

2 ∀B, (Y ⊂ B ⊂ Bm and ∃β ∈ m , ∀ν ∈ B, O (ν) ⊂ B) =⇒ Ymin , ⊂ B.

(21.7.1) (21.7.2) (21.7.3) (21.7.4)

We infer from Theorem 140, page 170 that (21.7.1) and (21.7.3) imply the 2-invariance of 2 2 × Ymin, relative to . So far so good. Xmin, Let now C, where X × Y ⊂ C ⊂ Bn × Bm , which is 2-invariant (α,β)

∃(α, β) ∈ n+m , ∀(μ, ν) ∈ C, O

(μ, ν) ⊂ C.

By using Theorem 140, page 170, we get that the sets C = {μ|μ ∈ Bn , ∃ν ∈ Bm , (μ, ν) ∈ C}, C = {ν|ν ∈ Bm , ∃μ ∈ Bn , (μ, ν) ∈ C}, are 2-invariant relative to ,  and that, consequently, (21.7.2), (21.7.4) imply 2 Xmin , ⊂ C ,

202

Boolean Systems

2 Ymin , ⊂ C . 2 2 The inclusion C ⊃ Xmin, × Ymin, should follow, but it does not. In Theorem 156, reasoning like 2 2 2 2 and C ⊂ Ymax, ) =⇒ C ⊂ Xmax, × Ymax, (C ⊂ Xmax,

works. Problem 12. What can we say about the Cartesian product of systems in the case of the minimal invariant superset?

22 Minimal invariant subset ◦1

Section 22.1 of this chapter defines for  : Bn → Bn the minimal invariant subsets X min , ..., ◦5

X min of a set X ⊂ Bn and Section 22.2 gives several examples. Some properties of these sets are addressed in Section 22.3. In Section 22.4, where minimality vs. nullclines is discussed, we show that if X ⊂ N Ci1 ∧ ◦k

λ

... ∧ N Cip , then a unique λ ∈ Bp exists with the property that Xmin ⊂ (Bn )λi11 ∧ ... ∧ (Bn )ipp . The isomorphisms transform minimal invariant subsets in minimal invariant subsets, and this is presented in Section 22.5. In Section 22.6 we leave as an open problem the situation of the Cartesian products of systems, similarly with the case of the minimal invariant superset.

22.1 Definition Definition 101. Let  : Bn → Bn , X ⊂ Bn and A ⊂ X nonempty. If one of 





∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A, ∀Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y ) =⇒ Y ⊃ A,

(22.1.1)

∃α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, ∀Y, (∅  Y ⊂ X and ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ Y ⊃ A,

(22.1.2)

∀α ∈ n , ∀μ ∈ A, O α (μ) ⊂ A, ∀Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ Y ⊃ A,

(22.1.3)





∀λ ∈ Bn , λ (A) ⊂ A, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) ⊂ Y ) =⇒ Y ⊃ A,

(22.1.4)

∀λ ∈ Bn , λ (A) = A, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ Y ⊃ A

(22.1.5)

is true, we say that A is the minimal 1-invariant, ..., minimal 5-invariant subset of X. ◦1

◦5

Notation 36. We denote with Xmin ,..., Xmin the minimal invariant subsets of X. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00028-3 Copyright © 2023 Elsevier Inc. All rights reserved.

203

204

Boolean Systems

◦1

◦5

Theorem 165. If Xmin ,..., Xmin exist, then ◦1

◦2

◦3

◦4

◦5

◦1

◦2

◦3

◦4

◦5

Xmin ⊂ X min ⊂ Xmin = Xmin = Xmin . Proof. The statements Xmin ⊂ Xmin ⊂ X min ⊂ Xmin ⊂ Xmin are obvious, from the way that these sets have been defined. ◦3

◦3

◦4

The set Xmin is 4-invariant and satisfies Xmin ⊂ X, thus Xmin ◦4

(22.1.4) ◦ 3 ⊂ Xmin . ◦4 ◦5

◦5

On the other hand, the set Xmin is 4-invariant and satisfies Xmin ⊂ Xmin , where Xmin is ◦4

◦4

5-invariant, thus Xmin is 5-invariant itself, from Theorem 135, page 165. We have X min ⊂ X ◦5

◦4

◦5

thus Xmin ⊂ Xmin , from the definition of X min .

22.2 Examples Example 121. We consider the state portrait - (1, 1) (0, 0)  Q  6 6 Q  Q Q  Q ? ?+  s Q (1, 0) (0, 1) The set A = {(0, 0), (0, 1)} is 1-invariant, as far as α = (0, 1), (1, 1), (0, 1), (1, 1), ... and β = (1, 1), (0, 1), (1, 1), (0, 1), ... fulfill O α (0, 0) = O β (0, 1) = A. A is not 2-invariant. For this, we see that  λ  (0, 0) ∈ {(0, 0), (0, 1)}, λ (0, 1) ∈ {(0, 1), (0, 0)} implies λ ∈ {(0, 0), (0, 1)} ∧ {(0, 0), (1, 0), (0, 1), (1, 1)} = {(0, 0), (0, 1)}. We get that any α ∈ 2 expressing the 2-invariance of A must satisfy ∀k ∈ N, α k ∈ {(0, 0), (0, 1)}, representing a contradiction with its request of progressiveness. The set X = {(0, 0), (0, 1), (1, 0)} is not invariant, because for any α ∈ 2 , we have ◦1

O α (1, 0)  X = ∅. Moreover, the only invariant subset of X is A. Then X min = A. A similar analysis is true by taking A = {(1, 1), (1, 0)} and X = {(1, 1), (1, 0), (0, 1)}, for ◦1

which Xmin = A again. Example 122. We continue to refer to the system from Example 121. The set A = {(0, 0), (1, 1)} is 2-invariant, because α = (1, 1), (1, 1), (1, 1), ... fulfills O α (0, 0) = O α (1, 1) = A.

Chapter 22 • Minimal invariant subset

205

A is not 3-invariant, as for any α with α 0 = (0, 1) we get O α (0, 0)  A = ∅. The set X = {(0, 1), (0, 0), (1, 1)} is 2-invariant, because α = (1, 1), (1, 1), (1, 1), ... satisfies ◦2

O α (0, 1) = X and A is a proper 2-invariant subset of X, the only one. We have Xmin = A. On the other hand X is 1-invariant, and has two proper 1-invariant subsets A and ◦1

{(0, 0), (0, 1)}, whose intersection {(0, 0)} is not 1-invariant: X min does not exist. A similar analysis is possible, resulted from the symmetry of the state portrait from Example 121, for A = {(0, 0), (1, 1)} and X = {(1, 0), (0, 0), (1, 1)}: A and X are 2-invariant, ◦2

◦1

X min = A and X min does not exist. Example 123. In the next state portrait (0, 0, 0) 6

- (0, 0, 1)

? (0, 1, 1)

(0, 1, 0) 

(1, 0, 0) 6 (1, 1, 0) 

- (1, 0, 1)

? (1, 1, 1)

the set A = {(0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0)} is 3-invariant: ∀α ∈ 3 , ∀μ ∈ A, O α (μ) = A. A is not 5-invariant, because (1,1,0) (0, 0, 1) = (1,1,0) (0, 1, 1) = (0, 1, 1), thus (1,1,0) : A → A (abusive notation!) is not injective. The set X = {(0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0), (1, 0, 0)} is not invariant, as far as ∀α ∈ 3 , O α (1, 0, 0)  X = ∅. The only invariant subset of X is A, ◦1

◦2

◦3

thus Xmin = Xmin = Xmin = A. The symmetry of the state portrait allows us to construct minimal invariant subsets starting from the 3-invariant set A = {(1, 0, 0), (1, 0, 1), (1, 1, 1), (1, 1, 0)} too. Example 124. The system (0, 0) 

(1, 0)

-

(0, 1)

- (1, 1)

satisfies the property that the set A = {(1, 1)} is 5-invariant, because (1, 1) is a fixed point. Let us take X = {(0, 0), (1, 0), (1, 1)}. We see that X is not invariant, because any α ∈ 2 satisfies O α (0, 0)  X = ∅. Two proper invariant subsets of X exist, {(1, 0), (1, 1)} that is 3◦k

invariant and {(1, 1)}. We get X min = A, for k ∈ {1, ..., 5}. Example 125. The previous system has the 5-invariant set A = {(0, 0), (0, 1)}. There, the set ◦k

X = {(0, 0), (0, 1), (1, 0)} is not invariant, and the minimal invariant subset of X is Xmin = A, k ∈ {1, ..., 5}.

206

Boolean Systems

Example 126. If in the case of the system from Example 124 we consider X = {(0, 0), (0, 1), (1, 1)}, then X is 5-invariant, together with two proper subsets, {(0, 0), (0, 1)} and {(1, 1)}. ◦k

But the intersection of these sets, which is empty, is not 5-invariant, thus Xmin does not exist, k ∈ {1, ..., 5}.

22.3 Properties Theorem 166. We consider  : Bn → Bn and the set X ⊂ Bn , X = ∅. Then ∀k ∈ {1, ..., 5}, ◦k

◦ ◦k

Xmin = (Xmin )kmin .

(22.3.1)

Proof. We take k ∈ {1, ..., 5} arbitrary. The double inclusion ◦ ◦k

◦k

(X min )kmin ⊂ Xmin ⊂ X

(22.3.2) ◦ ◦k

holds, as a consequence of the definition of the minimal invariant subset. In it, (X min )kmin is a k-invariant subset of X, thus ◦k

◦ ◦k

Xmin ⊂ (Xmin )kmin

(22.3.3)

is also true. (22.3.2) and (22.3.3) imply (22.3.1). Theorem 167. The system  and the nonempty set X ⊂ Bn are given. For any k ∈ {3, 4, 5}, (a) we suppose that k-invariant subsets of X exist. ◦k

◦k

(a.1) If X has exactly one k-invariant subset A, then Xmin exists and Xmin = A,

◦k

(a.2) if X has p ≥ 2 k-invariant subsets A1 , ..., Ap and A1 ∧ ... ∧ Ap = ∅, then Xmin exists ◦k

and X min = A1 ∧ ... ∧ Ap ,

◦k

(a.3) if X has p ≥ 2 k-invariant subsets A1 , ..., Ap and A1 ∧ ... ∧ Ap = ∅, then X min does not exist, ◦k

(b) we ask that Xmin exists. Then X has k-invariant subsets.

◦k

◦k

(b.1) In case that X has exactly one k-invariant subset A, then X min = A, (b.2) in case that X has p ≥ 2 k-invariant subsets A1 , ..., Ap , then A1 ∧ ... ∧ Ap = ∅ and

X min = A1 ∧ ... ∧ Ap . ◦k

(c) X min does not exist (c.1) if X has no k-invariant subsets, or

Chapter 22 • Minimal invariant subset

207

(c.2) if X has p ≥ 2 k-invariant subsets A1 , ..., Ap and A1 ∧ ... ∧ Ap = ∅.1 Proof. (a.2) A1 ∧ ... ∧ Ap is k-invariant, and for any Y k-invariant, ∅  Y ⊂ X, we have the existence of i ∈ {1, ..., p} with Y = Ai . We infer A1 ∧ ... ∧ Ap ⊂ Y ⊂ X. ◦k

Remark 147. Let k ∈ {1, 2}. If X has exactly one k-invariant subset A, then Xmin exists and ◦k

we have Xmin = A (like at Theorem 167, (a.1)). Theorem 168. The following property holds: ◦1

◦1

∀μ ∈ Xmin , ∃α ∈ n , ωα (μ) = O α (μ) = Xmin .

(22.3.4)

◦1

Proof. The set Xmin is 1-invariant: ◦1

◦1

◦1

◦1

∀μ ∈ Xmin , ∃α ∈ n , O α (μ) ⊂ Xmin , and we ask if the possibility ∃μ ∈ Xmin , ∃α ∈ n , O α (μ)  X min

(22.3.5)

exists. If (22.3.5) would be true, then the 1-invariance of O α (μ) (see Theorem 132, page 161) ◦1

would contradict the minimality of Xmin . We infer ◦1

◦1

∀μ ∈ X min , ∃α ∈ n , O α (μ) = Xmin . Similarly, we ask if the possibility ◦1

◦1

∃μ ∈ Xmin , ∃α ∈ n , ωα (μ)  O α (μ) = Xmin

(22.3.6)

exists. In case that (22.3.6) would be true, the 1-invariance of ωα (μ) (see Theorem 132 ◦1

again) would contradict the minimality of Xmin . (22.3.4) is true. ◦1

Remark 148. Two statements occur in (22.3.4): the equality O α (μ) = X min is one of topological transitivity, and the equality ωα (μ) = O α (μ) is of Poisson stability of μ. 1

(c.2) coincides with the hypothesis of (a.3).

208

Boolean Systems

22.4 Minimality vs. nullclines Theorem 169. The set X ⊂ Bn is given, X = ∅ and we suppose that X ⊂ N Ci1 ∧ ... ∧ N Cip , ◦k

where i1 , ..., ip ∈ {1, ..., n}, p ≥ 1. For an arbitrary k ∈ {1, ..., 5}, we ask that Xmin exists. Then a ◦k

λ

unique λ ∈ Bp exists with the property that X min ⊂ (Bn )λi11 ∧ ... ∧ (Bn )ipp . Proof. We fix an arbitrary k ∈ {1, ..., 5}, and we suppose against all reason that λ, λ ∈ Bp ◦k

◦k

λ

λ

exist, λ = λ with the property that Xmin ∧ (Bn )λi11 ∧ ... ∧ (Bn )ipp = ∅, Xmin ∧ (Bn )i11 ∧ ... ∧ ◦k

λ

λ

◦k

λ

λ

(Bn )ipp = ∅. Then X min ∧ (Bn )λi11 ∧ ... ∧ (Bn )ipp , X min ∧ (Bn )i11 ∧ ... ∧ (Bn )ipp are disjoint and ◦k

k-invariant, from Theorem 144 (a), page 180, therefore they are strictly included in X min , contradiction. ◦k

λ

We infer from here that for the unique λ ∈ Bp with Xmin ∧ (Bn )λi11 ∧ ... ∧ (Bn )ipp = ∅ we ◦k

◦k

◦k

λ

λ

have Xmin = Xmin ∧ (Bn )λi11 ∧ ... ∧ (Bn )ipp , i.e. Xmin ⊂ (Bn )λi11 ∧ ... ∧ (Bn )ipp .

22.5 Isomorphisms Theorem 170. We consider ,  : Bn → Bn , (h, h ) ∈ I so(φ, ψ) and the subset X ⊂ Bn . For ◦k

◦

k

any k ∈ {1, ..., 5}, we have h(X min, ) = h(X)min, . ◦1

Proof. We make the proof for k = 1. The hypothesis states that the set A = X min, satisfies α (μ) ⊂ A, ∀μ ∈ A, ∃α ∈ n , O

(22.5.1)

α ∀Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O (μ) ⊂ Y ) =⇒ Y ⊃ A,

(22.5.2)

and we must prove that β

∀ν ∈ h(A), ∃β ∈ n , O (ν) ⊂ h(A), β

∀Z, (∅  Z ⊂ h(X) and ∀ν ∈ Z, ∃β ∈ n , O (ν) ⊂ Z) =⇒ Z ⊃ h(A).

(22.5.3) (22.5.4)

Statement (22.5.3) is true from Theorem 138, page 167. Let now Z a set that satisfies β

∅  Z ⊂ h(X) and ∀ν ∈ Z, ∃β ∈ n , O (ν) ⊂ Z. We infer that ∅  h−1 (Z) ⊂ X and, from Theorem 138, α ∀μ ∈ h−1 (Z), ∃α ∈ n , O (μ) ⊂ h−1 (Z).

The hypothesis of (22.5.2) is fulfilled by Y = h−1 (Z) thus, from (22.5.2): h−1 (Z) ⊃ A. This fact implies Z ⊃ h(A), i.e. (22.5.4) is true.

Chapter 22 • Minimal invariant subset

209

22.6 Cartesian products Problem 13. (See Remark 146, page 201 and Problem 12.) What can we say about the Cartesian product of systems in the case of the minimal invariant subset?

23 Connectedness and separation In Section 23.1 we introduce connectedness or minimality, the property of a set X ⊂ Bn ◦k

that X = Xmin , k ∈ {1, ..., 5}. Separation or disconnectedness is the property of X of being invariant, with a proper subset which is invariant. This is presented in Section 23.2, and Section 23.3 contains examples. The topic of Section 23.4 is that of giving some simple properties of connectedness and separation. We notice in Section 23.5 that in certain circumstances connectedness implies topological transitivity and sometimes invariance together with topological transitivity implies connectedness. In Section 23.6 we see the way that connectedness implies path-connectedness, and also how invariance and path-connectedness imply connectedness. The connected relatively isolated sets are called connected components and they are presented in Section 23.7. In Section 23.8 we show how the isomorphisms bring connected sets in connected sets and disconnected sets in disconnected sets.

23.1 Connectedness Definition 102. The function  : Bn → Bn and X ⊂ Bn are given, X = ∅. If one of 





∀μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X, ∀Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y ) =⇒ (Y = X),

(23.1.1)

∃α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∀Y, (∅  Y ⊂ X and ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ (Y = X),

(23.1.2)

∀α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∀Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ (Y = X),

(23.1.3)





∀λ ∈ Bn , λ (X) ⊂ X, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) ⊂ Y ) =⇒ (Y = X),

(23.1.4)

∀λ ∈ Bn , λ (X) = X, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ (Y = X)

(23.1.5)

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00029-5 Copyright © 2023 Elsevier Inc. All rights reserved.

211

212

Boolean Systems

holds, then X is called connected, or minimal. We refer to k-connectedness, or kminimality, k ∈ {1, ..., 5}. Theorem 171. For X ⊂ Bn nonempty and k ∈ {1, ..., 5} arbitrary, the following statements are equivalent: (a) X is k-minimal, ◦k

(b) X = Xmin . Proof. We take for example k = 5, and we remark the equivalence between (23.1.5), stand◦5

ing for (a), and (22.1.5)page 203 , where A = Xmin was replaced by X: 

∀λ ∈ Bn , λ (X) = X, ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ (Y ⊃ X),

standing for (b). Remark 149. Theorem 171 makes clear the origin of the terminology of minimal set: X is minimal if it coincides with its minimal invariant subset. Theorem 172. We have (23.1.1) =⇒ (23.1.2) =⇒ (23.1.3) ⇐⇒ (23.1.4) ⇐⇒ (23.1.5). Proof. From the inclusions ◦1

◦2

◦3

◦4

◦5

Xmin ⊂ Xmin ⊂ Xmin = X min = Xmin ⊂ X, see Theorem 165, page 203, we have ◦1

◦2

◦3

◦4

◦5

X = Xmin =⇒ X = Xmin =⇒ X = X min ⇐⇒ X = X min ⇐⇒ X = X min . We take into account Theorem 171.

23.2 Separation Definition 103. For  : Bn → Bn and X ⊂ Bn nonempty, if one of 



∀μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X, ∃Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y and Y = X),

(23.2.1)

∃α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∃Y, (∅  Y ⊂ X and ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y and Y = X),

(23.2.2)

Chapter 23 • Connectedness and separation 

∀α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∃Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y and Y = X), 



213

(23.2.3)

∀λ ∈ Bn , λ (X) ⊂ X, ∃Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) ⊂ Y and Y = X),

(23.2.4)

∀λ ∈ Bn , λ (X) = X, ∃Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y and Y = X)

(23.2.5)

holds, we say that X is disconnected, or separated and each Y like previously is called a separation of X. We refer to k-disconnectedness, or k-separation, k ∈ {1, ..., 5}. Remark 150. The disconnectedness of the invariant set X means that a proper invariant subset Y of X exists; thus Y is not connected with X  Y , in the sense that the flows with the initial value in Y do not reach X  Y . A special case of disconnectedness is the situation when X, Y, X  Y are all invariant and Y, X  Y are both separations of X (they are relatively isolated). Theorem 173. We have the following implications: (23.2.5) =⇒ (23.2.4) ⇐⇒ (23.2.3) =⇒ (23.2.2) =⇒ (23.2.1). Proof. If X and Y  X are 5-invariant, then they are 4-invariant etc. Remark 151. To be compared Theorem 172 and Theorem 173, note that the implication (23.2.4) =⇒ (23.2.5) is false.

23.3 Examples Example 127. We consider the system from Example 119, page 196. We see first that the set X = {(0, 0, 1), (0, 0, 0)} is 1-invariant. For this it is enough to take α = (1, 0, 1), (0, 1, 1), (1, 0, 1), (0, 1, 1), ... and β = (0, 1, 1), (1, 0, 1), (0, 1, 1), (1, 0, 1), ... for which O α (0, 0, 0) = O β (0, 0, 1) = X. Second, we see that X is not 2-invariant. The request  λ  (0, 0, 1) ∈ {(0, 0, 1), (0, 0, 0)}, λ (0, 0, 0) ∈ {(0, 0, 0), (0, 0, 1)} gives λ ∈ {(0, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1)} ∧ {(0, 0, 0), (1, 0, 0), (0, 0, 1), (1, 0, 1)} = {(0, 0, 0), (0, 0, 1)}. In order that X be 2-invariant, some α must exist with ∀k ∈ N, α k ∈ {(0, 0, 0), (0, 0, 1)}, but such a computation function α is not progressive, contradiction. X is 1-connected, since it does not contain proper invariant subsets, that should be fixed points. We can prove that X = {(0, 0, 1), (0, 0, 0), (1, 0, 0)} is 1-invariant, and X is a 1-separation of X .

214

Boolean Systems

Example 128. We refer now to the system which was analyzed in Examples 121, 122, page 204. We have shown there that the set A = {(0, 0), (1, 1)} is 2-invariant. We see that A is also 2-connected, otherwise it would contain a fixed point. The set X = {(0, 1), (0, 0), (1, 1)} is 2-invariant as well, and A is a 2-separation of X. The reasoning is the same for A = {(0, 0), (1, 1)} and X = {(1, 0), (0, 0), (1, 1)}. Example 129. We have in the case of the system from Example 123, page 205, that the sets X = {(0, 0, 0), (0, 0, 1), (0, 1, 1), (0, 1, 0)}, Y = {(1, 0, 0), (1, 0, 1), (1, 1, 1), (1, 1, 0)} are 3connected. B3 is trivially 3-invariant, and X, Y are 3-separations of B3 . Example 130. For the system from Example 124, page 205, the sets X = {(1, 0), (1, 1)} and Y = {(1, 1)} are 3-invariant, and Y is a 3-separation of X. For that system, Y and X = {(0, 0), (0, 1)} are 5-connected.

23.4 Properties Theorem 174. We suppose that X ⊂ Bn , X = ∅ is k-invariant, k ∈ {1, ..., 5}: (a) if card(X) = 1, then X is 5-connected; (b) if card(X) ≥ 2 and μ ∈ X exists with (μ) = μ, then {μ} is a k-separation of X; (c) if card(X) ≥ 2 and X is k-connected, then it contains no fixed points of . Proof. (a) If X = {μ} and α ∈ n exists such that O α (μ) = {μ}, then (μ) = μ, X is 5invariant and has no invariant subsets. Theorem 175. If X1 ⊂ Bn , X2 ⊂ Bn are k-connected, k ∈ {3, 4, 5}, then two possibilities exist: either X1 ∧ X2 = ∅, or X1 = X2 . Proof. We fix k ∈ {3, 4, 5} arbitrary and we suppose, against all reason, that X1 ∧ X2 = ∅, and X1 = X2 . Then Theorem 134, page 163 states that X1 ∧ X2 is k-invariant and the fact that X1 ∧ X2  X1 contradicts the k-connectedness of X1 . Theorem 176. X is k + 1-invariant and k-connected =⇒ X is k + 1-connected, k ∈ {1, ..., 4}. Proof. We take k = 1. The first statement of (23.1.2) is true and let Y arbitrary, ∅  Y ⊂ X with ∃α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y . Then in (23.1.1), which is true from the hypothesis of the theorem, the first statement of 1-invariance of X is true, and the hypothesis of the second statement is true itself, therefore Y = X. It has resulted that the second statement of (23.1.2) is also true. Theorem 177. If X1 , X2 ⊂ Bn are k-invariant, k ∈ {1, 3, 4, 5}, and A is a k-separation of X1 , then A is a k-separation of X1 ∨ X2 . Proof. We make the proof for k = 1. We know from the hypothesis that ∀μ ∈ X1 , ∃α ∈ n , O α (μ) ⊂ X1 ,

Chapter 23 • Connectedness and separation

215

∀μ ∈ X2 , ∃α ∈ n , O α (μ) ⊂ X2 , ∅  A  X1 and ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A. These imply, see Theorem 134, page 163: ∀μ ∈ X1 ∨ X1 , ∃α ∈ n , O α (μ) ⊂ X1 ∨ X2 , ∅  A  X1 ∨ X2 and ∀μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A. Theorem 178. If X1 , X2 are k-invariant, k ∈ {1, 3, 4, 5}, and X1  X1 ∨ X2 , then X1 is a kseparation of X1 ∨ X2 ; in particular if X1 , X2 are k-connected, then X1 ∨ X2 is k-disconnected. Proof. This is similar with the proof of Theorem 177. Theorem 179. We consider  : Bn → Bn and X ⊂ Bn nonempty. For any k ∈ {3, 4, 5}, if A  X is a k-separation of X and B ⊂ X is k-connected, then B ⊂ A or B ⊂ X  A

(23.4.1)

holds. Proof. We fix k ∈ {3, 4, 5} arbitrary, and we suppose against all reason that (23.4.1) is false. Since B  (X  A) = B ∧ A, the negation of (23.4.1) gives: B  A = ∅ and B ∧ A = ∅. We have ∅  B ∧ A  B and B ∧ A is k-invariant, and this contradicts the k-minimality of B. Remark 152. We ask if the previous theorem holds for k ∈ {1, 2}. In order to show that it is false, we take a look at the next state portrait (1, 0) 6 ? (0, 0) 6

- (1, 1)

? (0, 1) where X = B2 , A = {(0, 0), (0, 1)}, B = {(0, 0), (1, 0)} are all 1-invariant and B is 1-minimal as far as one of (0, 0), (1, 0) should be a fixed point otherwise. The statement (23.4.1) is false.

216

Boolean Systems

By taking a look at the proof of Theorem 179 now, we note that it used the k-invariance of B ∧ A, k ∈ {3, 4, 5}, which is false in general for k ∈ {1, 2}. In the counterexample that we have just given, the set {(0, 0)} = A ∧ B is not 1-invariant. Theorem 180. We suppose that X ⊂ Bn , X = ∅ is 5-invariant, and X1 is a 5-separation of X: ∅  X1  X and ∀λ ∈ Bn , λ (X1 ) = X1 .

(23.4.2)

Then X  X1 is a 5-separation of X: ∅  X  X1  X and ∀λ ∈ Bn , λ (X  X1 ) = X  X1 .

(23.4.3)

Proof. This follows from Theorem 134, page 163.

23.5 Connectedness vs. topological transitivity Theorem 181. We consider  : Bn → Bn and the set X ⊂ Bn , X = ∅. The 1-connectedness of X: ∀μ ∈ X, ∃α ∈ n , O α (μ) ⊂ X, ∀Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O α (μ) ⊂ Y ) =⇒ (Y = X),

(23.5.1) (23.5.2)

implies the (15.2.1)page 136 -topological transitivity ∀μ ∈ X, ∃α ∈ n , O α (μ) = X.

(23.5.3)

Proof. We suppose against all reason that (23.5.3) is false, thus μ ∈ X exists such that ∀α ∈ n , O α (μ ) = X.

(23.5.4)

From (23.5.1) and (23.5.4), we have the existence of β ∈ n with O β (μ )  X.

(23.5.5)

We denote Y = O β (μ ). The set Y satisfies the hypothesis of (23.5.2), see Theorem 132, page 161, resulting that it satisfies its conclusion O β (μ ) = X.

(23.5.6)

The statements (23.5.5) and (23.5.6) are contradictory. (23.5.3) takes place. Remark 153. In Theorem 181, the strongest connectedness property implies the weakest topological transitivity property and improving the result does not seem to be possible.

Chapter 23 • Connectedness and separation

217

Theorem 182. If X ⊂ Bn nonempty is k-invariant, k ∈ {3, 4, 5}, and the topological transitivity property ∀μ ∈ X, ∃α ∈ n , O α (μ) = X

(23.5.7)

holds, then X is k-minimal. Proof. We make the proof for k = 3. We suppose against all reason that X is not 3-minimal, i.e. Y exists such that ∅  Y  X and ∀μ ∈ Y, ∀α ∈ n , O α (μ) ⊂ Y.

(23.5.8)

Let μ ∈ Y arbitrary. Then (23.5.7) shows the existence of α ∈ n with the property that O α (μ) = X. And we get also the existence of k ∈ N such that φ α (μ, k) ∈ Y and φ α (μ, k + 1) ∈ k X  Y , i.e. α (φ α (μ, k)) ∈ / Y . This is in contradiction with the 3-invariance of Y from (23.5.8). Theorem 183. For any k ∈ {1, 2, 3}, if X is (15.2.3)page 137 -topologically transitive ∀α ∈ n , ∀μ ∈ X, O α (μ) = X,

(23.5.9)

◦k

then X is k-invariant and X = Xmin . Proof. Case k = 2. The topological transitivity (23.5.9) of X from the hypothesis implies the 3-invariance of X and the 2-invariance of X too. Let Y arbitrary such that ∅  Y ⊂ X and we suppose that β ∈ n exists with ∀μ ∈ Y, O β (μ) ⊂ Y.

(23.5.10)

From (23.5.9) and (23.5.10) we infer ∀μ ∈ Y, O β (μ) = X ⊂ Y, thus Y = X.

23.6 Connectedness vs. path-connectedness Corollary 13. The 1-connectedness of X implies the path-connectedness ∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ .

(23.6.1)

Proof. This is a consequence of Theorem 181, page 216, which states that 1-connectedness implies the (15.2.1)page 136 -topological transitivity, and of Theorem 115, page 137, which states that the (15.2.1)page 136 -topological transitivity implies the (23.6.1)-path-connectedness.

218

Boolean Systems

Theorem 184. If X is k-invariant, k ∈ {3, 4, 5}, and (23.6.1)-path-connected, then X is kconnected. Proof. We make the proof for k = 5, when the request of connectedness of X is ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ (Y = X).

(23.6.2)

We suppose against all reason that (23.6.2) is false, thus Y exists satisfying ∅  Y  X and ∀λ ∈ Bn , λ (Y ) = Y.

(23.6.3)

We take μ ∈ Y and μ ∈ X  Y arbitrary. From (23.6.1), α ∈ n and k ∈ N exist with the property that ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X,

(23.6.4)

φ α (μ, k ) = μ

(23.6.5)

hold. As φ α (μ, ·) starts from μ ∈ Y and gets to μ ∈ X  Y , we have from (23.6.4), (23.6.5) the k

/ Y , i.e. α (φ α (μ, k

)) ∈ / existence of k

∈ {0, ..., k − 1} such that φ α (μ, k

) ∈ Y, φ α (μ, k

+ 1) ∈ Y . This is in contradiction with the 5-invariance of Y from (23.6.3). The minimality request (23.6.2) is fulfilled. Remark 154. Theorem 181, page 216 followed by Corollary 13 seem to indicate the possibility of a closer connection between connectedness and path-connectedness. For example, is the implication 

∀α ∈ n , ∀μ ∈ X, O α (μ) ⊂ X, ∀Y, (∅  Y ⊂ X and ∀α ∈ n , ∀μ ∈ Y, O α (μ) ⊂ Y ) =⇒ (Y = X)  ∀α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k ∈ N, =⇒ ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ

(23.6.6)

true? We give a counterexample. Example 131. For  : B2 → B2 defined by ∀μ ∈ B2 , (μ) = (μ1 , μ2 ), we have that X = B2 is ◦3

3-invariant and X = Xmin , therefore the hypothesis of (23.6.6) is fulfilled. On the other hand α = (1, 1), (1, 1), (1, 1), ... and μ = (0, 0) satisfy O α (μ) = {(0, 0), (1, 1)}, thus for μ = (0, 1) the conclusion μ ∈ O α (μ) of (23.6.6) is false.

23.7 Connected components Definition 104. Let X ⊂ Bn nonempty and X1 , ..., Xp ⊂ X a partition of k-connected subsets of X, p ≥ 2, k ∈ {1, ..., 5}. Then X1 , ..., Xp are called the k-connected components of X.

Chapter 23 • Connectedness and separation

219

Remark 155. The k-connected components of X are relatively k-isolated sets with the ◦k

property that Xi = Xi,min , i ∈ {1, ..., p}. Example 132. We consider the system which has the following state portrait: (0, 1, 0) 6

(0, 1, 1) (1, 0, 1) (1, 1, 0) 6 3 36  36              ? ? ? ?    - (0, 0, 1) - (1, 1, 1) - (1, 0, 0) (0, 0, 0) A partition of X = B3 consists in the sets X1 = {(0, 0, 0), (0, 1, 0)}, X2 = {(0, 0, 1), (0, 1, 1)}, X3 = {(1, 1, 1), (1, 0, 1)}, and X4 = {(1, 0, 0), (1, 1, 0)}. They are all 1-connected, except X4 which is 3-connected. X1 , X2 , X3 , X4 are the 1-connected components of X. Theorem 185. We suppose that X ⊂ Bn nonempty is 5-invariant. Then one of the following possibilities is true: (a) X is 5-connected, i.e. ∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ (Y = X), (b) the partition A1 , ..., Ap of X exists, p ≥ 2, such that each Ai is 5-connected: ∀i ∈ {1, ..., p}, ∀Y, (∅  Y ⊂ Ai and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ (Y = Ai ). Proof. If X is 5-connected the theorem is proved, so we suppose that it is not. This means the existence of a nonempty set Y  X which is 5-invariant. If Y is 5-connected, then we denote A1 = Y and we continue the reasoning with X  A1 which is 5-invariant, from Theorem 134, page 163. If Y is not 5-connected, then it contains a nonempty set Y  Y which is 5-invariant. If

Y is 5-connected, we denote A1 = Y and we continue the reasoning with X  A1 which is 5-invariant. If Y is not 5-connected, then it contains a nonempty set Y

 Y which is 5-invariant... In a finite number of steps we get the 5-connected set A1 and we continue the reasoning with X  A1 , which is 5-invariant. If it is also 5-connected, then A2 = X  A1 and the theorem is proved, so we suppose that it is not... In a finite number of steps we get the 5-connected sets A1 , ..., Ap . Problem 14. To be characterized the existence and the uniqueness of the k-connected components of a set X ⊂ Bn for k ∈ {1, ..., 4}.

220

Boolean Systems

23.8 Isomorphisms Theorem 186. The isomorphic systems ,  : Bn → Bn are given, where (h, h ) ∈ I so(φ, ψ). For any k ∈ {1, ..., 5}, (a) if X ⊂ Bn is k-minimal, then h(X) is k-minimal, (b) if X is k-disconnected, then h(X) is k-disconnected. ◦k

Proof. (a) For k ∈ {1, ..., 5} arbitrary, fixed, the hypothesis states that X = Xmin, . We have ◦k

h(X) = h(X min, )

Theorem 170, page 208

=

◦

k

h(X)min, .

(b) The proof for k = 1 looks like this. The hypothesis is α ∀μ ∈ X, ∃α ∈ n , O (μ) ⊂ X,

(23.8.1)

α ∃Y, (∅  Y ⊂ X and ∀μ ∈ Y, ∃α ∈ n , O (μ) ⊂ Y and Y = X),

(23.8.2)

and the conclusion to be proved is β

∀ν ∈ h(X), ∃β ∈ n , O (μ) ⊂ h(X), β

∃Z, (∅  Z ⊂ h(X) and ∀ν ∈ Z, ∃β ∈ n , O (ν) ⊂ Z and Z = h(X)).

(23.8.3) (23.8.4)

The invariance (23.8.3) follows from (23.8.1) and Theorem 138, page 167, with β = h (α). The separation (23.8.4) follows from (23.8.2) and Theorem 138, with Z = h(Y ) and β =

h (α).

24 Basins of attraction The basins of attraction of a set A ⊂ Bn are the sets of these points μ ∈ Bn for which ωα (μ) ⊂ A. They are defined in Section 24.1, and some examples are given in Section 24.2. The most important properties of the basins of attraction are given in Section 24.3. The special case of basins of attraction when A = {μ}, with (μ) = μ, is mentioned in Section 24.4, and a version of the definition of the basins of attraction, when A = {μ}, with (p) (μ) = μ, is addressed in Section 24.5. We show in Section 24.6 that the isomorphisms of flows bring basins of attraction in basins of attraction.

24.1 Definition Notation 37. Let  : Bn → Bn , μ ∈ Bn and A ⊂ Bn nonempty. We denote  ωα (μ), ω+ (μ) = α∈n

ω+ (A) =

 

ωα (ν).

ν∈Aα∈n

Definition 105. The computation function γ ∈ n and A ⊂ Bn , A = ∅ are given. The sets W 1 (A) = {μ|μ ∈ Bn , ∃α ∈ n , ωα (μ) ⊂ A}, Wγ2 (A) = {μ|μ ∈ Bn , ωγ (μ) ⊂ A}, W 3 (A) = {μ|μ ∈ Bn , ∀α ∈ n , ωα (μ) ⊂ A}, W 4 (A) = {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) ⊂ B}, W 5 (A) = {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) = B} are called basins (or kingdoms, or domains) of attraction1 of A. Remark 156. In literature, in general, when defining the basins of attraction, the set A is asked to be invariant, or rather to be an attractor. We preferred to give Definition 105 with A 1

Or stable sets.

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00030-1 Copyright © 2023 Elsevier Inc. All rights reserved.

221

222

Boolean Systems

arbitrary, and consider later the special cases when A is invariant or attractor (for example A = {μ}, where (μ) = μ). Attractors will be treated in Chapter 28. Theorem 187. For any A ⊂ Bn , A = ∅, γ ∈ n , μ ∈ Bn , (a) the following implications are true ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) = B =⇒ ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) ⊂ B ⇐⇒ ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A ⇐⇒ ∀α ∈ n , ωα (μ) ⊂ A =⇒ ωγ (μ) ⊂ A =⇒ ∃α ∈ n , ωα (μ) ⊂ A, (b) we have the inclusions W 5 (A) ⊂ W 4 (A) = W 3 (A) ⊂ Wγ2 (A) ⊂ W 1 (A). Proof. (a) We prove the implication ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A =⇒ ∀α ∈ n , ωα (μ) ⊂ A. The set B ⊂ Bn exists with μ ∈ B and ω+ (B) ⊂ A, wherefrom    ωα (μ) ⊂ ωα (ν) = ω+ (B) ⊂ A, α∈n

ν∈B α∈n

thus ∀α ∈ n , ωα (μ) ⊂ A. We prove now the truth of ∀α ∈ n , ωα (μ) ⊂ A =⇒ ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) ⊂ B for B = O + (μ). As μ ∈ O + (μ) is clear, we prove ω+ (B) ⊂ A.

(24.1.1)

We can write: ω+ (B) = ω+ (O + (μ)) =





ωα (ν) =

ν∈O + (μ)α∈n

=

   β∈n k∈Nα∈n

ω (φ (μ, k)) = α

β







 α∈

ωα (φ β (μ, k))

n

β∈n k∈N

ωα (μ) = ω+ (μ).

α∈n

But ω+ (μ) ⊂ A, from the hypothesis, therefore (24.1.1) is true. The 4-invariance of B was proved at Theorem 128, page 160. We just note that the implication ∀α ∈ n , ωα (μ) ⊂ A =⇒ ∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A

Chapter 24 • Basins of attraction

223

may be proved for B = {μ}. Remark 157. One of the consequences of Theorem 187 is that we can define W 4 (A) equivalently as W 4 (A) = {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A}. Remark 158. In order to make as clear as possible some differences, to be compared W 1 (A), W 3 (A) (Definition 105), W 4 (A) (Remark 157), with A1max , A3max , A4max (Corollary 10, page 189). Definition 106. If W ∈ {W 1 (A), Wγ2 (A), W 3 (A), W 5 (A)} is nonempty, A is said to be attractive. For a nonempty set B ⊂ W , we say that A is attractive for (the points of ) B and that B is attracted (the points of B are attracted) by A. The set A is by definition partially attractive if W = Bn and totally attractive whenever W = Bn . Remark 159. The basins of attraction W 1 (A), Wγ2 (A), ..., W 5 (A) have been defined in a manner which is compatible with the five definitions of invariance of a set. They represent the sets of these points μ which are attracted by A, in the sense that the omega-limit set of μ is included in A.

24.2 Examples Example 133. We consider the state portrait - (0, 1, 0) (0, 0, 0) Q Q Q Q Q ? s ? Q (1, 1, 0)  (1, 0, 0)

- (0, 0, 1) (0, 1, 1)  6 3         ?   + - (1, 1, 1)  (1, 0, 1)

The point (0, 0, 0) is a source and the set A = {(0, 0, 0)} is not attractive: W 1 (A) = ∅, meaning that ∀μ ∈ B3 , ∀α ∈ 3 , finitely many terms φ α (μ, k) fulfill φ α (μ, k) = (0, 0, 0), k ∈ N. For γ ∈ 3 given by γ k = (1, 1, 1), k ∈ N, the set A = {(0, 0, 0), (1, 1, 0), (1, 1, 1)} fulfills the attractiveness properties W 1 (A) = B3 \ {(0, 0, 1)}, Wγ2 (A) = W 3 (A) = {(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1)}, W 5 (A) = {(1, 1, 0), (1, 1, 1)}. We take now A = {(1, 1, 0), (1, 1, 1), (0, 0, 1)}. It is totally attractive, in the sense that W 1 (A) = B3 , and also partially attractive, in the sense that Wγ2 (A) = W 3 (A) = B3 \ {(0, 1, 1), (1, 0, 1)}, and W 5 (A) = {(1, 1, 0), (1, 1, 1), (0, 0, 1)}. The set A = {(0, 1, 1), (1, 0, 1)} has the basin of attraction W 1 (A) = Wγ2 (A) = A. The set A = {(1, 1, 0), (1, 1, 1), (0, 1, 1), (0, 0, 1), (1, 0, 1)} is totally attractive: W 1 (A) = 2 Wγ (A) = W 3 (A) = B3 , and partially attractive as W 5 (A) = {(1, 1, 0), (1, 1, 1), (0, 0, 1)}.

224

Boolean Systems

Example 134. In the following state portrait, (0, 1, 0) 6

(0, 0, 0)

(0, 1, 1) 6 - (1, 1, 0)

- (1, 1, 1)

? (1, 0, 0)

- (0, 0, 1)

? (1, 0, 1)

for γ = (1, 1, 1), (1, 0, 1), (1, 1, 1), (1, 1, 1), ... we have the basins of attraction Wγ2 ({(0, 0, 1)}) = {(0, 0, 0), (1, 1, 1), (0, 0, 1)}, W 5 ({(0, 0, 1)}) = {(0, 0, 1)}, Wγ2 ({(0, 1, 0), (0, 1, 1), (0, 0, 1), (1, 0, 1), (1, 0, 0)}) = B3 , W 5 ({(0, 1, 0), (0, 1, 1), (0, 0, 1), (1, 0, 1), (1, 0, 0)}) = {(0, 1, 0), (0, 1, 1), (0, 0, 1), (1, 0, 1), (1, 0, 0)}.

24.3 Properties Lemma 2. Let  : Bn → Bn and A ⊂ Bn nonempty, 5-invariant. Then ω+ (A) = A. Proof. We consider the computation function γ ∈ n defined by ∀k ∈ N, γ k = (1, ..., 1). We notice that, from Theorem 136, page 165, ∀μ ∈ A, the state φ γ (μ, ·) is periodic, thus μ is Poisson stable O γ (μ) = ωγ (μ).

(24.3.1)

On the other hand A is 3-invariant, thus for any α ∈ n ,  O α (μ) ⊂ A, A⊂

(24.3.2)

μ∈A

in particular A⊂



O γ (μ) ⊂ A,

(24.3.3)

μ∈A

wherefrom



ωγ (μ)

μ∈A

(24.3.1)

=



O γ (μ)

μ∈A

(24.3.3)

=

A.

(24.3.4)

Chapter 24 • Basins of attraction

We infer A

(24.3.2)

=

 

O α (μ) ⊃

α∈n μ∈A

 

ωα (μ) ⊃

α∈n μ∈A



ωγ (μ)

(24.3.4)

=

225

A.

μ∈A

Theorem 188. For any μ ∈ Bn and α, γ ∈ n , we have: (i) W 1 (O α (μ)) = W 1 (ωα (μ)), (ii) W 1 (Bn ) = Wγ2 (Bn ) = W 3 (Bn ) = W 4 (Bn ) = Bn , (iii) if A ⊂ A ⊂ Bn , A = ∅, then ∀k ∈ {1, 3, 4, 5}, W k (A) ⊂ W k (A ) and Wγ2 (A) ⊂ Wγ2 (A ) hold, (iv) if A ⊂ Bn , A = ∅ fulfills that one of W 1 (A), Wγ2 (A), ..., W 5 (A) is nonempty, then B ⊂ A nonempty, 1-invariant exists, (v) let A ⊂ Bn nonempty; if B ⊂ A, B = ∅, 5-invariant exists, then W 1 (A), Wγ2 (A), ..., 5 W (A) are all nonempty. Special case: A contains a fixed point of . Proof. (i) We prove W 1 (O α (μ)) ⊂ W 1 (ωα (μ)) only, but see (iii) also, together with its proof, for the inverse inclusion. Let μ ∈ W 1 (O α (μ)) arbitrary, thus β ∈ n , k  , k  ∈ N exist such that ωβ (μ ) ⊂ O α (μ), φ β (μ , k  ) = φ α (μ, k  ). Case k  = 0 This case corresponds to μ ∈ O α (μ). We define 

δ k = α k+k , k ≥ 0 and we infer φ δ (μ , k) = φ α (μ, k + k  ), k ≥ 0. Case k  ≥ 1 We define

 δ = k

and we obtain φ δ (μ , k) =



(24.3.5)

β k , if k ∈ {0, ..., k  − 1},   α k−k +k , if k ≥ k  φ β (μ , k), if k ∈ {0, ..., k  }, φ α (μ, k − k  + k  ), if k ≥ k  .

From (24.3.5), (24.3.6) we get that ωδ (μ ) = ωα (μ). We have proved the existence of δ with ωδ (μ ) ⊂ ωα (μ), i.e. μ ∈ W 1 (ωα (μ)).

(24.3.6)

226

Boolean Systems

(ii) The equality W 3 (Bn ) = Bn means the truth of ∀μ ∈ Bn , ∀α ∈ n , ωα (μ) ⊂ Bn , which is obvious. We apply Theorem 187 (b), page 222. (iii) Let k = 1 and μ ∈ W 1 (A) arbitrary. From ∃α ∈ n , ωα (μ) ⊂ A and A ⊂ A we infer that μ ∈ W 1 (A ). (iv) By taking into account Theorem 187 (b), the hypothesis states that W 1 (A) = ∅, and this means the existence of μ ∈ Bn , α ∈ n with ωα (μ) ⊂ A. The set B = ωα (μ) is nonempty and 1-invariant. (v) The hypothesis states the existence of the nonempty, 5-invariant set B ⊂ A. As far as ω+ (B)

Lemma 2

=

B, we infer B ⊂ W 5 (A) and we take into account Theorem 187 (b).

Theorem 189. Given  : Bn → Bn and A ⊂ Bn nonempty, we have ∀k ∈ {1, 3, 4, 5}, W k (A) = W k (Akmax ). Proof. We must prove ∀k ∈ {1, 3, 4, 5} that W k (A) ⊂ W k (Akmax ),

(24.3.7)

since the inverse inclusion results from Akmax ⊂ A and from Theorem 188 (iii). k = 1: We know from Corollary 10, page 189 that A1max = {μ|μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A}, and the statement (24.3.7) ∀μ ∈ W 1 (A), μ ∈ W 1 (A1max ) has the following hypothesis, with μ ∈ Bn arbitrary, fixed: α ∈ n exists, such that ωα (μ) ⊂ A,

(24.3.8)

∃β ∈ n , ωβ (μ) ⊂ A1max .

(24.3.9)

and the following conclusion:

We prove (24.3.9) with β = α, i.e. we prove the truth of ∀λ ∈ ωα (μ), λ ∈ A and ∃γ ∈ n , O γ (λ) ⊂ A.

(24.3.10)

We take λ ∈ ωα (μ) arbitrary, fixed. The fact that λ ∈ A results from (24.3.8). On the other hand, we have the existence of k  ∈ N with λ = φ α (μ, k  ), ωα (μ) = {φ α (μ, k)|k ≥ k  }.

Chapter 24 • Basins of attraction

227



For γ ∈ n defined as γ = σ k (α), we infer ∀k ∈ N, φ γ (λ, k) = φ σ

k  (α)

(φ α (μ, k  ), k) = φ α (μ, k + k  ),

therefore O γ (λ) = ωα (μ)

(24.3.8)

⊂

A.

(24.3.10) is proved. k = 4: First, we recall from Corollary 10 that A4max = {μ|∃B ⊂ A, μ ∈ B and ∀λ ∈ n B , λ (B) ⊂ B}, and we must prove, see (24.3.7): ∀μ ∈ W 4 (A), μ ∈ W 4 (A4max ).

(24.3.11)

In (24.3.11) the hypothesis is, for an arbitrary, fixed μ ∈ Bn , that B ⊂ Bn exists such that, see Remark 157, page 223 μ ∈ B,

(24.3.12)

ω+ (B) ⊂ A

(24.3.13)

and the conclusion to be proved is that B  ⊂ Bn exists such that μ ∈ B ,

(24.3.14)

ω+ (B  ) ⊂ {ν|∃C ⊂ A, ν ∈ C and ∀λ ∈ Bn , λ (C) ⊂ C}.

(24.3.15)

We prove the conclusion for B  = B and C = ω+ (B) when, taking into account (24.3.12), (24.3.13), we must show the truth of ω+ (B) ⊂ {ν|ν ∈ ω+ (B) and ∀λ ∈ Bn , λ (ω+ (B)) ⊂ ω+ (B)}, i.e. all that we must prove is the 4-invariance of ω+ (B): ∀λ ∈ Bn , λ (ω+ (B)) ⊂ ω+ (B).   ωα (δ), we infer the existence For this, let λ ∈ Bn and μ ∈ ω+ (B) arbitrary. As ω+ (B) = α∈n δ∈B

of α ∈ n , δ ∈ B and k  ∈ N such that μ = φ α (δ, k  ),

ωα (δ) = {φ α (δ, k)|k ≥ k  }, λ = αk



228

Boolean Systems

hold. In these circumstances, k

λ (μ ) = λ (φ α (δ, k  )) = α (φ α (δ, k  )) = φ α (δ, k  + 1) ∈ ωα (δ) ⊂ ω+ (B). (24.3.11) is proved. Theorem 190. For any k ∈ {1, 3, 4, 5} and A ⊂ Bn , A = ∅, we have Akmax ⊂ W k (A). Proof. We use Corollary 10, page 189. For k = 1, the inclusion to prove is A1max = {μ|μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A} ⊂ {μ|μ ∈ Bn , ∃α ∈ n , ωα (μ) ⊂ A} = W 1 (A). Indeed, for any μ ∈ A1max , we get μ ∈ A and the existence of α ∈ n such that O α (μ) ⊂ A. As ωα (μ) ⊂ O α (μ) ⊂ A, we have obtained that μ ∈ W 1 (A). For k = 4, we must prove A4max = {μ|∃B ⊂ A, μ ∈ B and ∀λ ∈ Bn , λ (B) ⊂ B} ⊂ {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A} = W 4 (A) and let μ ∈ A4max arbitrary, for which B ⊂ A exists such that μ ∈ B and ∀λ ∈ Bn , λ (B) ⊂ B. Then O α (μ) ⊂ B is true, with arbitrary α ∈ n , therefore ωα (μ) ⊂ O α (μ) ⊂ B ⊂ A. The fact that previously μ and α were arbitrary makes us conclude that ω+ (B) ⊂ A, i.e. μ ∈ W 4 (A). For k = 5, the inclusion to be proved is A5max = {μ|∃B ⊂ A, μ ∈ B and ∀λ ∈ Bn , λ (B) = B} ⊂ {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) = B} = W 5 (A). We take μ ∈ A5max arbitrary, thus B ⊂ A exists with μ ∈ B and ∀λ ∈ Bn , λ (B) = B. Because ω+ (B)

Lemma 2, page 224

=

B ⊂A

we get μ ∈ W 5 (A). Theorem 191. We consider the set A ⊂ Bn , A = ∅, and the computation function γ ∈ n . Then ∀k ∈ {1, 3, 4, 5}, if A is k-invariant then A ⊂ W k (A). In addition, if A fulfills the 2invariance ∀μ ∈ A, O γ (μ) ⊂ A, then A ⊂ Wγ2 (A).

(24.3.16)

Chapter 24 • Basins of attraction

229

Proof. For k ∈ {1, 3, 4, 5}, the k-invariance of A implies A = Akmax , see Theorem 148, page 186. We apply Theorem 190. We suppose now that (24.3.16) is true, and let μ ∈ A arbitrary. We have ωγ (μ) ⊂ O γ (μ) ⊂ A, thus μ ∈ Wγ2 (A). Theorem 192. Let A ⊂ Bn , A = ∅. If A is attractive, then its basin of attraction is invariant: ∀k ∈ {1, 3, 4, 5}, W k (A) = ∅ =⇒ W k (A) is k-invariant. Proof. k = 1: We suppose that W 1 (A) = ∅ and we must show that ∀μ ∈ W 1 (A), ∃α ∈ n , O α (μ) ⊂ W 1 (A), i.e. ∀μ ∈ W 1 (A), ∃α ∈ n , ∀μ ∈ O α (μ), μ ∈ W 1 (A). Let μ ∈ W 1 (A) arbitrary, thus from the definition of W 1 (A), α ∈ n exists such that ωα (μ) ⊂ A.

(24.3.17)

We have to show the existence of β ∈ n with ∀μ ∈ O β (μ), ∃γ ∈ n , ωγ (μ ) ⊂ A.

(24.3.18)

We prove (24.3.18) for β = α and let us take μ ∈ O α (μ) arbitrary, for which k  ∈ N exists such  that μ = φ α (μ, k  ). We define γ by γ = σ k (α). Then ∀k ≥ k  , φ α (μ, k) = φ γ (μ , k − k  ), thus ωα (μ) = ωγ (μ ) and (24.3.18) is a consequence of (24.3.17). k = 4: We suppose that W 4 (A) = ∅. In order to prove ∀λ ∈ Bn , λ (W 4 (A)) ⊂ W 4 (A),

(24.3.19)

let first λ, μ ∈ Bn and β ∈ n arbitrary. Then α ∈ n exists, namely  α = k

λ, if k = 0, β k−1 , if k ≥ 1,

for which we have ∀k ≥ 1, φ β (λ (μ), k − 1) = φ α (μ, k), therefore ωβ (λ (μ)) = ωα (μ). We have just proved that   ωβ (λ (μ)) ⊂ ωα (μ). (24.3.20) β∈n

α∈n

We take now in (24.3.19) λ ∈ Bn and μ ∈ W 4 (A) arbitrary, thus B ⊂ Bn exists with the property that μ ∈ B and ω+ (B) ⊂ A.

(24.3.21)

230

Boolean Systems

Then for B  = {λ (μ)} we have λ (μ) ∈ B  and ω+ (B  ) =

 β∈n

ωβ (λ (μ))

(24.3.20)

⊂



ωα (μ) ⊂ ω+ (B)

(24.3.21)

⊂

A,

α∈n

showing that λ (μ) ∈ W 4 (A). k = 5: We ask that W 5 (A) = ∅ and we must prove ∀λ ∈ Bn , λ (W 5 (A)) = W 5 (A). For this, we notice the existence of B1 , ..., Bp ⊂ Bn with W 5 (A) ⊂ B1 ∨ ... ∨ Bp and ∀i ∈ {1, ..., p}, ω+ (Bi ) ⊂ A,

(24.3.22)

∀λ ∈ Bn , λ (Bi ) = Bi .

(24.3.23)

The inclusion B1 ∨...∨Bp ⊂ W 5 (A) follows from Lemma 2, page 224, (24.3.22) and (24.3.23), which give ∀i ∈ {1, ..., p}, Bi ⊂ A. The 5-invariance of W 5 (A) is a consequence of the 5-invariance of B1 , ..., Bp . Remark 160. In Theorem 192, γ ∈ n exists such that the property Wγ2 (A) = ∅ =⇒ ∀μ ∈ Wγ2 (A), O γ (μ) ⊂ Wγ2 (A)

(24.3.24)

does not hold, see Example 134, page 223, where for γ = (1, 1, 1), (1, 0, 1), (1, 1, 1), (1, 1, 1), ... and A = {(0, 0, 1)}, we have Wγ2 (A) = {(0, 0, 0), (1, 1, 1), (0, 0, 1)}, O γ (0, 0, 0) = {(0, 0, 0), (1, 1, 0), (1, 1, 1), (0, 0, 1)} ⊂ Wγ2 (A). Corollary 14. For any k ∈ {1, 3, 4, 5}, if the nonempty set A ⊂ Bn is k-invariant, then its basin of attraction is k-invariant, and we have W k (A) ⊂ W k (W k (A)). Proof. We fix an arbitrary k ∈ {1, 3, 4, 5}. If A = ∅ is k-invariant then, from Theorem 191, page 228, A ⊂ W k (A). This means that W k (A) = ∅ thus, from Theorem 192, W k (A) is kinvariant. We infer, from Theorem 191, that W k (A) ⊂ W k (W k (A)). Theorem 193. The function  : Bn → Bn and A ⊂ Bn , A = ∅ are given. If W 5 (A) = ∅, we have W 5 (A) = A5max .

Chapter 24 • Basins of attraction

231

Proof. This is a consequence of W 5 (A) = {μ|∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ A and ∀λ ∈ Bn , λ (B) = B} Lemma 2, page 224

=

{μ|∃B ⊂ A, μ ∈ B and ∀λ ∈ Bn , λ (B) = B} = A5max .

Corollary 15. If A = ∅ is 5-invariant, then W 5 (A) = A. Proof. If A is 5-invariant, then A = A5max , thus W 5 (A)

Theorem 193

=

A5max = A.

24.4 The basin of attraction of the fixed points Notation 38. For arbitrary μ ∈ Bn and γ ∈ n we use the simpler notations W 1 (μ), Wγ2 (μ), ..., W 5 (μ) instead of W 1 ({μ}), Wγ2 ({μ}), ..., W 5 ({μ}). Furthermore, if the point μ is identified with the n-tuple (μ1 , ..., μn ), it is usual to write W 1 (μ1 , ..., μn ), Wγ2 (μ1 , ..., μn ), ..., W 5 (μ1 , ..., μn ) for these sets. Remark 161. This section is dedicated to the special case when in Definition 105, page 221 the nonempty set A ⊂ Bn consists in a point μ, in other words W 1 (μ) = {μ |μ ∈ Bn , ∃α ∈ n , ωα (μ ) ⊂ {μ}}, Wγ2 (μ) = {μ |μ ∈ Bn , ωγ (μ ) ⊂ {μ}}, W 3 (μ) = {μ |μ ∈ Bn , ∀α ∈ n , ωα (μ ) ⊂ {μ}}, W 4 (μ) = {μ |∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ {μ}}, W 5 (μ) = {μ |∃B ⊂ Bn , μ ∈ B and ω+ (B) ⊂ {μ} and ∀λ ∈ Bn , λ (B) = B}. We shall prove that attractiveness coincides with the fact that (μ) = μ. Theorem 194. Let2 μ ∈ Bn and γ ∈ n . The following statements are true: (a) We have W 1 (μ) = {μ |μ ∈ Bn , ∃α ∈ n , lim φ α (μ , k) = μ}, k→∞

Wγ2 (μ) = {μ |μ ∈ Bn , lim φ γ (μ , k) = μ}, k→∞

2

This theorem is a version of the stable manifold theorem, see for example [6], page 82.

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Boolean Systems

W 3 (μ) = W 4 (μ) = {μ |μ ∈ Bn , ∀α ∈ n , lim φ α (μ , k) = μ}, k→∞

 W 5 (μ) =

∅, if (μ) = μ, {μ}, otherwise.

(b) The attractiveness of {μ}, expressed by any of W 1 (μ) = ∅, Wγ2 (μ) = ∅, ..., W 5 (μ) = ∅ is equivalent with the fact that μ is a fixed point of . (c) We suppose that the attractiveness proprieties W 1 (μ) = ∅, Wγ2 (μ) = ∅, ..., W 5 (μ) = ∅ hold. Then {μ} = W 5 (μ) ⊂ W 3 (μ) ⊂ Wγ2 (μ) ⊂ W 1 (μ)

(24.4.1)

take place and the following invariance statements are also fulfilled: ∀μ ∈ W 1 (μ), ∃α ∈ n , O α (μ ) ⊂ W 1 (μ),

(24.4.2)

∀μ ∈ Wγ2 (μ), O γ (μ ) ⊂ Wγ2 (μ),

(24.4.3)

∀μ ∈ W 3 (μ), ∀α ∈ n , O α (μ ) ⊂ W 3 (μ),

(24.4.4)

∀λ ∈ Bn , λ (μ) = μ.

(24.4.5)

Proof. (a) Case k = 2. Let μ ∈ Wγ2 (μ) arbitrary, thus ωγ (μ ) ⊂ {μ}. As ωγ (μ ) has at least one element and at most one element, we get that ωγ (μ ) = {μ}. This means that ∃k  ∈ N, ∀k ≥ k  , φ γ (μ , k) = μ. (b) Case k = 2. =⇒ We take μ ∈ Wγ2 (μ) arbitrary satisfying lim φ γ (μ , k) = μ. Theok→∞

rem 89, page 104 implies that (μ) = μ. ⇐= If (μ) = μ, then ωγ (μ) = {μ}, thus μ ∈ Wγ2 (μ). (c) We have W 5 (μ) = {μ}, and on the other hand (24.4.1) results from Theorem 187 (b), page 222. The fact that the basins of attraction are invariant was stated at Theorem 192, page 229. Example 135. In the next state portrait we have the identity 1B2 : B2 −→ B2 where all the points are fixed points and ∀μ ∈ B2 , ∀γ ∈ 2 , (0, 0)

(0, 1)

(1, 0)

(1, 1)

W 1 (μ) = Wγ2 (μ) = W 3 (μ) = W 5 (μ) = {μ}. Any μ ∈ B2 is (partially) attractive.

Chapter 24 • Basins of attraction

233

Example 136. The points (0, 0), (1, 0) are fixed in the state portrait (0, 0) 

(0, 1)      ?  + (1, 1) (1, 0)  and W 1 (0, 0) = Wγ2 (0, 0) = {(0, 0), (0, 1)}, W 3 (0, 0) = W 5 (0, 0) = {(0, 0)}, where we have denoted with γ the sequence γ = (0, 1), (1, 1), (1, 1), ..., respectively W 1 (1, 0) = Wγ2 (1, 0) = {(0, 1), (1, 1), (1, 0)}, W 3 (1, 0) = {(1, 0), (1, 1)}, W 5 (1, 0) = {(1, 0)}, where γ k = (1, 1), k ∈ N. Example 137. We have also the example of the following system, where for arbitrary γ ∈ 2 we get: (0, 0)

(1, 0) 

- (0, 1)

? (1, 1)

W 1 (1, 0) = Wγ2 (1, 0) = W 3 (1, 0) = B2 , thus the fixed point (1, 0) is totally attractive in three different ways. On the other hand W 5 (1, 0) = {(1, 0)}. Remark 162. An interesting special case of Definition 105, page 221 is the one when all the points of A are fixed points of . This is also a version of the basins of attraction of the present section, when instead of a fixed point, we have a set of fixed points.

24.5 The basin of attraction of the periodic points Definition 107. The point μ ∈ Bn is said to be a periodic point of  : Bn → Bn with the period p ≥ 1, if (p) (μ) = μ.

234

Boolean Systems

Remark 163. So far, the concept of periodicity referred in general to a signal, see Definition 52, page 50, and we have briefly studied the case when the signal was the state of a system. We have related periodicity also, in Definition 84, page 133, with the possibility that the values μ ∈ O α (μ) of the state repeat in an asynchronous framework, these are the periodic points. Note that Definition 107 has essentially a synchronous origin. Remark 164. Note that, from a synchronous point of view, all the points μ ∈ Bn are eventually periodic (∃p ≥ 1, ∃k  ∈ N, ∀k ≥ k  , (p) ((k) (μ)) = (k) (μ)), and that periodic points always exist. Definition 108. Let  : Bn → Bn , γ ∈ n and the periodic point μ ∈ Bn : (p) (μ) = μ, with p ≥ 1. We define the following basins of attraction of μ: W 1 (μ) = {μ |μ ∈ Bn , ∃α ∈ n , lim ψ α (μ , k) = μ}, k→∞

Wγ2 (μ) = {μ |μ ∈ Bn , lim ψ γ (μ , k) = μ}, k→∞

W 3 (μ) = W 4 (μ) = {μ |μ ∈ Bn , ∀α ∈ n , lim ψ α (μ , k) = μ},  W 5 (μ) =

k→∞

∅, if (μ) = μ, {μ}, otherwise,

and we have used the notation  = (p) . Remark 165. The periodic points of  are fixed points of  and their basins of attraction from Definition 108 coincide with those of the fixed points from Theorem 194, page 231. Example 138. We notice that (0, 1, 0), (0, 0, 1) ∈ B3 are periodic points of  : B3 → B3 , ∀μ ∈ B3 , (μ1 , μ2 , μ3 ) = (μ1 μ2 , μ3 , μ2 ) and they have the period 2. The state portrait of  =  ◦ , ∀μ ∈ B3 , (μ1 , μ2 , μ3 ) = (μ1 μ2 μ3 , μ2 , μ3 ) was drawn below. (1, 0, 0)

(1, 1, 0)

(1, 0, 1)

(1, 1, 1)

? (0, 0, 0)

? (0, 1, 0)

? (0, 0, 1)

(0, 1, 1)

For any γ ∈ 3 , we have: W 1 (0, 1, 0) = Wγ2 (0, 1, 0) = W 3 (0, 1, 0) = W 4 (0, 1, 0) = {(0, 1, 0), (1, 1, 0)}, W 1 (0, 0, 1) = Wγ2 (0, 0, 1) = W 3 (0, 0, 1) = W 4 (0, 0, 1) = {(1, 0, 1), (0, 0, 1)}, W 5 (0, 1, 0) = {(0, 1, 0)}, W 5 (0, 0, 1) = {(0, 0, 1)}.

Chapter 24 • Basins of attraction

235

24.6 Isomorphisms Theorem 195. The functions ,  : Bn −→ Bn , the set A ⊂ Bn , A = ∅ and the isomorphism (h, h ) : φ → ψ are given. Then ∀γ ∈ n , h(Wγ2, (A)) = Wh2 (γ ), (h(A)) and ∀k ∈ {1, 3, 4, 5}, we have h(Wk (A)) = Wk (h(A)). Proof. Case k = 1. In order to prove that h(W1 (A)) ⊂ W1 (h(A)), we take an arbitrary μ ∈ W1 (A), and this means the existence of α ∈ n such that α (μ) ⊂ A. We infer h(ωα (μ)) ⊂ h(A). From Theorem 78, page 91, we know that h(ωα (μ)) = ω   h (α)

ω (h(μ)), therefore β ∈ n exists, β = h (α), with ω (h(μ)) ⊂ h(A). We get h(μ) ∈ W1 (h(A)). We prove β

W1 (h(A)) ⊂ h(W1 (A)), β

and we take an arbitrary ν ∈ W1 (h(A)). We have the existence of β ∈ n such that ω (ν) ⊂ h(A). Furthermore, μ ∈ Bn and α ∈ n exist with h(μ) = ν and h (α) = β, for which h (α) α (μ)) ⊂ h(A), thus ωα (μ) ⊂ A. It ω (h(μ)) ⊂ h(A). We infer from Theorem 78 that h(ω  1 has resulted that μ ∈ W (A). Case k = 5. We prove h(W5 (A)) ⊂ W5 (h(A)),

(24.6.1)

and we take an arbitrary μ ∈ W5 (A). We have the existence of B ⊂ Bn such that μ ∈ B, + ω (B) ⊂ A,

(24.6.2)

∀λ ∈ Bn , λ (B) = B.

(24.6.3)

We denote B  = h(B), for which we have h(μ) ∈ B  and we must prove that + ω (B  ) ⊂ h(A),

(24.6.4)

∀λ ∈ Bn ,  λ (B  ) = B  .

(24.6.5)

α ω (ν) ⊂ A

(24.6.6)

We infer from (24.6.2) that

236

Boolean Systems

h (α)

α (ν)) = ω is true for any ν ∈ B and any α ∈ n . As h(ω  inclusion (24.6.6) implies that h (α)

ω B

(h(ν)), from Theorem 78, page 91,

(h(ν)) ⊂ h(A).

h (α)

(24.6.7) h

In (24.6.7), h(ν) runs in and runs in n , because h and are bijections. We infer that (24.6.4) is true. The fact that (24.6.3) implies (24.6.5) is a consequence of Theorem 138, page 167. In order to prove the inclusion W5 (h(A)) ⊂ h(W5 (A)), we take μ ∈ W5 (h(A)) arbitrary, and this means the existence of B  ⊂ Bn with μ ∈ B  , (24.6.4) and (24.6.5) true. We must prove that μ = h−1 (μ) belongs to W5 (A), i.e. B ⊂ Bn exists such that μ ∈ B, (24.6.2) and (24.6.3) hold. We follow the same steps like in the proof of (24.6.1).

25 Basins of attraction of the states The basin of attraction of the state φ α (μ, ·) is introduced in Section 25.1 as the set of the initial values μ of the states φ β (μ , ·) which are omega-limit equivalent with φ α (μ, ·) and several examples are given in Section 25.2. The most important properties of these basins of attraction are found in Section 25.3. Section 25.4 shows that the isomorphisms map basins of attraction in basins of attraction.

25.1 Definition Definition 109. Let  : Bn → Bn , μ ∈ Bn and α, γ ∈ n . We define the basins (or kingdoms, or domains) of attraction of φ α (μ, ·) by W 1 [φ α (μ, ·)] = {μ |μ ∈ Bn , ∃β ∈ n , φ β (μ , ·) ≈ φ α (μ, ·)},

(25.1.1)

Wγ2 [φ α (μ, ·)] = {μ |μ ∈ Bn , φ γ (μ , ·) ≈ φ α (μ, ·)},

(25.1.2)

W 3 [φ α (μ, ·)] = {μ |μ ∈ Bn , ∀β ∈ n , φ β (μ , ·) ≈ φ α (μ, ·)},

(25.1.3)

W 4 [φ α (μ, ·)] = {μ |∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) ⊂ B},

(25.1.4)

W 5 [φ α (μ, ·)] = {μ |∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) = B}.

(25.1.5)

Remark 166. In (25.1.1)–(25.1.5) we have used the symbol ≈ of the omega-limit equivalence of the signals, see Definition 48, page 46. We have also used the brackets [φ α (μ, ·)] instead of the parentheses (φ α (μ, ·)) in order to make these basins of attraction as distinct as possible from the previous basins, so that confusions are not likely to happen. Remark 167. The basins of attraction and the attractiveness of the states1 of a system are defined in the spirit of Definitions 105, 106, page 221. The interpretation of (25.1.1)–(25.1.5) is: a point μ ∈ Bn belongs to the basin of attraction of a state, if it is the initial value of an omega-limit equivalent state. 1

This is the attractiveness of an orbit through a point, see [12], page 133, which in our framework refers to the state φ α (μ, ·). Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00031-3 Copyright © 2023 Elsevier Inc. All rights reserved.

237

238

Boolean Systems

Remark 168. If φ α (μ, ·) ≈ φ β (μ , ·), then their basins of attraction are equal. Theorem 196. Let α, γ ∈ n and μ, μ ∈ Bn arbitrary. (a) We have the implications ∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) = B =⇒ ∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) ⊂ B ⇐⇒ ∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) ⇐⇒ ∀β ∈ n , φ β (μ , ·) ≈ φ α (μ, ·) =⇒ φ γ (μ , ·) ≈ φ α (μ, ·) =⇒ ∃β ∈ n , φ β (μ , ·) ≈ φ α (μ, ·). (b) The following inclusions take place W 5 [φ α (μ, ·)] ⊂ W 4 [φ α (μ, ·)] = W 3 [φ α (μ, ·)] ⊂ Wγ2 [φ α (μ, ·)] ⊂ W 1 [φ α (μ, ·)]. Proof. (a) We prove the only implication which is not obvious ∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) =⇒ ∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) ⊂ B. Sets B ⊂ Bn exist such that μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and let B1 , ..., Bq be these sets. We denote B  = B1 ∨ ... ∨ Bq and we see that μ ∈ B  and ∀ν ∈ B  , ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·)

(25.1.6)

hold. We propose to prove that ∀λ ∈ Bn , λ (B  ) ⊂ B  .

(25.1.7)

We take in (25.1.7) λ ∈ Bn and ν  ∈ B  arbitrary. At the same time, we take δ ∈ n arbitrary, such that δ 0 = λ. From (25.1.6) written for ν = ν  and β = δ, we infer φ δ (ν  , ·) ≈ φ α (μ, ·).

(25.1.8)

Because ∀k ∈ N, φ δ (ν  , k + 1) = φ σ

1 (δ)

(φ δ (ν  , 1), k) = φ σ

1 (δ)

(λ (ν  ), k),

Chapter 25 • Basins of attraction of the states

239

we infer φ δ (ν  , ·)

Remark 63, page 64

≈

φσ

1 (δ)

(λ (ν  ), ·),

(25.1.9)

thus φσ From the way that B 

1 (δ)

(λ (ν  ), ·)

(25.1.8),(25.1.9)

≈

φ α (μ, ·).

(25.1.10)

was defined, (25.1.10) implies λ (ν  ) ∈ B  . (25.1.7) is true.

Statement (b) is a consequence of (a). Remark 169. Theorem 196 shows that W 4 [φ α (μ, ·)] may be equivalently defined as W 4 [φ α (μ, ·)] = {μ |∃B ⊂ Bn , μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·)}.

25.2 Examples Example 139. In the next state portrait (0, 0)  6 Q  Q  Q  Q  Q  + s Q (1, 0) (0, 1) Q k 3  Q  Q  Q  Q ?  Q  (1, 1) for α k = (1, 1), k ∈ N we get W 1 [φ α ((0, 0), ·)] = {(0, 0), (1, 1)},  Wγ2 [φ α ((0, 0), ·)] =

{(0, 0), (1, 1)}, if ∀k ∈ N, γ1k = γ2k , ∅, otherwise,

W 3 [φ α ((0, 0), ·)] = W 4 [φ α ((0, 0), ·)] = W 5 [φ α ((0, 0), ·)] = ∅. If α = (0, 1), (1, 0), (0, 1), (1, 0), ... then W 1 [φ α ((0, 0), ·)] = {(0, 0), (0, 1), (1, 1)},

Wγ2 [φ α ((0, 0), ·)] =

⎧ ⎨

?, if γ k1 = (1, 1), {(0, 0), (0, 1)}, if γ k1 = (0, 1), ⎩ {(1, 1), (0, 1)}, if γ k1 = (1, 0),

240

Boolean Systems

where k1 = min {k|k ∈ N, γ1k ∪ γ2k = 1}, and W 3 [φ α ((0, 0), ·)] = W 4 [φ α ((0, 0), ·)] = W 5 [φ α ((0, 0), ·)] = {(0, 1)}. On the other hand, for an arbitrary α we infer W 1 [φ α ((1, 0), ·)] = {(0, 0), (1, 1), (1, 0)},

Wγ2 [φ α ((1, 0), ·)] =

⎧ ⎨

?, if γ k1 = (1, 1), {(0, 0), (1, 0)}, if γ k1 = (1, 0), ⎩ {(1, 1), (1, 0)}, if γ k1 = (0, 1),

with k1 = min {k|k ∈ N, γ1k ∪ γ2k = 1} again, and W 3 [φ α ((1, 0), ·)] = W 4 [φ α ((1, 0), ·)] = W 5 [φ α ((1, 0), ·)] = {(1, 0)}. Example 140. We take for the following system (0, 0)

(1, 0)

- (0, 1) 6 ? (1, 1)

α, γ ∈ 2 arbitrary. We get ∀μ ∈ {(0, 0), (0, 1), (1, 1)}, W 1 [φ α (μ, ·)] = Wγ2 [φ α (μ, ·)] = W 3 [φ α (μ, ·)] = W 4 [φ α (μ, ·)] = {(0, 0), (0, 1), (1, 1)}, W 5 [φ α (μ, ·)] = {(0, 1), (1, 1)}. Example 141. Let the system (0, 0)

- (0, 1)

(1, 0)

? (1, 1)

We take α, γ ∈ 2 arbitrary, μ ∈ {(0, 0), (0, 1), (1, 1)} and we see that W 1 [φ α (μ, ·)] = Wγ2 [φ α (μ, ·)] = W 3 [φ α (μ, ·)] = W 4 [φ α (μ, ·)] = {(0, 0), (0, 1), (1, 1)}, W 5 [φ α (μ, ·)] = {(1, 1)}. These basins of attraction are to be compared with the basins of attraction from Example 140.

Chapter 25 • Basins of attraction of the states

241

25.3 Properties Theorem 197. We consider  : Bn → Bn , the point μ ∈ Bn and the function α ∈ n . We have O α (μ) ⊂ W 1 [φ α (μ, ·)], in particular W 1 [φ α (μ, ·)] is nonempty. Proof. We take μ ∈ O α (μ) arbitrary, thus k  ∈ N exists such that μ = φ α (μ, k  ). We define  β = σ k (α), and we see that ∀k ≥ k  , φ β (μ , k − k  ) = φ α (μ, k). We have (see Example 34, page 46) that φ β (μ , ·) ≈ φ α (μ, ·), thus μ ∈ W 1 [φ α (μ, ·)]. Theorem 198. For any μ ∈ Bn and any α, γ ∈ n , (a) the basin of attraction W 1 [φ α (μ, ·)] is 1-invariant; (b) any of the attractiveness properties Wγ2 [φ α (μ, ·)] = ∅, ..., W 5 [φ α (μ, ·)] = ∅ implies the k-invariance of that basin of attraction, k ∈ {2, ..., 5}. Proof. (a) We prove that ∀μ ∈ W 1 [φ α (μ, ·)], ∃β ∈ n , ∀μ ∈ O β (μ ), μ ∈ W 1 [φ α (μ, ·)]. Let μ ∈ W 1 [φ α (μ, ·)] arbitrary, fixed, thus ρ ∈ n exists such that φ ρ (μ , ·) ≈ φ α (μ, ·).

(25.3.1)

We must show the existence of β ∈ n with ∀μ ∈ O β (μ ), ∃δ ∈ n , φ δ (μ , ·) ≈ φ α (μ, ·),

(25.3.2)

and we prove the truth of (25.3.2) for β = ρ. Indeed, let μ ∈ O ρ (μ ) arbitrary; k  ∈ N exists  then having the property that μ = φ ρ (μ , k  ). We define δ = σ k (ρ) and we can see that ∀k ≥ k  , φ δ (μ , k − k  ) = φ ρ (μ , k). φ δ (μ , ·)

(25.3.3)

φ ρ (μ , ·)

≈ and, taking into account (25.3.1), we infer Eq. (25.3.3) shows that φ δ (μ , ·) ≈ φ α (μ, ·). We conclude that μ ∈ W 1 [φ α (μ, ·)]. (b) Case k = 5. The hypothesis states the existence of μ ∈ W 5 [φ α (μ, ·)], thus B ⊂ Bn exists with μ ∈ B, ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) = B satisfied. We denote with B1 , ..., Bp ⊂ Bn the sets that satisfy ∀i ∈ {1, ..., p}, ∀ν ∈ Bi , ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·),

(25.3.4)

∀λ ∈ Bn , λ (Bi ) = Bi .

(25.3.5)

W 5 [φ α (μ, ·)] ⊂ B1 ∨ ... ∨ Bp

(25.3.6)

The inclusion

242

Boolean Systems

results from the way that these sets were defined. In order to prove that B1 ∨ ... ∨ Bp ⊂ W 5 [φ α (μ, ·)],

(25.3.7)

we take an arbitrary μ ∈ B1 ∨ ... ∨ Bp , thus i ∈ {1, ..., p} exists with μ ∈ Bi and, in addition, (25.3.4), (25.3.5) hold. This means that ∀ν ∈ Bi , ν ∈ W 5 [φ α (μ, ·)], i.e. Bi ⊂ W 5 [φ α (μ, ·)], therefore (25.3.7) follows. From (25.3.6) and (25.3.7) we get W 5 [φ α (μ, ·)] = B1 ∨ ... ∨ Bp .

(25.3.8)

Then W 5 [φ α (μ, ·)] is 5-invariant, from (25.3.8) and the truth ∀i ∈ {1, ..., p} of (25.3.5). Theorem 199. The point μ ∈ Bn and α, γ ∈ n are given. We have ∀k ∈ {1, 3, 4, 5}, W k [φ α (μ, ·)] ⊂ W k (O α (μ)), and Wγ2 [φ α (μ, ·)] ⊂ Wγ2 (O α (μ)). Proof. Case k = 1. We take an arbitrary μ ∈ W 1 [φ α (μ, ·)], meaning that β ∈ n exists such that φ β (μ , ·) ≈ φ α (μ, ·). Then ωβ (μ )

Theorem 41, page 46

=

ωα (μ) ⊂ O α (μ),

therefore μ ∈ W 1 (O α (μ)). Case k = 5. For an arbitrary μ ∈ W 5 [φ α (μ, ·)], the set B ⊂ Bn exists such that μ ∈ B, ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·),

(25.3.9)

∀λ ∈ Bn , λ (B) = B. (25.3.9) implies that ω+ (B) = ωα (μ) ⊂ O α (μ), wherefrom μ ∈ W 5 (O α (μ)). Theorem 200. We suppose that μ, μ ∈ Bn and α ∈ n fulfill lim φ α (μ, k) = μ . For any γ ∈ k→∞

n and any k ∈ {1, 3, 4, 5}, W k (μ ) = W k [φ α (μ, ·)], and on the other hand Wγ2 (μ ) = Wγ2 [φ α (μ, ·)].

Proof. Case k = 2. We prove that Wγ2 (μ ) ⊂ Wγ2 [φ α (μ, ·)] by noting that an arbitrary, fixed μ ∈ Wγ2 (μ ) satisfies ωγ (μ ) ⊂ {μ }, i.e. we have lim φ γ (μ , k) = μ . Then k  ∈ N exists with k→∞

∀k ≥ k  , φ γ (μ , k) = μ . As far as ∃k  ∈ N, ∀k ≥ k  , φ α (μ, k) = μ , we infer that φ γ (μ , ·) ≈ φ α (μ, ·), meaning that μ ∈ Wγ2 [φ α (μ, ·)].

Chapter 25 • Basins of attraction of the states

243

The inclusion Wγ2 [φ α (μ, ·)] ⊂ Wγ2 (μ ) is proved similarly. Case k = 5. We have (μ ) = μ (from lim φ α (μ, k) = μ ), Theorem 194, page 231 implies k→∞

W 5 (μ ) = {μ } and we prove that W 5 [φ α (μ, ·)] = {μ } too. We prove first that μ ∈ W 5 [φ α (μ, ·)]: indeed, μ ∈ {μ }, ∀ν ∈ {μ }, ∀β ∈ n , lim φ β (ν, k) = k→∞

μ , thus φ β (ν, ·) ≈ φ α (μ, ·) and on the other hand ∀λ ∈ Bn , λ ({μ }) = {μ }, thus {μ } is 5invariant. We suppose now against all reason that μ = μ exists, μ ∈ W 5 [φ α (μ, ·)]. We have the existence of B ⊂ Bn with the property that μ ∈ B, ∀ν ∈ B, ∀β ∈ n , lim φ β (ν, k) = μ and k→∞

∀λ ∈ Bn , λ (B) = B. We infer that the set B ∨ {μ } satisfies the same properties, including ∀ν ∈ B ∨ {μ }, ∀β ∈ n , lim φ β (ν, k) = μ ,

(25.3.10)

∀λ ∈ Bn , λ (B ∨ {μ }) = B ∨ {μ }.

(25.3.11)

k→∞

As in (25.3.10) we can take ν = μ and β ∈ n given by ∀k ∈ N, β k = (1, ..., 1), we get the existence of k  ∈ N and μ = μ such that μ = φ β (μ , k  ) = φ β (μ , k  + 1) = (μ ) = μ = (μ ).

(25.3.12)

But (25.3.12) represents a contradiction with (25.3.11), where B ∨ {μ }  μ −→ (μ) ∈ B ∨ {μ } is bijective. Corollary 16. If μ ∈ Bn is a fixed point of  and α, γ ∈ n are arbitrary, then ∀k ∈ {1, 3, 4, 5}, W k (μ) = W k [φ α (μ, ·)], and Wγ2 (μ) = Wγ2 [φ α (μ, ·)]. Proof. This is a special case of Theorem 200 when μ = μ.

25.4 Isomorphisms Theorem 201. We consider the functions ,  : Bn −→ Bn , the isomorphism (h, h ) : φ → ψ and let α, γ ∈ n , μ ∈ Bn . We have 

h(Wγ2, [φ α (μ, ·)]) = Wh2 (γ ), [ψ h (α) (h(μ), ·)] and ∀k ∈ {1, 3, 4, 5}, we can write 

h(Wk [φ α (μ, ·)]) = Wk [ψ h (α) (h(μ), ·)].

(25.4.1)

244

Boolean Systems

Proof. We make the proof for k = 1. In order to prove 

h(W1 [φ α (μ, ·)]) ⊂ W1 [ψ h (α) (h(μ), ·)], we take an arbitrary μ ∈ W1 [φ α (μ, ·)]. The hypothesis states the existence of β ∈ n such  that φ β (μ , ·) ≈ φ α (μ, ·) and we must prove that h(μ ) ∈ W1 [ψ h (α) (h(μ), ·)], i.e. that β  ∈ n   exists with ψ β (h(μ ), ·) ≈ ψ h (α) (h(μ), ·). But this is true indeed for β  = h (β), if we take into account Theorem 82, page 94. The proof of 

W1 [ψ h (α) (h(μ), ·)] ⊂ h(W1 [φ α (μ, ·)]) is similar. For k = 5 we prove the inclusion 

h(W5 [φ α (μ, ·)]) ⊂ W5 [ψ h (α) (h(μ), ·)], and let μ ∈ W5 [φ α (μ, ·)] arbitrary, fixed. This means the existence of B ⊂ Bn with μ ∈ B and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·),

(25.4.2)

∀λ ∈ Bn , λ (B) = B.

(25.4.3)

In order to prove that h(μ ) ∈ W5 [ψ with h(μ ) ∈ B  and

h (α)

(h(μ), ·)], we must show the existence of B  ⊂ Bn 



∀ν  ∈ B  , ∀β  ∈ n , ψ β (ν  , ·) ≈ ψ h (α) (h(μ), ·), 

∀λ ∈ Bn ,  λ (B  ) = B  .

(25.4.4) (25.4.5)

We take B  = h(B) and we remark that the implication (25.4.2)=⇒(25.4.4) is a consequence of Theorem 82, while the implication (25.4.3)=⇒(25.4.5) is a consequence of Theorem 138, page 167.

26 Local basins of attraction The local basins of attraction are defined in Section 26.1 by analogy with the local stable manifolds of the equilibrium points from literature. In Section 26.2 we give some properties of the local basins of attraction, and in Section 26.3 we prove that the isomorphisms bring local basins of attraction in local basins of attraction.

26.1 Definition Definition 110. Let  : Bn → Bn , the nonempty sets A, B ⊂ Bn and γ ∈ n . The following local basins of attraction are defined: W 1 (A, B) = {μ|μ ∈ A, ∃α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B}, Wγ2 (A, B) = {μ|μ ∈ A, O γ (μ) ⊂ A and ωγ (μ) ⊂ B}, W 3 (A, B) = {μ|μ ∈ A, ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B}, W 4 (A, B) = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) ⊂ C}, W 5 (A, B) = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) = C}. Remark 170. The occurrence in Definition 110 of the requests O α (μ) ⊂ A, O γ (μ) ⊂ A, μ ∈ C and O + (C) ⊂ A is justified by the fact that A is not invariant, in general. Remark 171. The local basins of attraction which are introduced in this chapter are suggested by what is called in the real numbers’ literature local stable manifold of the equiBoolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00032-5 Copyright © 2023 Elsevier Inc. All rights reserved.

245

246

Boolean Systems

librium point x0 ∈ X.1 This gives a generalization of the basins of attraction from Definition 105, page 221 by the restriction of the orbits to a certain set. Theorem 202. The nonempty sets A, B ⊂ Bn , γ ∈ n and μ ∈ A are given. (a) The following implications hold: ∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) = C =⇒ ∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) ⊂ C ⇐⇒ ∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B ⇐⇒ ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B =⇒ O γ (μ) ⊂ A and ωγ (μ) ⊂ B =⇒ ∃α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B. (b) We have: W 5 (A, B) ⊂ W 4 (A, B) = W 3 (A, B) ⊂ Wγ2 (A, B) ⊂ W 1 (A, B). Proof. (a) We prove that ∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B =⇒ ∃C ⊂ Bn , μ ∈ C and O + (C ) ⊂ A and ω+ (C ) ⊂ B and ∀λ ∈ Bn , λ (C ) ⊂ C is true for C = O + (C). As C ⊂ Bn and μ ∈ C are obvious, we prove first that O + (C ) ⊂ O + (C) ⊂ A.

(26.1.1)

For ν ∈ O + (C ) arbitrary, we have the existence of ν ∈ C , ρ ∈ Bn , ..., δ ∈ Bn such that



ν = (ρ ◦ ... ◦ δ )(ν), thus the existence of ξ ∈ C, ρ ∈ Bn , ..., δ ∈ Bn with ν = (ρ ◦ ... ◦ δ )(ξ ),

(26.1.2)

and we conclude that



ν = (ρ ◦ ... ◦ δ ) ◦ (ρ ◦ ... ◦ δ )(ξ ), 1

In [7], page 27 and [17], page 3 for example, if U ⊂ X is a neighborhood of x0 , we have the definition W (x0 , U ) = {x|x ∈ U, ∀t ≥ 0, t (x) ∈ U and lim t (x) = x0 }. t→∞

Chapter 26 • Local basins of attraction

247

i.e. ν ∈ O + (C). The inclusion O + (C) ⊂ A is true from the hypothesis, thus (26.1.1) is true. We prove now that ω+ (C ) ⊂ ω+ (C) ⊂ B

(26.1.3)

and let ζ ∈ ω+ (C ) arbitrary. We get the existence of α ∈ n , ν ∈ C , k ∈ N and k1 ≥ k such that ωα (ν) = {φ α (ν, k)|k ≥ k },

(26.1.4)

ζ = φ α (ν, k1 ).

(26.1.5)

We have also the existence of ξ ∈ C, ρ ∈ Bn , ..., δ ∈ Bn making (26.1.2) true. Thus we infer the existence of β ∈ n and k ∈ N with ν = φ β (ξ, k ), ∀k ≥ k , φ α (ν, k − k ) = φ β (ξ, k), ωα (ν)

(26.1.6)

= ζ

ωβ (ξ )

(26.1.4)

=

(26.1.5),(26.1.6)

=

{φ β (ξ, k)|k ≥ k + k },

φ β (ξ, k1 + k ).

(26.1.6) (26.1.7) (26.1.8)

As k1 ≥ k , (26.1.7), (26.1.8) show that ζ ∈ ωβ (ξ ), therefore ζ ∈ ω+ (C). This, together with the hypothesis ω+ (C) ⊂ B, implies the truth of (26.1.3). In order to prove ∀λ ∈ Bn , λ (C ) ⊂ C , let λ ∈ Bn , ν ∈ C arbitrary. Then ξ ∈ C, ρ ∈ Bn , ..., δ ∈ Bn exist such that (26.1.2) is true, and obviously λ (ν) = λ ((ρ ◦ ... ◦ δ )(ξ )) = (λ ◦ ρ ◦ ... ◦ δ )(ξ ) fulfills λ (ν) ∈ O + (C) = C , thus C is 4-invariant. On the other hand, we see that the implication ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B =⇒ ∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B is true for C = {μ}. Remark 172. Due to the previous theorem, we can write equivalently: W 4 (A, B) = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ B}.

248

Boolean Systems

26.2 Properties Lemma 3. Let B ⊂ A ⊂ Bn , B = ∅ with the property that ∀λ ∈ Bn , λ (B) ⊂ B. Then O + (B) ⊂ A.

(26.2.1)

Proof. We take an arbitrary ν ∈ O + (B). Then ξ ∈ B, ρ ∈ Bn , ..., δ ∈ Bn exist with the property that ν = (ρ ◦ ... ◦ δ )(ξ ). We can prove by induction that δ (ξ ) ∈ B, ..., (ρ ◦ ... ◦ δ )(ξ ) ∈ B, in other words ν ∈ A. Statement (26.2.1) is true. Theorem 203.  : Bn → Bn , and A, B ∈ Bn nonempty are given. We have W 1 (A, B) ⊂ W 1 (B) ∧ A1max , and ∀k ∈ {3, 4, 5}, W k (A, B) = W k (B) ∧ Akmax . Proof. Case k = 1: W 1 (A, B) = {μ|μ ∈ A and ∃α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B} = {μ|μ ∈ Bn and μ ∈ A and ∃α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B} ⊂ {μ|μ ∈ Bn and ∃α ∈ n , ωα (μ) ⊂ B and μ ∈ A and ∃β ∈ n , O β (μ) ⊂ A} = {μ|μ ∈ Bn and ∃α ∈ n , ωα (μ) ⊂ B} ∧ {μ|μ ∈ A and ∃β ∈ n , O β (μ) ⊂ A} = W 1 (B) ∧ A1max . Case k = 3: W 3 (A, B) = {μ|μ ∈ A and ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B} = {μ|μ ∈ Bn and μ ∈ A and ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ B} = {μ|μ ∈ Bn and ∀α ∈ n , ωα (μ) ⊂ B and μ ∈ A and ∀β ∈ n , O β (μ) ⊂ A} = {μ|μ ∈ Bn and ∀α ∈ n , ωα (μ) ⊂ B} ∧ {μ|μ ∈ A and ∀β ∈ n , O β (μ) ⊂ A} = W 3 (B) ∧ A3max . Case k = 5. We prove W 5 (A, B) ⊂ W 5 (B) ∧ A5max and we take μ ∈ W 5 (A, B) arbitrary. This means that C ⊂ Bn exists such that μ ∈ C,

(26.2.2)

Chapter 26 • Local basins of attraction

249

O + (C) ⊂ A,

(26.2.3)

ω+ (C) ⊂ B,

(26.2.4)

∀λ ∈ Bn , λ (C) = C

(26.2.5)

hold and from C ⊂ O + (C), (26.2.3) we get also C ⊂ A. The existence of C ⊂ Bn that fulfills the truth of (26.2.2), (26.2.4), (26.2.5) shows that μ ∈ W 5 (B), and the existence of C ⊂ A that fulfills the truth of (26.2.2), (26.2.5) implies μ ∈ A5max . In order to prove W 5 (B) ∧ A5max ⊂ W 5 (A, B), let μ ∈ W 5 (B) ∧ A5max arbitrary. The hypothesis states the existence of C ⊂ Bn such that (26.2.2), (26.2.4), (26.2.5) are true and also the existence of C with C ⊂ A,

(26.2.6)

μ ∈ C ,

(26.2.7)

∀λ ∈ Bn , λ (C ) = C

(26.2.8)

fulfilled. We claim that the set C ∧ C satisfies μ

(26.2.2), (26.2.7)

∈

C ∧ C ,

(26.2.9)

O + (C ∧ C ) ⊂ O + (C ) ⊂ A, ω+ (C ∧ C ) ⊂ ω+ (C) ∀λ ∈ Bn , λ (C ∧ C )

(26.2.4)

⊂

B,

(26.2.5), (26.2.8)

=

C ∧ C .

(26.2.10) (26.2.11) (26.2.12)

The only statement that must be justified in (26.2.9)–(26.2.12) is O + (C ) ⊂ A that occurs in (26.2.10); it results from (26.2.6), (26.2.8) and Lemma 3. Theorem 204. The nonempty subsets A, A , B, B of Bn , and γ ∈ n are given. The following properties of the local basins of attraction hold: ∀k ∈ {1, 3, 4, 5}, (i) A ∧ B = ∅ =⇒ W k (A, B) = ∅, (ii) W k (Bn , B) = W k (B), (iii) W k (A, A) = Akmax , (iv) if A is k-invariant, then W k (A, A) = A, (v) if A ⊂ A ⊂ Bn , A = ∅, and B ⊂ B ⊂ Bn , B = ∅ then W k (A, B) ⊂ W k (A , B ), and similar (i), (ii), (v) properties are true for Wγ2 (A, B) also.

250

Boolean Systems

Proof. (i) Case k = 1. We suppose that A ∧ B = ∅ and we suppose also, against all reason, that μ ∈ W 1 (A, B) exists. This means that μ ∈ A and ∃α ∈ n such that O α (μ) ⊂ A, ωα (μ) ⊂ B are true. We infer the contradiction: ωα (μ) = O α (μ) ∧ ωα (μ) ⊂ A ∧ B = ∅. (ii) Case k = 5. W 5 (Bn , B) = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ Bn and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) = C} = {μ|∃C ⊂ Bn , μ ∈ C and ω+ (C) ⊂ B and ∀λ ∈ Bn , λ (C) = C} = W 5 (B). (iii) Case k = 3. We can write W 3 (A, A) = {μ|μ ∈ A, ∀α ∈ n , O α (μ) ⊂ A and ωα (μ) ⊂ A} = {μ|μ ∈ A, ∀α ∈ n , O α (μ) ⊂ A} = A3max . Case k = 5. We note that W 5 (A, A) = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ω+ (C) ⊂ A and ∀λ ∈ Bn , λ (C) = C} = {μ|∃C ⊂ Bn , μ ∈ C and O + (C) ⊂ A and ∀λ ∈ Bn , λ (C) = C}, A5max = {μ|∃C ⊂ A, μ ∈ C and ∀λ ∈ Bn , λ (C) = C} and we prove that W 5 (A, A) ⊂ A5max . Let μ ∈ W 5 (A, A) arbitrary, fixed, thus we have the existence of C ⊂ Bn such that μ ∈ C,

(26.2.13)

O + (C) ⊂ A,

(26.2.14)

∀λ ∈ Bn , λ (C) = C

(26.2.15)

are true. From (26.2.14), as C ⊂ O + (C), we infer C ⊂ A. Statements (26.2.13), (26.2.15) and (26.2.16) show that μ ∈ A5max . In order to prove A5max ⊂ W 5 (A, A)

(26.2.16)

Chapter 26 • Local basins of attraction

251

we take μ ∈ A5max arbitrary. We get the existence of C such that (26.2.16), (26.2.13) and (26.2.15) be true. From Lemma 3 we infer the truth of (26.2.14), therefore μ ∈ W 5 (A, A). (iv) If A is k-invariant, k ∈ {1, 3, 4, 5}, then A = Akmax and we apply item (iii). (v) Case k = 2. We take an arbitrary μ ∈ Wγ2 (A, B), i.e. μ ∈ A, O γ (μ) ⊂ A and ωγ (μ) ⊂ B hold. We infer that μ ∈ A , O γ (μ) ⊂ A and ωγ (μ) ⊂ B , therefore μ ∈ Wγ2 (A , B ). Remark 173. We could not write which are the items (iii), (iv) of the theorem in the case k = 2 because of a gap in Theorem 152, page 188: we do not know which the maximal 2invariant subset of a set A is, in case that it exists. Remark 174. We can try to find other properties of the local basins of attraction W k (A, B) starting from Theorems 188, page 225 to 194, page 231.

26.3 Isomorphisms Theorem 205. We consider ,  : Bn −→ Bn , A, B ⊂ Bn nonempty both, γ ∈ n and the isomorphism (h, h ) : φ → ψ. Then h(Wγ2, (A, B)) = Wh2 (γ ), (h(A), h(B)) and ∀k ∈ {1, 3, 4, 5}, we have h(Wk (A, B)) = Wk (h(A), h(B)). Proof. The proof is similar with the proof of Theorem 195, page 235. Case k = 1. We prove h(W1 (A, B)) ⊂ W1 (h(A), h(B)), and we take an arbitrary μ ∈ W1 (A, B). This implies that μ ∈ A and α ∈ n exists such that α (μ) ⊂ A and ωα (μ) ⊂ B. We infer that h(μ) ∈ h(A), and on the other hand β ∈  exists, O n  i.e. β = h (α), with the property that β

h (α)

β

h (α)

O (h(μ)) = O ω (h(μ)) = ω

(h(μ))

(h(μ))

Theorem 78, page 91

=

α h(O (μ)) ⊂ h(A),

Theorem 78, page 91

α h(ω (μ)) ⊂ h(B),

=

therefore h(μ) ∈ W1 (h(A), h(B)). Case k = 5. We prove h(W5 (A, B)) ⊂ W5 (h(A), h(B)), and we take μ ∈ W5 (A, B). We have from the hypothesis that C ⊂ Bn exists such that μ ∈ C, + (C) ⊂ A, O

(26.3.1)

252

Boolean Systems

+ (C) ⊂ B, ω

(26.3.2)

∀λ ∈ Bn , λ (C) = C

(26.3.3)

are true. With the notation C = h(C), we have h(μ) ∈ C and we must prove that + O (C ) ⊂ h(A),

(26.3.4)

+ ω (C ) ⊂ h(B),

(26.3.5)

∀λ ∈ Bn ,  λ (C ) = C

(26.3.6)

hold. We take α ∈ n , ν ∈ C arbitrary, fixed. We infer from (26.3.1), (26.3.2) that α (ν) ⊂ A, O

(26.3.7)

α ω (ν) ⊂ B.

(26.3.8)

But h (α)

O

h (α)

ω

(h(ν)) (h(ν))

Theorem 78

=

α h(O (ν))

(26.3.7)

Theorem 78

α h(ω (ν))

(26.3.8)

=

⊂

⊂

h(A), h(B),

where h (α) runs in n and h(ν) runs in C , because h, h are bijections. (26.3.4) and (26.3.5) are true. The fact that (26.3.6) is a consequence of (26.3.3) results from Theorem 138, page 167.

27 Local basins of attraction of the states The group of chapters dedicated to the basins of attraction ends with the local basins of attraction of the states, which are defined in Section 27.1. Section 27.2 gives some properties of these basins of attraction, and Section 27.3 shows how these basins of attraction behave relative to the isomorphisms.

27.1 Definition Definition 111. Let the function  : Bn → Bn , the set A ⊂ Bn , A = ∅, α, γ ∈ n and μ ∈ Bn . The following local basins of attraction of φ α (μ, ·) are defined: W 1 [φ α (μ, ·), A] = {μ |μ ∈ A, ∃β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·)}, Wγ2 [φ α (μ, ·), A] = {μ |μ ∈ A, O γ (μ ) ⊂ A and φ γ (μ , ·) ≈ φ α (μ, ·)}, W 3 [φ α (μ, ·), A] = {μ |μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·)}, W 4 [φ α (μ, ·), A] = {μ |∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) ⊂ B}, W 5 [φ α (μ, ·), A] = {μ |∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) = B}. Remark 175. In Definition 111 no assumption was made on the invariance of A. Theorem 206. We consider A ⊂ Bn nonempty, α, γ ∈ n , μ ∈ Bn and μ ∈ A. (a) The following implications hold: ∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) = B =⇒ ∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B) ⊂ B ⇐⇒ ∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) ⇐⇒ ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·) Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00033-7 Copyright © 2023 Elsevier Inc. All rights reserved.

253

254

Boolean Systems

=⇒ O γ (μ ) ⊂ A and φ γ (μ , ·) ≈ φ α (μ, ·) =⇒ ∃β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·). (b) We have W 5 [φ α (μ, ·), A] ⊂ W 4 [φ α (μ, ·), A] = W 3 [φ α (μ, ·), A] ⊂ Wγ2 [φ α (μ, ·), A] ⊂ W 1 [φ α (μ, ·), A]. Proof. (a) We can prove the truth of the implication ∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) =⇒ ∃B  ⊂ Bn , μ ∈ B  and O + (B  ) ⊂ A and ∀ν ∈ B  , ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) and ∀λ ∈ Bn , λ (B  ) ⊂ B  for B  = O + (B), and the truth of the implication ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·) =⇒ ∃B  ⊂ Bn , μ ∈ B  and O + (B  ) ⊂ A and ∀ν ∈ B  , ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·) for B  = {μ }. Remark 176. The previous theorem shows that we can write W 4 [φ α (μ, ·), A] = {μ |∃B ⊂ Bn , μ ∈ B and O + (B) ⊂ A and ∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·)}.

27.2 Properties Theorem 207.  : Bn → Bn , A ⊂ Bn , A = ∅, α ∈ n and μ ∈ Bn are given. Then W 1 [φ α (μ, ·), A] ⊂ W 1 [φ α (μ, ·)] ∧ A1max , and ∀k ∈ {3, 4, 5}, W k [φ α (μ, ·), A] = W k [φ α (μ, ·)] ∧ Akmax are true. Proof. Case k = 1: W 1 [φ α (μ, ·), A] = {μ |μ ∈ A, ∃β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·)} ⊂ {μ |μ ∈ A, ∃β ∈ n , O β (μ ) ⊂ A and ∃γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·)}

Chapter 27 • Local basins of attraction of the states

255

= {μ |μ ∈ Bn , ∃γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·) and μ ∈ A, ∃β ∈ n , O β (μ ) ⊂ A} = {μ |μ ∈ Bn , ∃γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·)} ∧ {μ |μ ∈ A, ∃β ∈ n , O β (μ ) ⊂ A} = W 1 [φ α (μ, ·)] ∧ A1max . Case k = 3: W 3 [φ α (μ, ·), A] = {μ |μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·)} = {μ |μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A and ∀γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·)} = {μ |μ ∈ Bn , ∀γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·) and μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A} = {μ |μ ∈ Bn , ∀γ ∈ n , φ γ (μ , ·) ≈ φ α (μ, ·)} ∧{μ |μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A} = W 3 [φ α (μ, ·)] ∧ A3max , Case k = 5: In order to prove W 5 [φ α (μ, ·), A] ⊂ W 5 [φ α (μ, ·)] ∧ A5max , we take an arbitrary μ ∈ W 5 [φ α (μ, ·), A]. This means the existence of B ⊂ Bn with μ ∈ B,

(27.2.1)

O + (B) ⊂ A,

(27.2.2)

∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·),

(27.2.3)

∀λ ∈ Bn , λ (B) = B.

(27.2.4)

The existence of B such that (27.2.1), (27.2.3), (27.2.4) are true shows that μ ∈ W 5 [φ α (μ, ·)]. On the other hand, as B ⊂ O + (B), we have from (27.2.2) that B ⊂ A. This, together with (27.2.1), (27.2.4), shows that μ ∈ A5max . We prove the inclusion W 5 [φ α (μ, ·)] ∧ A5max ⊂ W 5 [φ α (μ, ·), A] and let for this μ ∈ W 5 [φ α (μ, ·)] ∧ A5max arbitrary. On one hand we have the existence of B such that (27.2.1), (27.2.3), (27.2.4) are true and on the other hand we have the existence of B  with B  ⊂ A,

(27.2.5)

μ ∈ B  ,

(27.2.6)

∀λ ∈ Bn , λ (B  ) = B  ,

(27.2.7)

256

Boolean Systems

O + (B  )

Lemma 3, page 248

⊂

(27.2.8)

A.

We infer: μ

(27.2.1),(27.2.6)

∈

B ∧ B ,

O + (B ∧ B  ) ⊂ O + (B  )

(27.2.8)

∀ν ∈ B ∧ B  , ∀β ∈ n , φ β (ν, ·) ∀λ ∈ Bn , λ (B ∧ B  )

⊂

A,

(27.2.3)

≈

(27.2.4),(27.2.7)

=

φ α (μ, ·),

B ∧ B ,

showing that μ ∈ W 5 [φ α (μ, ·), A]. Theorem 208. The nonempty sets A, A ⊂ Bn , α, γ ∈ n and μ ∈ Bn are considered. The following properties are true ∀k ∈ {1, 3, 4, 5}: (i) ωα (μ) ⊂ A =⇒ W k [φ α (μ, ·), A] = ∅, (ii) W k [φ α (μ, ·), Bn ] = W k [φ α (μ, ·)], (iii) A ⊂ A =⇒ W k [φ α (μ, ·), A] ⊂ W k [φ α (μ, ·), A ] and similar properties hold for Wγ2 [φ α (μ, ·), A] too. Proof. (i) k = 1. We suppose that ωα (μ) ⊂ A and we suppose also, against all reason, that μ ∈ W 1 [φ α (μ, ·), A] exists. Then μ ∈ A and β ∈ n exists such that O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·). We have obtained the contradiction ωα (μ) = ωβ (μ ) = ωβ (μ ) ∧ A = ωα (μ) ∧ A = ωα (μ). (ii) k = 2. We get Wγ2 [φ α (μ, ·), Bn ] = {μ |μ ∈ Bn , O γ (μ ) ⊂ Bn and φ γ (μ , ·) ≈ φ α (μ, ·)} = {μ |μ ∈ Bn , φ γ (μ , ·) ≈ φ α (μ, ·)} = Wγ2 [φ α (μ, ·)]. (iii) k = 3. We suppose that A ⊂ A and we take an arbitrary μ ∈ W 3 [φ α (μ, ·), A]. We have μ ∈ A, ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·). We infer μ ∈ A , ∀β ∈ n , O β (μ ) ⊂ A and φ β (μ , ·) ≈ φ α (μ, ·), thus μ ∈ W 3 [φ α (μ, ·), A ]. Remark 177. We can try getting other properties of these basins of attraction if we start from Theorems 197, page 241 to 200, page 242.

27.3 Isomorphisms Theorem 209. Let the systems ,  : Bn −→ Bn , the isomorphism (h, h ) : φ → ψ and also A ⊂ Bn , A = ∅, α, γ ∈ n , μ ∈ Bn . Then 

h(Wγ2, [φ α (μ, ·), A]) = Wh2 (γ ), [ψ h (α) (h(μ), ·), h(A)]

Chapter 27 • Local basins of attraction of the states

257

and ∀k ∈ {1, 3, 4, 5}, 

h(Wk [φ α (μ, ·), A]) = Wk [ψ h (α) (h(μ), ·), h(A)]

(27.3.1)

hold. Proof. The proof is similar with the proof of Theorem 201, page 243. Case k = 1. In order to prove the inclusion 

h(W1 [φ α (μ, ·), A]) ⊂ W1 [ψ h (α) (h(μ), ·), h(A)], we take an arbitrary μ ∈ W1 [φ α (μ, ·), A], thus μ ∈ A and β ∈ n exist such that O (μ ) ⊂ A,

(27.3.2)

φ β (μ , ·) ≈ φ α (μ, ·).

(27.3.3)

β

Then h(μ ) ∈ h(A) and we must show the existence of β  ∈ n that satisfies β

O (h(μ )) ⊂ h(A), 

(27.3.4)



ψ β (h(μ ), ·) ≈ ψ h (α) (h(μ), ·).

(27.3.5)

More exactly, we shall prove that (27.3.4), (27.3.5) hold for β  = h (β). (27.3.4) results from β

h (β)

O (h(μ )) = O

(h(μ ))

Theorem 78, page 91

=

h(O (μ )) β

(27.3.2)

⊂

h(A),

while (27.3.5) is a consequence of (27.3.3) and Theorem 82, page 94. Case k = 5. We prove the inclusion 

h(W5 [φ α (μ, ·), A]) ⊂ W5 [ψ h (α) (h(μ), ·), h(A)], and let μ ∈ W5 [φ α (μ, ·), A] arbitrary. This means the existence of B ⊂ Bn , such that μ ∈ B, + O (B) ⊂ A,

(27.3.6)

∀ν ∈ B, ∀β ∈ n , φ β (ν, ·) ≈ φ α (μ, ·),

(27.3.7)

∀λ ∈ Bn , λ (B) = B.

(27.3.8)

We denote B  = h(B). We remark that h(μ ) ∈ B  and we must prove that + O (B  ) ⊂ h(A), 

(27.3.9) 

∀ν  ∈ B  , ∀β  ∈ n , ψ β (ν  , ·) ≈ ψ h (α) (h(μ), ·),

(27.3.10)

258

Boolean Systems



∀λ ∈ Bn ,  λ (B  ) = B  .

(27.3.11)

Proving (27.3.9), (27.3.10) is made in similar terms with the proof of (27.3.4), (27.3.5). On the other hand (27.3.11) is a consequence of (27.3.8) and Theorem 138, page 167.

28 Attractors The 5-minimal sets X ⊂ Bn are called attractors. They are introduced in Section 28.1 and several examples are given in Section 28.2. Some interesting properties of the attractors are addressed in Section 28.3, such as: if X is 5-invariant, then ∀μ ∈ X, O + (μ) is attractor, and if X is attractor, then ∀μ ∈ X, X = O + (μ) = {μ}5min . In Section 28.4 we prove that X is attractor iff it is 5-invariant and topologically transitive. And in Section 28.5 we prove that X is attractor iff it is 5-invariant and path-connected. The isomorphisms bring attractors in attractors and this is shown in Section 28.6. Section 28.7 shows that X is attractor if and only if it is attractive W 5 (X) = ∅ and it is the omega-limit set of some point X = ωα (μ). In Section 28.8 we prove that if X is attractor and Y is attractor, then the Cartesian product X × Y is attractor. The attractors X with card(X) > 1 have also a property of chaos that is stated in Section 28.9. In Section 28.10 we define the repellers and in Section 28.11 we define the weak attractors. The sources of inspiration are [15] and [8].

28.1 Definition Definition 112. The function  : Bn → Bn and X ⊂ Bn , X = ∅ are given. If X is 5-minimal: ◦5

X = Xmin , i.e. if ∀λ ∈ Bn , λ (X) = X,

(28.1.1)

∀Y, (∅  Y ⊂ X and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ X = Y,

(28.1.2)

then it is called attractor. Definition 113. An attractor X with Wγ2 (X) = Bn , γ ∈ n or with ∃k ∈ {1, 3, 4, 5}, W k (X) = Bn is called global. Remark 178. For any μ ∈ Bn we have the equivalence: X = {μ} is attractor ⇐⇒ (μ) = μ. By identifying abusively μ with {μ}, we can say that the fixed points are the simplest example of attractors. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00034-9 Copyright © 2023 Elsevier Inc. All rights reserved.

259

260

Boolean Systems

Remark 179. Let X ⊂ Bn nonempty and 5-invariant. Recall Theorem 185, page 219 stating the existence of the following possibilities: either X is an attractor, or a partition X1 , ..., Xp of X exists, p ≥ 2, such that each Xi is attractor. Remark 180. We know also, see Corollary 15, page 231, that if X is an attractor, then X = W 5 (X). This is one of the meanings of Definition 113, if we would have called the attractor X global for W 5 (X) = Bn , this would have meant that the only global attractor is Bn , which is somehow non interesting.

28.2 Examples Example 142. The fixed point (1, 0) of the next system (0, 0)

- (0, 1)

- (1, 1)

- (1, 0)

is a global attractor: W 4 (1, 0) = B2 . Example 143. The state portrait of the following function 

(0, 0, 0)

(1, 0, 0)

(0, 1, 0) 6 ? (1, 1, 0)

(0, 0, 1)  6 3     ?  (1, 1, 1)  - (0, 1, 1)

     ?+  (1, 0, 1) 6

indicates the existence of four attractors, A = {(0, 0, 0)}, B = {(1, 0, 0)}, C = {(0, 1, 0), (1, 1, 0)} and D = {(1, 0, 1), (1, 1, 1), (0, 0, 1), (0, 1, 1)}. We can see by replacing  with (0,1,1) (0, 0, 1) 6 (0, 0, 0)

(0, 1, 0)

(1, 0, 0)

(1, 1, 0)

(1, 1, 1) 6

? (0, 1, 1)

? (1, 0, 1)

that the attractor C is split in two attractors C = {(0, 1, 0)}, C = {(1, 1, 0)}, and D is split in two attractors also, D = {(1, 0, 1), (1, 1, 1)}, D = {(0, 0, 1), (0, 1, 1)}. And if we consider the

Chapter 28 • Attractors

261

function (1,0,1) , (0, 0, 1)

(0, 0, 0)

(1, 0, 0)

(0, 1, 0) 6 ? (1, 1, 0)

(1, 1, 1) 

-

(0, 1, 1)

(1, 0, 1)

then the attractor C of  is unchanged, but the attractor D is split in three attractors, D1 = {(1, 0, 1)}, D2 = {(0, 0, 1)}, and D3 = {(1, 1, 1), (0, 1, 1)}. In the cases of , (0,1,1) , (1,0,1) , the basins of attraction of the attractors coincide with the attractors, under the form: ∀γ ∈ 3 , ∀k ∈ {1, 3, 4, 5}, Wγ2 (A) = W k (A) = A, Wγ2 (D) = W k (D) = D, etc. Example 144. Here is the state portrait of a function for which D = {(1, 0, 1), (1, 1, 1), (0, 0, 1), (0, 1, 1)} is attractor, like previously

(0, 0, 0)

(1, 0, 0)

(0, 0, 1)  6 3     ?  (1, 1, 1)  (0, 1, 1) (0, 1, 0) 3 6          ? ?+  (1, 0, 1) (1, 1, 0)

but the basins of attraction of this attractor include the points (0, 1, 0), (1, 1, 0): ∀γ ∈ 3 , ∀k ∈ {1, 3, 4}, Wγ2 (D) = W k (D) = D ∨ {(0, 1, 0), (1, 1, 0)}. In addition, W 5 (D) = D. Example 145. We give the example of a system (1, 0, 0)

(1, 0, 1)

- (1, 1, 0) Q k 6 Q Q Q Q ? Q - (1, 1, 1)  - (0, 1, 1)  Q 6 Q Q Q Q ? s Q (0, 1, 0) 

(0, 0, 1)

(0, 0, 0)

262

Boolean Systems

that has a global attractor E = {(1, 1, 0), (1, 1, 1), (0, 1, 1), (0, 1, 0)} : ∀γ ∈ 3 , ∀k ∈ {1, 3, 4}, Wγ2 (E) = W k (E) = B3 . Remark 181. Small obvious changes of the systems from Examples 144, 145 can make D, E become global/not global.

28.3 Properties Theorem 210. We suppose that X ⊂ Bn is nonempty and 5-invariant. (a) If card(X) = 1, then X is an attractor, (b) if card(X) ≥ 2 and X contains sources, sinks or isolated fixed points, then X is not an attractor, (c) if card(X) ≥ 2 and X is an attractor, then all its points are transient. Proof. (b) We suppose that card(X) ≥ 2. If μ ∈ X exists with μ− = {μ}, μ+ = {μ}, as μ+ = [μ, (μ)], we infer that (μ) = μ. We have obtained that μ ∈ / (X), the function  : X → X is not surjective, contradiction with the 5-invariance of X. If some μ ∈ X exists such that μ− = {μ}, μ+ = {μ}, then we obtain the existence of μ ∈ − μ , μ = μ, and of λ ∈ Bn , that satisfy λ (μ ) = μ = λ (μ). The function λ : X → X is not injective, contradiction with the 5-invariance of X. In case that μ ∈ X exists with μ− = {μ}, μ+ = {μ}, we get (μ) = μ and the set {μ} is 5-invariant, satisfying {μ}  X. This contradicts the 5-minimality of X. Theorem 211. The function  : Bn → Bn is given. For any μ ∈ Bn , the 5-minimal set containing μ, denoted {μ}5min , which is defined conformally with (21.1.5)page 195 by  ∀λ ∈ Bn , λ ({μ}5min ) = {μ}5min , (28.3.1) ∀Y, (μ ∈ Y and ∀λ ∈ Bn , λ (Y ) = Y ) =⇒ {μ}5min ⊂ Y is an attractor. Proof. We fix an arbitrary μ ∈ Bn and we know that {μ}5min is 5-invariant. We suppose against all reason that a 5-invariant set Y  {μ}5min exists. Case μ ∈ Y . This contradicts the second statement of (28.3.1). Case μ ∈ / Y . In this case the set {μ}5min  Y  {μ}5min is 5-invariant and contains μ, contradiction with the second statement of (28.3.1). Theorem 212. Let  : Bn → Bn and the nonempty set X ⊂ Bn . X is attractor if and only if ∀μ ∈ X, X = {μ}5min . Proof. If. This is a consequence of Theorem 211. Only if. X is attractor and we suppose against all reason that μ ∈ X exists such that X = {μ}5min . This implies the 5-invariance of the set X  {μ}5min  X, representing a contradiction with the request of 5-minimality of X.

Chapter 28 • Attractors

263

Theorem 213. If X is 5-invariant, then ∀μ ∈ X, O + (μ) is attractor. Proof. We take μ ∈ X arbitrary and fixed. The 5-invariance of X: ∀λ ∈ Bn , λ (X) = X shows that for any α ∈ n , we have O α (μ) ⊂ X (this is the 3-invariance of X), therefore  O + (μ) = O α (μ) ⊂ X. α∈n

Theorem 128, page 160, shows that O + (μ) is 4-invariant. As O + (μ) is a 4-invariant subset of the 5-invariant set X we get, taking into account Theorem 135, page 165, that O + (μ) is 5-invariant. We prove that O + (μ) is 5-minimal. For this, we suppose against all reason that Y exists satisfying ∅  Y  O + (μ) and ∀λ ∈ Bn , λ (Y ) = Y.

(28.3.2)

Case μ ∈ Y . The 5-invariance of Y shows that ∀α ∈ n , O α (μ) ⊂ Y and the request O + (μ)  Y implies the contradiction ∃α ∈ n , O α (μ)  Y = ∅. Case μ ∈ O + (μ)  Y . Both O + (μ), Y are 5-invariant, thus O + (μ)  Y is 5-invariant, therefore ∀α ∈ n , O α (μ) ⊂ O + (μ)  Y, ∀α ∈ n , O α (μ) ∧ Y = ∅, and we infer O + (μ) ∧ Y = ∅. We have obtained a contradiction with the supposition that ∅  Y ⊂ O + (μ). Theorem 214. We consider the attractors X, Y ⊂ Bn . (a) If X ∧ Y = ∅, then X ∨ Y, X ∧ Y are not attractors. (b) If X ∧ Y = ∅, then X = Y . Proof. (a) The set X ∨ Y is 5-invariant, but it is not 5-minimal, thus X ∨ Y is not attractor. X ∧ Y = ∅ cannot be attractor. (b) We suppose that X ∧ Y = ∅. If, against all reason X = Y , then at least one of X  Y = ∅, Y  X = ∅ is true. In the first case for example X  Y is 5-invariant and X  Y  X, contradiction with the 5-minimality of X. We conclude that X = Y .

264

Boolean Systems

Corollary 17. (a) If X is 5-invariant, then ∀μ ∈ X, the set O + (μ) = {μ}5min is attractor. (b) If X is attractor, then ∀μ ∈ X, X = O + (μ) = {μ}5min . Proof. (a) We suppose that X is 5-invariant and let μ ∈ X arbitrary. O + (μ) is attractor from Theorem 213, {μ}5min is attractor from Theorem 211 and μ ∈ O + (μ) ∧ {μ}5min , thus O + (μ) ∧ {μ}5min = ∅. Then Theorem 214 implies O + (μ) = {μ}5min . Theorem 215. We suppose that X, Y ⊂ Bn nonempty fulfill: X is a global attractor, in the sense that W 4 (X) = Bn , and Y is 4-invariant. Then X ⊂ Y . Proof. We want to show first that X ∧ Y = ∅. In this respect we take μ ∈ Y , α ∈ n arbitrary and we have ωα (μ) ⊂ O α (μ) ⊂ Y, ωα (μ) ⊂ X, therefore ωα (μ) ⊂ X ∧ Y . It has resulted that X ∧ Y is nonempty and it is also the intersection of two 4-invariant sets, thus it is 4-invariant, from Theorem 134, page 163. We know at this moment that X ∧ Y is a 4-invariant subset of the 5-invariant set X, and Theorem 135, page 165 implies the 5invariance of X ∧ Y . But X ∧ Y is a 5-invariant subset of the 5-minimal set X, wherefrom X ∧ Y = X. We have obtained that X ⊂ Y . Theorem 216. The nonempty the set X ⊂ Bn with card(X) = p, p ≥ 1 is given. If (a) ∀λ ∈ Bn , the partition X1 , ..., Xk of X exists, k ≥ 1, such that ∀i ∈ {1, ..., k}, ∃pi ∈ {1, ..., p}, ∀μ ∈ Xi , Xi = {μ, (λ )(μ), ..., (λ )(pi −1) (μ)} and (λ )(pi ) (μ) = μ,

(28.3.3)

(b) ∀μ ∈ X, ∃λ ∈ Bn ,





X = {μ, (λ )(μ), ..., (λ )(p−1) (μ)} and (λ )(p) (μ) = μ

(28.3.4)

are true, then X is an attractor. Proof. In case that X = {μ}, the implication expressed by the theorem holds with (μ) = μ, X attractor and (a), (b) both trivially true. We suppose in the rest of the proof that card(X) > 1. The request (a) is equivalent with the 5-invariance of X, from Theorem 136, page 165. We suppose now against all reason that the 5-minimality of X is false, i.e. Y X

(28.3.5)

∀λ ∈ Bn , λ (Y ) = Y.

(28.3.6)

exists such that

Chapter 28 • Attractors

265

Let us take μ ∈ Y arbitrary. As μ ∈ X, we have from (b) the existence of λ ∈ Bn with the property that



X = {μ, (λ )(μ), ..., (λ )(p−1) (μ)}

(28.3.6)

⊂

(28.3.7)

Y.

The statements (28.3.5), (28.3.7) are contradictory. X is attractor. Remark 182. The property (b) from Theorem 216 is one of path-connectedness. The relation between 5-invariance, path-connectedness and attractors is treated in Section 5.

28.4 Topological transitivity Theorem 217. The function  : Bn → Bn and X ⊂ Bn nonempty are given. If X is attractor, then X is topologically transitive: ∀μ ∈ X, ∃α ∈ n , O α (μ) = X.

(28.4.1)

Proof. If X = {μ}, (μ) = μ then the theorem is proved, thus we can suppose in the rest of the proof that card(X) ≥ 2. We infer from Corollary 17, page 264, in succession,  O α (μ), ∀μ ∈ X, X = O + (μ) = α∈n

∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , μ ∈ O α (μ), ∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k ∈ N, μ ∈ φ α (μ, k). We put X under the form X

(28.4.2)

= {μ0 , μ1 , ..., μp } and we infer from (28.4.2) that ij

∀i ∈ {0, 1, ..., p}, ∀j ∈ {0, 1, ..., p}, ∃α ij ∈ n , ∃kij ∈ N, μj = φ α (μi , kij ). We take without losing the generality μ = μ0 and we define α ∈ n in the following manner: ⎧ ⎪ α 01,k , if k ∈ {0, ..., k01 − 1}, ⎪ ⎪ ⎪ ⎪ α 12,k−k01 , if k ∈ {k01 , ..., k01 + k12 − 1}, ⎨ k 23,k−k −k12 , if k ∈ {k + k , ..., k + k + k − 1}, 01 α = α 01 12 01 12 23 ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎩ α p−1p,k−k01 −...−kp−2p−1 , if k ≥ k + ... + k 01 p−2p−1 . We conclude that 01

φ α (μ, k01 ) = φ α (μ0 , k01 ) = μ1 , φ α (μ, k01 + k12 ) = φ σ

k01 (α)

12

(μ1 , k12 ) = φ α (μ1 , k12 ) = μ2 ,

266

Boolean Systems

... φ α (μ, k01 + ... + kp−1p ) = φ σ = φα

p−1p

k01 +...+kp−2p−1 (α)

(μp−1 , kp−1p )

(μp−1 , kp−1p ) = μp .

Since X ⊂ O α (μ) ⊂ X, (28.4.1) is proved. Corollary 18. The following statements are equivalent: (a) X is attractor, (b) X is 5-invariant and the topological transitivity property (28.4.1) holds. Proof. (a)=⇒(b) was stated in Theorem 217 and (b)=⇒(a) was stated in Theorem 182, page 217. Corollary 19. If X is 5-invariant and one of the topological transitivity properties ∃α ∈ n , ∀μ ∈ X, O α (μ) = X, ∀α ∈ n , ∀μ ∈ X, O α (μ) = X is true, then X is attractor. Proof. Any of these two properties implies (28.4.1), see Theorem 115, page 137. We use Corollary 18.

28.5 Path-connectedness Theorem 218. Let  : Bn → Bn and X ⊂ Bn , X = ∅. If X is attractor, then it is pathconnected: ∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ .

(28.5.1)

Proof. Let μ ∈ X arbitrary, fixed. We know from Theorem 217, page 265, that α ∈ n exists with the property that O α (μ) = X. Then for any μ ∈ X, some k ∈ N exists such that ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ . We have just proved that ∀μ ∈ X, ∃α ∈ n , ∀μ ∈ X, ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ

(28.5.2)

holds. The equivalence between (28.5.1) and (28.5.2) follows from Theorem 115, page 137.

Chapter 28 • Attractors

267

Corollary 20. The following statements are equivalent: (a) X is attractor, (b) X is 5-invariant and the path-connectedness property (28.5.1) is true. Proof. (a)=⇒(b) follows from Theorem 218 and the truth of the implication (b)=⇒(a) is inferred from Theorem 184, page 218. Corollary 21. If X is 5-invariant and one of the path-connectedness properties ∃α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ , ∀α ∈ n , ∀μ ∈ X, ∀μ ∈ X, ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ is true, then X is attractor. Proof. Theorem 115, page 137 shows that any of these two path-connectedness properties implies (28.5.1). We use Corollary 20. Remark 183. We get back to Theorem 210, page 262 for a moment, and we consider the attractor X ⊂ Bn with card(X) > 1. The theorem states that any μ ∈ X is a transient point. We notice that if X contains against all reason a source, or a sink, or an isolated fixed point, then Theorem 116 (a), page 140 shows that X does not fulfill (28.5.1), contradiction.

28.6 Isomorphisms Corollary 22. Let , : Bn → Bn be two systems and (h, h ) ∈ I so(φ, ψ). If X is an attractor of , then h(X) is an attractor of . Proof. This is a consequence of Theorem 186, page 220.

28.7 Attractors as omega-limit sets Theorem 219. The set X ⊂ Bn , X = ∅ is given.1 (a) If X is attractive W 5 (X) = ∅

(28.7.1)

and it is the omega-limit set of some point 1

In [15], page 399, S is supposed to be a compact invariant set; then X ⊂ S is called an attractor in S if a neighborhood U of X exists such that ω(U ∧ S) = X.

268

Boolean Systems

∃μ ∈ Bn , ∃α ∈ n , X = ωα (μ),

(28.7.2)

then it is attractor. (b) If X is attractor, then (28.7.1), (28.7.2) hold. Proof. (a) We suppose that μ ∈ Bn , α ∈ n fulfill X = ωα (μ), from (28.7.2). The hypothesis (28.7.1) states, see Definition 105, page 221, the existence of a 5-invariant set B ⊂ Bn fulfilling ω+ (B) ⊂ X and 5-invariance, i.e.   ωβ (ν) ⊂ ωα (μ), (28.7.3) ν∈B β∈n

∀λ ∈ Bn , λ (B) = B

(28.7.4)

are true. We get ∀ν ∈ B, ∀β ∈ n , ωβ (ν) ⊂ O β (ν) We prove first that

 

(28.7.4)

⊂

(28.7.5)

B.

ωβ (ν) = B.

(28.7.6)

ν∈B β∈n

The set B can be written as a disjoint union of sets B1 , ..., Bk such that ∀i ∈ {1, ..., k}, ∃pi ≥ 1, ∀ν i ∈ Bi , Bi = {ν i , (ν i ), ..., (pi −1) (ν i )}, (pi ) (ν i ) = ν i , thus Bi = ωγ (ν i ), where γ k = (1, ..., 1) ∈ Bn , ∀k ∈ N. We have for arbitrary ν 1 ∈ B1 , ..., ν k ∈ Bk : B = B1 ∨ ... ∨ Bk = ωγ (ν 1 ) ∨ ... ∨ ωγ (ν k ) ⊂

 

ωβ (ν)

(28.7.5)

⊂

B,

ν∈B β∈n

thus (28.7.6) is true. (28.7.3) becomes B ⊂ ωα (μ).

(28.7.7)

ωα (μ) ⊂ B,

(28.7.8)

We prove that

and let μ ∈ ωα (μ) arbitrary. We take ν ∈ B arbitrary itself and from (28.7.7) we have the existence of k ∈ N with α (μ, k ) = ν.

Chapter 28 • Attractors

269

As the set {k|k ∈ N, φ α (μ, k) = μ } is infinite, k > k exists such that φ α (μ, k ) = μ . We define

 ρk =



α k+k , if k ∈ {0, ..., k − k − 1}, arbitrary, if k ≥ k − k

and ρ k ∈ Bn , k ≥ k − k are chosen in such a way that ρ ∈ n . We have ∀k ∈ {0, ..., k − k }, φ ρ (ν, k) = φ α (μ, k + k ) thus μ = φ α (μ, k ) = φ ρ (ν, k − k )

(28.7.4)

∈

B.

(28.7.8) is true and, taking into account (28.7.7) also, we have B = ωα (μ).

(28.7.9)

∀ν ∈ B, ∀ν ∈ B, ∃δ ∈ n , ∃k ∈ N, ∀k ∈ {0, ..., k }, φ δ (ν, k) ∈ B and φ δ (ν, k ) = ν

(28.7.10)

We prove that B is path-connected:

and let ν ∈ B, ν ∈ B arbitrary. As the sets {k|k ∈ N, φ α (μ, k) = ν}, {k|k ∈ N, φ α (μ, k) = ν } are infinite, k1 < k2 exist such that φ α (μ, k1 ) = ν, φ α (μ, k2 ) = ν . We define

 δk =

α k+k1 , if k ∈ {0, ..., k2 − k1 − 1}, arbitrary, if k ≥ k2 − k1 ,

where the arbitrary values δ k ∈ Bn , k ≥ k2 − k1 are chosen in such a way that δ ∈ n . We have ∀k ∈ {0, ..., k2 − k1 }, φ δ (ν, k) = φ α (μ, k + k1 ), thus (28.7.10) is true, with k = k2 − k1 . Corollary 20, page 267 implies that B is attractor. From (28.7.9), X is attractor. (b) We note first that ω+ (X) ⊂ X,

(28.7.11)

∀λ ∈ Bn , λ (X) = X.

(28.7.12)

270

Boolean Systems

The 5-invariance (28.7.12) results from the fact that X is attractor, and (28.7.11) results from ω+ (X) =

 

 

ωβ (ν) ⊂

ν∈Xβ∈n

O β (ν)

(28.7.12)

⊂

X.

ν∈Xβ∈n

The truth of (28.7.11) and (28.7.12) implies the truth of (28.7.1). Statement (28.7.2) that we must still prove holds for card(X) = 1 due to (28.7.12), and for this reason we shall suppose for the rest of this proof that card(X) ≥ 2. We put X under the form X = {μ0 , μ1 , ..., μp }, p ≥ 1 and we suppose without losing the generality that μ = μ0 is arbitrary, fixed. We infer the existence of p > 1 such that μ, (μ), ..., (p −1) (μ) are distinct and (p ) (μ) = μ. We recall now, by taking into account Theorem 217, page 265 and the notations that occurred in its proof, that X is topologically transitive, i.e. ∀i ∈ {0, 1, ..., p}, ∀j ∈ {0, 1, ..., p}, ij

∃α ij ∈ n , ∃kij ∈ N, μj = φ α (μi , kij ) holds. We define α : N → Bn by the sequence α 01,0 , α 01,1 , ..., α 01,k01 −1 , α 12,0 , α 12,1 , ..., α 12,k12 −1 , ..., α p−1p,0 , α p−1p,1 , ..., α p−1p,kp−1p −1 , α p0,0 , α p0,1 , ..., α p0,kp0 −1 , (1, ..., 1), ..., (1, ..., 1), ... 

 p

that repeats periodically, thus α ∈ n . We have: 01

φ α (μ, k01 ) = φ α (μ0 , k01 ) = μ1 , ... φ α (μ, k01 + ... + kp−1p ) = φ σ = φα

p−1p

k01 +...+kp−2p−1 (α)

(μp−1 , kp−1p )

(μp−1 , kp−1p ) = μp ,

φ α (μ, k01 + ... + kp−1p + kp0 ) = φ σ

k01 +...+kp−2p−1 +kp−1p (α)

(μp , kp0 )

p0

= φ α (μp , kp0 ) = μ0 , φ α (μ, k01 + ... + kp0 + p ) = φ σ

k01 +...+kp0 (α)

= φ (1,...,1),...,(1,..,1),... (μ0 , p ) = μ0 , ... which is repeated periodically. We have proved that X = O α (μ) = ωα (μ).

(μ0 , p )

Chapter 28 • Attractors

271

28.8 Cartesian products Theorem 220. The systems  : Bn −→ Bn , : Bm −→ Bm , and X ⊂ Bn , Y ⊂ Bm nonempty are given. If X is an attractor of , and Y is an attractor of , then X × Y is an attractor of  × . Proof. We suppose that X, Y are attractors, i.e., from Corollary 20, page 267, ∀λ ∈ Bn , λ (X) = X,

(28.8.1)

∀μ ∈ X, ∀μ ∈ X, ∃α ∈ n , ∃k ∈ N, ∀k ∈ {0, ..., k }, φ α (μ, k) ∈ X and φ α (μ, k ) = μ ,

(28.8.2)

∀ν ∈ Bm , ν (Y ) = Y,

(28.8.3)

∀ν ∈ Y, ∀ν ∈ Y, ∃β ∈ m , ∃k ∈ N, ∀k ∈ {0, ..., k }, ψ β (ν, k) ∈ Y and ψ β (ν, k ) = ν ,

(28.8.4)

and we must prove that ∀(λ, ν) ∈ Bn × Bm , ( × )(λ,ν) (X × Y ) = X × Y,

(28.8.5)

∀(μ, ν) ∈ X × Y, ∀(μ , ν ) ∈ X × Y, ∃(α, β) ∈ n × m , ∃k1 ∈ N, ∀k ∈ {0, ..., k1 }, (φ α (μ, k), ψ β (ν, k)) ∈ X × Y and (φ α (μ, k1 ), ψ β (ν, k1 )) = (μ , ν ).

(28.8.6)

(28.8.1), (28.8.3) and Theorem 140, page 170, imply (28.8.5). At the same time, (28.8.2), (28.8.4) and Theorem 121, page 143, imply (28.8.6). Remark 184. Note that in the proof of the previous theorem we could not start from the ◦

5

◦5

◦5

initial Definition 112, page 259 of the attractors, since the result (X × Y )min = Xmin × Y min is missing. Problem 15. To be studied the inverse implication: if X × Y is an attractor of  × , then X is an attractor of , and Y is an attractor of .

28.9 Chaos Theorem 221. If X ⊂ Bn is attractor of  : Bn → Bn and card(X) > 1, then the following property of chaos is true: ∀μ ∈ X, ∃α ∈ n such that O α (μ) = X,

(28.9.1)

∃p ≥ 1, ∀k ∈ N, φ α (μ, k) = φ α (μ, k + p),

(28.9.2)

272

Boolean Systems

∃μ ∈ X, ∀k ∈ N, φ α (μ, k) = φ α (μ , k).

(28.9.3)

Proof. Let μ ∈ X arbitrary, fixed. Theorem 217, page 265 shows the existence of β ∈ n such that O β (μ) = X and we denote with k ≥ 1 the number k = min{k1 |k1 ∈ N, {φ β (μ, k)|k ∈ {0, ..., k1 }} = X}. We use the notation μ = φ β (μ, k ), and we obviously have μ = μ. As X contains no fixed points, thus (μ ) = μ , p ≥ 2 exists with the property that μ , (μ ), ..., (p −1) (μ ) are distinct and (p ) (μ ) = μ . We know from Theorem 218, page 266 that X is also path-connected, and this shows the existence of γ ∈ n , k ≥ 1 with the property that φ γ (μ , k ) = μ. We define α ∈ n by the sequence

β 0 , ..., β k −1 , (1, ..., 1), ..., (1, ..., 1), γ 0 , ..., γ k

 

−1

, ...

p

that repeats periodically. We have φ α (μ, k ) = φ β (μ, k ) = μ , φ α (μ, k + p ) = φ σ

k (α)

φ α (μ, k + p + k ) = φ σ



(μ , p ) = (p ) (μ ) = μ ,

k +p (α)

(μ , k ) = φ γ (μ , k ) = μ,

... (28.9.1) and (28.9.2) are obviously true, with the period p = k + p + k . (28.9.3) is also true, as far as φ α (μ, 0) = μ = μ = φ α (μ , 0) and, for any k, if φ α (μ, k) = φ α (μ , k), then φ α (μ, k + 1) = α (φ α (μ, k)) = α (φ α (μ , k)) = φ α (μ , k + 1) k

k

k

holds, from the bijectivity of α : X → X.

28.10 Repellers Definition 114. For  : Bn → Bn and the attractor X ⊂ Bn , R(X) = {μ|μ ∈ Bn , ∃α ∈ n , ωα (μ) ∧ X = ∅}, C(X) = {μ|μ ∈ Bn  X, ∀α ∈ n , ωα (μ) ⊂ X}

Chapter 28 • Attractors

273

are called2 the repeller of X, and the set of connecting orbits from R(X) to X. Theorem 222. X, R(X), C(X) are a partition of Bn . Proof. In order to prove that Bn ⊂ X ∨ R(X) ∨ C(X),

(28.10.1)

we take an arbitrary μ ∈ Bn and the following possibilities exist. Case (a) μ ∈ X, when (28.10.1) holds. Case (b) μ ∈ Bn  X, Case (b.1) ∃α ∈ n , ωα (μ) ∧ X = . Then μ ∈ R(X) and (28.10.1) holds. Case (b.2) ∀α ∈ n , ωα (μ) ∧ X = . Let α ∈ n arbitrary, thus k ∈ N exists with the properties that φ α (μ, k ) ∈ X, ωα (μ) = {φ α (μ, k)|k ≥ k }. As ω+ (X)

Lemma 2, page 224

=

(28.10.2)

X,

k

we note that β ∈ n defined by β = σ (α) satisfies ωα (μ) = ωσ

k (α)

(φ α (μ, k )) = ωβ (φ α (μ, k ))

(28.10.2)

⊂

X,

i.e. μ ∈ C(X) and (28.10.1) is true. We prove the truth of X ∧ R(X) = , and we suppose against all reason that μ ∈ X ∧ R(X) exists. Then μ ∈ R(X) shows the existence of α ∈ n such that ωα (μ) ∧ X = , while μ ∈ X gives ωα (μ)

(28.10.2)

⊂

X,

contradiction. The other statements are easily proved as well. Example 146. We consider for the system (1, 0, 0)  6     ?  + (1, 1, 0) (0, 0, 0)  (0, 0, 1) 6

2

(1, 0, 1) 

- (0, 1, 0)

(1, 1, 1) 6 - (0, 1, 1)

The definition was inspired by the work of Mischaikow and Mrozek, from [15].

274

Boolean Systems

the attractor X = {(0, 0, 0), (0, 0, 1)}, that defines R(X) = {(1, 1, 0), (0, 1, 0), (0, 1, 1), (1, 1, 1), (1, 0, 1)} and C(X) = {(1, 0, 0)}. We want to show which is the position of an arbitrary 1invariant set Y ⊂ B3 relative to X, R(X), C(X). We get: (i) Y = X, when Y ⊂ X,3 (ii) Y = {(1, 1, 1), (1, 0, 1)}, when Y ⊂ R(X), (iii) Y = {(1, 1, 0), (0, 0, 0), (0, 0, 1)}, when Y ∧ X = ∅, Y ∧ R(X) = ∅, (iv) Y = {(1, 0, 0), (0, 0, 0), (0, 0, 1)}, when Y ∧ X = ∅, Y ∧ C(X) = ∅, (v) Y = {(1, 1, 0), (1, 0, 0), (0, 0, 0), (0, 0, 1)}, when Y ∧ X = ∅, Y ∧ R(X) = ∅, Y ∧ C(X) = ∅ and these are all the possibilities. We understand also, from the state portrait, why C(X) is called set of connecting orbits from R(X) to X. This connection of orbits is however just a possibility, since in the case of the system (1, 0, 0)      ?  + (1, 1, 0) (0, 0, 0) (0, 0, 1) 6

(1, 0, 1) 

- (0, 1, 0)

(1, 1, 1) 6 - (0, 1, 1)

X, R(X), C(X) are the same as previously, but C(X) connects no orbits from R(X) to X.

28.11 Weak attractors Definition 115. The system  : Bn → Bn is given. If K ⊂ Bn exists such that one of ∀μ ∈ Bn , ∃α ∈ n , O α (μ) ∧ K = ∅, ∃α ∈ n , ∀μ ∈ Bn , O α (μ) ∧ K = ∅, ∀α ∈ n , ∀μ ∈ Bn , O α (μ) ∧ K = ∅ is true, φ is called weakly dissipative, and K is called weak attractor. Problem 16. This definition was inspired by the analogue definition from [8], page 16 and such concepts, if they prove to be interesting, may be also investigated.

3

Indeed, the general case includes the existence of Y , ∅  Y  X when X is 5-invariant and Y is 1-invariant. An example for this is given by  : B2 → B2 , (μ1 , μ2 ) = (μ1 , μ2 ), X = B2 and Y = {(0, 0), (1, 1)} (Y is 2-invariant, thus 1-invariant, but it is not 5-invariant).

29 Stability The global asymptotic stability represents the property of a system that its states have final values. Several versions of this definition are presented in Section 29.1, together with the relations between them, and in Section 29.2 we give examples. Section 29.3 relates stability with the basins of attraction of the fixed points. Morphisms bring stable systems in stable systems and this is the topic of Section 29.4. In Section 29.5 we show the way that a stable subsystem makes the system be partially stable, while the subsystems of a stable system are stable. In Section 29.6 we remark that stability implies independence on the initial conditions.

29.1 Definition Remark 185. We consider the function  : Bn → Bn . The attribute stable has in this work several meanings that, hopefully, will create no confusions: (a) the coordinate i ∈ {1, ..., n} of μ ∈ Bn is stable if i (μ) = μi , (b) the signal x ∈ S (n) is stable if lim x(k) exists, in particular the state φ α (μ, k) is stable if lim φ α (μ, k) exists, α ∈ n ,

k→∞

k→∞

(c) we introduce now the concept of stability of the system . Theorem 223. The statements ∀μ ∈ Bn , ∃μ ∈ Bn , ∃α ∈ n , lim φ α (μ , k) = μ,

(29.1.1)

∃μ ∈ Bn , ∀μ ∈ Bn , ∃α ∈ n , lim φ α (μ , k) = μ,

(29.1.2)

∃α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , lim φ α (μ , k) = μ,

(29.1.3)

∃α ∈ n , ∃μ ∈ Bn , ∀μ ∈ Bn , lim φ α (μ , k) = μ,

(29.1.4)

∀α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , lim φ α (μ , k) = μ,

(29.1.5)

∀α ∈ n , ∃μ ∈ Bn , ∀μ ∈ Bn , lim φ α (μ , k) = μ,

(29.1.6)

∀μ ∈ Bn , ∃μ ∈ Bn , ∀α ∈ n , lim φ α (μ , k) = μ,

(29.1.7)

k→∞

k→∞

k→∞

k→∞

k→∞

k→∞

k→∞

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00035-0 Copyright © 2023 Elsevier Inc. All rights reserved.

275

276

Boolean Systems

∃μ ∈ Bn , ∀μ ∈ Bn , ∀α ∈ n , lim φ α (μ , k) = μ

(29.1.8)

k→∞

fulfill the following implications (29.1.8) =================================⇒ (29.1.6)   ==  = =  == ==    = =   = ==   == ==   = =   = = ⇒   =   ⇐   (29.1.4) = = = = ⇒ (29.1.2)                                         (29.1.3) = = = = ⇒ (29.1.1)   ⇐=     ⇒ = =   = = =   = = =   = = =   = = =  == ==  = == (29.1.7) =================================⇒ (29.1.5) Proof. (29.1.7)=⇒(29.1.3) A family of functions fα : Bn → Bn is given, α ∈ n , which is defined by ∀μ ∈ Bn , fα (μ ) = lim φ α (μ , k),

(29.1.9)

k→∞

and has the property that μ = fα (μ ) do not depend on α. Then α ∈ n exists (α is arbitrary) and the (unique) function fα : Bn → Bn , defined by (29.1.9). (29.1.8)=⇒(29.1.4) The hypothesis states the existence of μ ∈ Bn with ∀μ ∈ Bn , ∀α ∈ n , lim φ α (μ , k) = μ, in other words the functions fα from (29.1.9) depend on neither of α k→∞

and μ . Then α ∈ n and μ ∈ Bn exist (α is arbitrary) such that ∀μ ∈ Bn , lim φ α (μ , k) = μ. k→∞

(29.1.3)=⇒(29.1.1) From the hypothesis, we have the existence of α ∈ n such that ∀μ ∈ Bn , ∃μ ∈ Bn , lim φ α (μ , k) = μ.

(29.1.10)

k→∞

Let μ ∈ Bn arbitrary, fixed. As (29.1.10) shows the existence of μ ∈ Bn with lim φ α (μ , k) = k→∞

μ, (29.1.1) takes place. (29.1.6)=⇒(29.1.5) Let α ∈ n , μ ∈ Bn arbitrary, fixed. The hypothesis states the existence of μ ∈ Bn (which does not depend on μ ) such that lim φ α (μ , k) = μ. (29.1.5) holds. k→∞

(29.1.6)=⇒(29.1.2) The hypothesis states that for any α ∈ n , we have the existence of μα ∈ Bn such that ∀μ ∈ Bn , lim φ α (μ , k) = μα . k→∞

We suppose against all reason that μα really depends on α, meaning that β = α and μβ = μα exist with the property ∀μ ∈ Bn , lim φ β (μ , k) = μβ . k→∞

Chapter 29 • Stability

277

We know about μα , μβ that they are fixed points of , from Theorem 89, page 104. We have obtained the contradiction μα = lim φ β (μα , k) = μβ . k→∞

We infer that (29.1.6) means after all the existence of μ with the property ∀μ ∈ Bn , lim φ α (μ , k) = μ k→∞

true for any α. (29.1.2) holds. Definition 116. The statements (29.1.1)–(29.1.8) are called of (global, asymptotic) stability. A system  that fulfills one of them is called (globally, asymptotically) stable. Remark 186. In Definition 116, (a) the attribute global refers to the quantification ∀μ ∈ Bn . In the absence of globality, the set X ⊂ Bn , X = ∅ is given and ∀μ ∈ Bn is replaced by ∀μ ∈ X. We can replace (29.1.2) for example with ∃μ ∈ Bn , ∀μ ∈ X, ∃α ∈ n , lim φ α (μ , k) = μ. k→∞

(29.1.11)

(b) the attribute asymptotic refers to the presence of the limit lim φ α (μ , k). A version k→∞

of this definition replaces the asymptotic request lim φ α (μ , k) = μ, i.e. ωα (μ ) ⊂ {μ}, with k→∞

O α (μ ) ⊂ X, and we get the invariance of X. This was anticipated at Remark 125, page 157. A stronger version of stability than (29.1.11) which is derived from (29.1.2) requires the 1-invariance of X and the existence of the final value of the state: ∃μ ∈ X, ∀μ ∈ X, ∃α ∈ n , O α (μ ) ⊂ X and lim φ α (μ , k) = μ. k→∞

Remark 187. The previous remark recalls Definition 110, page 245, of the local basins of attraction. The connection between stability and the basins of attraction of the fixed points is the topic of the third section. Remark 188. In (29.1.2), (29.1.4), (29.1.6), (29.1.8) ∃μ means ∃!μ, i.e. unique existence.1 Indeed, let us suppose against all reason that this is not true in (29.1.2). We have the existence of μ, ν ∈ Bn , μ = ν, such that ∀μ ∈ Bn , ∃α ∈ n , lim φ α (μ , k) = μ and ∀μ ∈ Bn , ∃β ∈ k→∞

n , lim φ β (μ , k) = ν. Then μ, ν are both fixed points and the statement k→∞

μ = lim φ α (μ, k) = lim φ β (μ, k) = ν k→∞

k→∞

is a contradiction. An example for the situation when in (29.1.1) ∃μ is not ∃!μ will be given in Example 147. 1

We have already used this type of reasoning in the proof of Theorem 223, implication (29.1.6)=⇒(29.1.2).

278

Boolean Systems

Remark 189. We conclude that a system is stable if it fulfills the weakest stability property, which is (29.1.1). This means that for any initial value μ ∈ Bn , a fixed point μ ∈ Bn , a computation function α ∈ n , and a final time instant k  ∈ N exist, such that ∀k ≥ k  , φ α (μ , k) = μ.

(29.1.12)

In (29.1.12), the point μ is called sometimes eventually fixed point, or eventually equilibrium. Remark 190. The concept of instability comes, in its most general sense, by negating the strongest stability property, which is (29.1.8), rewritten under the form ∃μ ∈ Bn , ∀μ ∈ Bn , ∀α ∈ n , ωα (μ ) = {μ}. We obtain ∀μ ∈ Bn , ∃μ ∈ Bn , ∃α ∈ n , ωα (μ ) = {μ}. Problem 17. An interesting issue is: to be formulated a Lyapunov-Lagrange type stability theorem, similar with Theorem 142, page 172.

29.2 Examples Example 147. For the system whose state portrait is drawn below (0, 0, 0) 6 (1, 0, 0) 

- (1, 1, 0)  6

- (0, 1, 0)

? (1, 0, 1)

- ?  - (1, 1, 1) 

- ? (0, 1, 1)

(0, 0, 1)

property (29.1.1) is true, and properties (29.1.2), (29.1.4) are false. (29.1.3) is also true,  (0, 0, 1), if μ = (0, 0, 0), Properties (29.1.5)–(29.1.8) are with α k = (1, 1, 1), k ∈ N and μ = (0, 0, 0), if μ = (0, 0, 0). false. Note that in the statement of (29.1.1) for this system ∃μ ∈ B3 means ∃μ ∈ {(0, 0, 0), (0, 0, 1)}, and not ∃!μ ∈ B3 .

Chapter 29 • Stability

279

Example 148. The following system fulfills (29.1.1) and also (29.1.2), with μ = (0, 0, 0). (0, 0, 0) 6 (1, 0, 0) 

? (1, 0, 1)

- (1, 1, 0)  6

- (0, 1, 0)

-

- ? (0, 1, 1) 

?  - (1, 1, 1) 

(0, 0, 1)

We cannot say if (29.1.3), (29.1.4) are true or false. (29.1.5)–(29.1.8) are false. Example 149. In this example (1, 0)



- (0, 0) 

- (0, 1)

? (1, 1) the system satisfies the properties (29.1.1)–(29.1.4). We can take in (29.1.4) α = (1, 1), (1, 1), (1, 1), ... and μ = (1, 1). We see also that (29.1.5) is false. Example 150. In the next state portrait (0, 0, 0)

? (0, 1, 0)

- (0, 0, 1)

(0, 1, 1)

(1, 0, 0)

? (1, 1, 0)

- (1, 0, 1)

(1, 1, 1)

we have a system that fulfills (29.1.5), and (29.1.6), (29.1.7) are false. Example 151. In the next state portrait (0, 0)

- (0, 1)

(1, 0)

- (1, 1)

we have a system for which we notice that (29.1.7) is true and (29.1.6), (29.1.8) are false.

280

Boolean Systems

Example 152. The system - (0, 1)

(0, 0)

? (1, 1)

(1, 0) 

fulfills all the stability properties (29.1.1)–(29.1.8) with μ = (1, 0). Problem 18. We did not give examples in this section for all the possibilities related with the truth of (29.1.1)–(29.1.8), perhaps finding them might be useful. We just mention that we do not know an example when (29.1.1) is true, and all the other stability properties are false.

29.3 Stability vs. the basins of attraction of the fixed points Remark 191. The basins of attraction of the point μ ∈ Bn were characterized in Theorem 194, page 231 by  ∅, if (μ) = μ, W 1 (μ) = {μ |μ ∈ Bn , ∃α ∈  , lim φ α (μ , k) = μ}, otherwise, n k→∞

 Wγ2 (μ) =

∅, if (μ) = μ, lim φ γ (μ , k) = μ}, otherwise,

{μ |μ

∈ Bn ,

{μ |μ

∈ Bn , ∀α

k→∞

 W (μ) = W (μ) = 3

4

 W (μ) = 5

∅, if (μ) = μ, ∈ n , lim φ α (μ , k) = μ}, otherwise, k→∞

∅, if (μ) = μ, {μ}, otherwise,

where γ ∈ n . Here the attractiveness of {μ}, meaning that any of W 1 (μ) = ∅, Wγ2 (μ) = ∅, ..., W 5 (μ) = ∅ holds, is equivalent with (μ) = μ. These are connected with stability in the following way. Theorem 224. We have:



W 1 (μ) = Bn ⇐⇒ (29.1.1),

(29.3.1)

μ∈Bn

∃μ ∈ Bn , W 1 (μ) = Bn ⇐⇒ (29.1.2), ∃α ∈ n ,

 μ∈Bn

Wα2 (μ) = Bn ⇐⇒ (29.1.3),

(29.3.2) (29.3.3)

Chapter 29 • Stability

∃α ∈ n , ∃μ ∈ Bn , Wα2 (μ) = Bn ⇐⇒ (29.1.4), ∀α ∈ n ,



Wα2 (μ) = Bn ⇐⇒ (29.1.5),

281

(29.3.4) (29.3.5)

μ∈Bn

∀α ∈ n , ∃μ ∈ Bn , Wα2 (μ) = Bn ⇐⇒ (29.1.6), 

W 3 (μ) = Bn ⇐⇒ (29.1.7),

(29.3.6) (29.3.7)

μ∈Bn

∃μ ∈ Bn , W 3 (μ) = Bn ⇐⇒ (29.1.8).

(29.3.8)

Proof. (29.3.1) =⇒ We suppose that μ1 , ..., μp ∈ Bn are fixed points, and ∀μ ∈ Bn {μ1 , ..., μp }, W 1 (μ) = ∅. The hypothesis states that W 1 (μ1 ) ∨ ... ∨ W 1 (μp ) = Bn , where the sets W 1 (μ1 ), ..., W 1 (μp ) are not necessarily disjoint. Then ∀μ ∈ Bn , ∃μ ∈ {μ1 , ..., μp }, μ ∈ W 1 (μ), and this means the existence of α ∈ n with lim φ α (μ , k) = μ. k→∞

⇐= For any μ ∈ Bn , we have the existence of the fixed point μ ∈ Bn such that μ ∈ W 1 (μ). We get the existence of the fixed points μ1 , ..., μp with W 1 (μ1 ) ∨ ... ∨ W 1 (μp ) = Bn . (29.3.3) =⇒ We have the existence of α ∈ n and of the fixed points μ1 , ..., μp ∈ Bn such that Wα2 (μ1 ) ∨ ... ∨ Wα2 (μp ) = Bn (the sets Wα2 (μ1 ), ..., Wα2 (μp ) are disjoint). Then ∀μ ∈ Bn , ∃μ ∈ {μ1 , ..., μp } (a unique μ exists) such that μ ∈ Wα2 (μ), i.e. lim φ α (μ , k) = μ. k→∞

⇐= The hypothesis states the existence of α ∈ n such that ∀μ ∈ Bn , ∃μ ∈ Bn , lim φ α (μ , k) = μ. Let μ ∈ Bn arbitrary. We infer the existence of the fixed points μ1 , ..., μp ∈

k→∞ Bn and of μ ∈ {μ1 , ..., μp } depending on μ

such that μ ∈ Wα2 (μ) thus, as far as μ is arbitrary, 2 1 2 we have obtained that Wα (μ ) ∨ ... ∨ Wα (μp ) = Bn . (29.3.7) =⇒ The fixed points μ1 , ..., μp ∈ Bn exist having the property that W 3 (μ1 ) ∨ ... ∨

W 3 (μp ) = Bn (the sets W 3 (μ1 ), ..., W 3 (μp ) are disjoint) and let μ ∈ Bn arbitrary. There exists then μ ∈ {μ1 , ..., μp } (a unique μ exists) with μ ∈ W 3 (μ). We have ∀α ∈ n , lim φ α (μ , k) = k→∞ μ. ⇐= For μ ∈ Bn arbitrary, the fixed point μ ∈ Bn (uniquely) exists, (μ depends on μ ), such that μ ∈ W 3 (μ). This fact shows the existence of the fixed points μ1 , ..., μp ∈ Bn with W 3 (μ1 ) ∨ ... ∨ W 3 (μp ) = Bn .

29.4 Morphisms Theorem 225. The functions  : Bn → Bn , : Bp → Bp and the morphism (h, h ) : φ → ψ are given, with h surjective. (a) If φ is k-stable, k ∈ {(29.1.1), ..., (29.1.4)}, then ψ is k-stable. (b) If φ is k-stable, k ∈ {(29.1.5), ..., (29.1.8)} and h is surjective, then ψ is k-stable.

282

Boolean Systems

Proof. (29.1.1) Let ν  ∈ Bp arbitrary, fixed. As h is surjective, μ ∈ Bn exists with h(μ ) = ν  . The hypothesis (29.1.1) shows the existence of μ ∈ Bn and α ∈ n such that lim φ α (μ , k) = μ.

(29.4.1)

k→∞

We prove that ∃ν ∈ Bp , ∃β ∈ p , lim ψ β (ν  , k) = ν k→∞

holds, with ν = h(μ) and β = h (α). Indeed, Eq. (29.4.1) states the existence of k  ∈ N such that ∀k ≥ k  , φ α (μ , k) = μ.

(29.4.2)

We have ∀k ≥ k  , 

ψ β (ν  , k) = ψ h (α) (h(μ ), k) = h(φ α (μ , k))

(29.4.2)

=

h(μ) = ν.

(29.4.3)

(29.1.3) The hypothesis states the existence of α ∈ n such that ∀μ ∈ Bn , ∃μ ∈ Bn , lim φ α (μ , k) = μ, k→∞

(29.4.4)

and we prove that ∀ν  ∈ Bp , ∃ν ∈ Bp , lim ψ β (ν  , k) = ν k→∞

holds, with β = h (α). Let ν  ∈ Bp arbitrary, fixed. The surjectivity of h gives the existence of μ ∈ Bn with h(μ ) = ν  . We have from (29.4.4) the existence of μ ∈ Bn such that lim φ α (μ , k) = μ. We must prove that lim ψ β (ν  , k) = ν holds, with ν = h(μ). As k  ∈ N ex-

k→∞

k→∞

ists with ∀k ≥ k  , (29.4.2) is true, we can write that ∀k ≥ k  , (29.4.3) takes place. (29.1.7) Let ν  ∈ Bp arbitrary, fixed. We have the existence of μ ∈ Bn such that ∀α ∈ n , lim φ α (μ , k) = μ k→∞

is true, with μ chosen such that h(μ ) = ν  (from the surjectivity of h), and we prove ∀β ∈ p , lim ψ β (ν  , k) = ν, k→∞

with ν = h(μ). We take β ∈ p arbitrary. The surjectivity of h implies the existence of α with β = h (α), and we get also the existence of k  ∈ N such that ∀k ≥ k  , (29.4.2) is true. We conclude that for ∀k ≥ k  , (29.4.3) holds.

Chapter 29 • Stability

283

29.5 Subsystems Theorem 226. We consider the systems : Bn+m → Bn+m and  : Bn → Bn with the property that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, i (μ, ν) = i (μ). (a) If  fulfills the (29.1.1) stability property, then ∀ξ  ∈ Bn+m , ∃ξ ∈ Bn+m , ∃β ∈ n+m , ∀i ∈ {1, ..., n}, lim γi (ξ  , k) = ξi . β

k→∞

(b) If is (29.1.1)-stable, then  is (29.1.1)-stable. Proof. These statements are consequences of Theorem 69, page 75. Remark 192. Similar properties concerning the stability of the subsystems result if we replace in the previous theorem the (29.1.1) request with any of (29.1.2)–(29.1.8). Remark 193. Treating the Cartesian product of stable systems is made similarly with Theorem 226. Remark 194. In the sense of the previous reasoning, we can say that a system with a stable subsystem has a partial stability, and also that the subsystems of a stable system are stable.

29.6 (In)dependence on the initial conditions Theorem 227. The stability properties imply the following statements of independence on the initial conditions:  (29.1.1)page 275 (29.6.1) =⇒ ∀μ ∈ Bn , ∃μ ∈ Bn , ∃α ∈ n , ∃k ∈ N, φ α (μ , k) = φ α (μ, k), ⎧ ⎨

(29.1.2)page 275 =⇒ ∃μ ∈ Bn , ∀μ ∈ Bn , ∃α ∈ n , ∃k ∈ N, φ α (μ , k) = φ α (μ, k) ⎩ ⇐⇒ non (13.1.6)page 120 , 

⎧ ⎨ ⎩

(29.6.2)

(29.1.3)page 275 =⇒ ∃α ∈ n , ∀μ ∈ Bn , ∃μ ∈ Bn , ∃k ∈ N, φ α (μ , k) = φ α (μ, k),

(29.6.3)

(29.1.4)page 275 ∈ Bn , ∃k ∈ N, φ α (μ , k) = φ α (μ, k) ⇐⇒ non (13.1.5)page 120 ,

(29.6.4)

(29.1.5)page 275 n ∈ B , ∃μ ∈ Bn , ∃k ∈ N, φ α (μ , k) = φ α (μ, k),

(29.6.5)

=⇒ ∃α ∈ n

, ∃μ ∈ Bn , ∀μ

 =⇒ ∀α ∈ n

, ∀μ

284

Boolean Systems ⎧ ⎨

(29.1.6)page 275 =⇒ ∀α ∈ n , ∃μ ∈ Bn , ∀μ ∈ Bn , ∃k ∈ N, φ α (μ , k) = φ α (μ, k) ⎩ ⇐⇒ non (13.1.4)page 119 , 

⎧ ⎨ ⎩

(29.6.6)

(29.1.7)page 275 =⇒ ∀μ ∈ Bn , ∃μ ∈ Bn , ∀α ∈ n , ∃k ∈ N, φ α (μ , k) = φ α (μ, k),

(29.6.7)

(29.1.8)page 276 ∈ n , ∃k ∈ N, φ α (μ , k) = φ α (μ, k) ⇐⇒ non (13.1.3)page 119 .

(29.6.8)

=⇒ ∃μ ∈ Bn , ∀μ

∈ Bn , ∀α

Proof. Obvious, since μ in (29.1.1)page 275 , ..., μ in (29.1.8)page 276 is a fixed point and it fulfills ∀α ∈ n , ∀k ∈ N, φ α (μ, k) = μ. Remark 195. A slight unbalance occurs in the previous implications, since (13.1.3)page 119 and (13.1.4)page 119 are equivalent.2

2

Which is the source of this unbalance?

30 Time-reversal symmetry The time-reversal symmetrical systems ,  : Bn −→ Bn are these that run through their orbits in inverse senses, and their definition is given in Section 30.1. Section 30.2 gives examples of time-reversal symmetrical systems. The uniqueness of the symmetrical function is addressed in Section 30.3. We prove in Section 30.4 some properties of these systems, for example if ,  are time− + + − reversal symmetrical, then O (μ) = O (μ) and O (μ) = O (μ). The way that the morphisms act on time-reversal symmetrical systems is shown in Section 30.5. In Section 30.6 we present the relation that exists between time-reversal symmetry and the Cartesian products.

30.1 Definition Theorem 228. Let ,  : Bn −→ Bn . The following statements are equivalent: (a) ∀μ ∈ Bn , + μ−  = μ ,

(30.1.1)

+ μ−  = μ ,

(30.1.2)

∀ν ∈ Bn , ∃λ ∈ Bn , (λ ◦  ν )(μ) = μ,

(30.1.3)

∀λ ∈ Bn , ∃ν ∈ Bn , ( ν ◦ λ )(μ) = μ,

(30.1.4)

∀k ∈ N, ∀β ∈ n , ∃α ∈ n , φ α (ψ β (μ, k), k) = μ,

(30.1.5)

∀k ∈ N, ∀α ∈ n , ∃β ∈ n , ψ β (φ α (μ, k), k) = μ,

(30.1.6)

(b) ∀μ ∈ Bn ,

(c) ∀μ ∈ Bn ,

Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00036-2 Copyright © 2023 Elsevier Inc. All rights reserved.

285

286

Boolean Systems

(d) ∀μ ∈ Bn , ∀k ≥ 1, ∀β ∈ n , ∃α ∈ n , ⎧ ⎪ y(0) = μ, ⎪ ⎪ p ⎪ ⎪ ∀p ∈ {0, ..., k − 1}, y(p + 1) =  β (y(p)), ⎨ x(0) = y(k), ⎪ p ⎪ ⎪ ∀p ∈ {0, ..., k − 1}, x(p + 1) = α (x(p)), ⎪ ⎪ ⎩ x(k) = μ,

(30.1.7)

∀k ≥ 1, ∀α ∈ n , ∃β ∈ n , ⎧ ⎪ x(0) = μ, ⎪ ⎪ p ⎪ ⎪ ∀p ∈ {0, ..., k − 1}, x(p + 1) = α (x(p)), ⎨ y(0) = x(k), ⎪ p ⎪ ⎪ ∀p ∈ {0, ..., k − 1}, y(p + 1) =  β (y(p)), ⎪ ⎪ ⎩ y(k) = μ,

(30.1.8)

(e) ∀μ ∈ Bn , ∀ν ∈ Bn , (μ, ν) ∈ E ⇐⇒ (ν, μ) ∈ E ,

(30.1.9)

where G = (Bn , E ), G = (Bn , E ) are the state portraits of , . n Proof. (a)=⇒(b) We fix μ ∈ Bn , ν ∈ Bn arbitrary. As  ν (μ) ∈ μ+ = μ−   , ∃λ ∈ B such that λ ν ( ◦  )(μ) = μ. (30.1.3) is proved and the proof of (30.1.4) is similar. (b)=⇒(c) We take μ ∈ Bn , k ∈ N arbitrary. If k = 0, then (30.1.5) is true under the form (30.1.1)

∀β ∈ n , ∃α ∈ n , φ α (ψ β (μ, 0), 0) = φ α (μ, 0) = μ, thus we can suppose that k ≥ 1 and let β ∈ n arbitrary, meaning that β 0 , β 1 , ..., β k−1 ∈ Bn are given. We get from the hypothesis: 0

k−1

)(( β

k−2

◦ ... ◦  β )(μ)) = ( β

1

k−2

)(( β

k−3

◦ ... ◦  β )(μ)) = ( β

∃α 0 ∈ Bn , (α ◦  β ∃α 1 ∈ Bn , (α ◦  β

0

k−2

◦ ... ◦  β )(μ),

0

0

k−3

◦ ... ◦  β )(μ),

0

(30.1.10) (30.1.11)

... ∃α k−1 ∈ Bn , (α

k−1

0

◦  β )(μ) = μ,

therefore α ∈ n exists with α 0 , α 1 , ..., α k−1 like in (30.1.10)–(30.1.12) such that φ α (ψ β (μ, k), k) = φ α (( β = (α

k−1

= (α

0

k−1

◦ ... ◦ α )(( β

k−1

0

0

◦ ... ◦  β )(μ), k) 0

k−1

◦ ... ◦  β )(μ))

k−1

◦ ... ◦  β )(μ)

◦ ... ◦ α ◦  β

0

(30.1.12)

Chapter 30 • Time-reversal symmetry

(30.1.10)

=

(α

(30.1.11)

=

k−1

1

◦ ... ◦ α ◦  β

... = (α

k−1

k−2

287

0

◦ ... ◦  β )(μ)

0

◦  β )(μ)

(30.1.12)

=

μ.

(30.1.5) is proved and the proof of (30.1.6) is made similarly. (c)=⇒(d) Let μ ∈ Bn , k ≥ 1, and β ∈ n arbitrary, fixed. We have, by taking into account the hypothesis, the existence of α ∈ n with x(k) = α = (α

k−1

k−1

(x(k − 1)) = ... = (α 0

◦ ... ◦ α )(y(k)) = (α α k−1

= ... = ( α k−1

= (

α0

◦ ... ◦  )(

k−1

α0

◦ ... ◦  )(

β k−1

k−1

0

◦ ... ◦ α )(x(0)) 0

◦ ... ◦ α )( β

β k−1

k−1

(y(k − 1))

β0

◦ ... ◦  )(y(0))

β0

◦ ... ◦  )(μ) = φ α (( β

k−1

0

◦ ... ◦  β )(μ), k)

= φ α (ψ β (μ, k), k) = μ. This completes the proof of (30.1.7). The proof of (30.1.8) is similar. (d)=⇒(e) We take μ ∈ Bn , ν ∈ Bn , arbitrary, fixed, such that (μ, ν) ∈ E . This means that μ = ν, and λ ∈ Bn exists, with ν = λ (μ). We use (30.1.8) written for k = 1 and α ∈ n arbitrary with α 0 = λ. We have the existence of β ∈ n with 0

0

0

0

0

0

μ = y(1) =  β (y(0)) =  β (x(1)) =  β (α (x(0))) =  β (λ (μ)) =  β (ν), i.e. (ν, μ) ∈ E . The other implication is proved similarly. + − n λ (e)=⇒(a) We show that μ−  ⊂ μ , and let ν ∈ μ , i.e. ∃λ ∈ B ,  (ν) = μ. If μ = ν, then + ν ∈ μ , thus we can suppose that μ = ν. We have (ν, μ) ∈ E , thus, from the hypothesis, (μ, ν) ∈ E . This fact implies the existence of δ ∈ Bn with  δ (μ) = ν, i.e. ν ∈ μ+ . + − + n λ We show that μ ⊂ μ and we take ν ∈ μ arbitrary, i.e. ∃λ ∈ B ,  (μ) = ν. If μ = ν, then ν ∈ μ−  , thus we suppose that μ = ν. We have (μ, ν) ∈ E , thus (ν, μ) ∈ E . This means that ∃δ ∈ Bn , δ (ν) = μ, i.e. ν ∈ μ− . The proof of (30.1.2) is similar. Definition 117. If one of the properties (a)–(e) of Theorem 228 is true, we say that  and  (φ and ψ, Eq and Eq , G and G ) are time-reversal symmetrical. Remark 196. The Boolean systems (functions, flows, equations of evolution, state portraits) have several types of symmetry and antisymmetry, of which the time-reversal symmetry introduced by Theorem 228 and Definition 117 is interpreted in the following way: in (a) the predecessors/successors of the points μ relative to  coincide with the successors/predecessors of the points μ relative to , in (b) ,  seem to be ‘inverse’ to each other, in (c) φ, ψ are ‘inverse’ flows, in (d) the equations Eq , Eq give solutions x, y that run in reversed senses of time and in (e) G , G suggest the inversion of causality, as E = (E )−1 .

288

Boolean Systems

30.2 Examples Example 153. The identity function 1Bn : Bn −→ Bn is time-reversal symmetrical with it− self: we have ∀μ ∈ Bn , μ+ 1Bn = μ1Bn = {μ}. Example 154. The constant functions ,  : Bn −→ Bn , ∃ξ ∈ Bn , ∀μ ∈ Bn , (μ) = ξ, (μ) = ξ are time-reversal symmetrical, see Theorem 231 to follow. (0, 0)

- (1, 0)

? (0, 1)

- ? - (1, 1)

(0, 0) 6

(1, 0) 6

(0, 1) 

(1, 1)





We note that in the two state portraits, where n = 2 and ξ = (1, 1), the arrows are the same, and the sense of the arrows is inverted, property (30.1.9). Example 155. To be noticed the time-reversal symmetry of the functions  and  from the following state portraits (0, 0, 0)

- (0, 1, 0)

? (0, 0, 1)

- ? - (0, 1, 1)

(1, 0, 0) 6

(1, 1, 0)

? (1, 0, 1)

? (1, 1, 1)

  (0, 0, 0)  6

(0, 1, 0) 6

(1, 0, 0) 6

(1, 1, 0) 6

(0, 0, 1) 

(0, 1, 1)

? (1, 0, 1)

(1, 1, 1)

 Here the property may be analyzed on three disjoint invariant sets: {(0, 0, 0), (0, 1, 0), (0, 1, 1), (0, 0, 1)}, where the symmetry works like in Example 154; {(1, 0, 0), (1, 0, 1)} where there is an auto-symmetry; and {(1, 1, 0), (1, 1, 1)}, where in E , E the arrows are inverted.

Chapter 30 • Time-reversal symmetry

289

Example 156. The function  defined by the following state portrait (0, 0, 1) 

 (0, 1, 1) 



(0, 0, 0) 6

 (0, 1, 0)

? (1, 1, 1)

-

(1, 0, 1) -

-

-

(1, 0, 0)

(1, 1, 0) accepts a time-reversal symmetrical function . The state portrait of  results by reversing in E the arrows and by underlining the appropriate coordinates. Example 157. The following function  (0, 0)

? (0, 1)

- (1, 0)

(1, 1)

does not accept a time-reversal symmetrical function , because reversing the arrow (0, 0) → (1, 1) is not possible.

30.3 The uniqueness of the symmetrical function Theorem 229. Let the functions , , : Bn −→ Bn . If , are time-reversal symmetrical functions of , then  = . Proof. We suppose against all reason that , are the time-reversal symmetrical functions of  and  = , i.e. μ ∈ Bn exists such that (μ) = (μ), wherefrom μ+  = [μ, (μ)] = + − + [μ, (μ)] = μ+ . We have obtained the contradiction μ = μ = μ .

 

30.4 Properties Remark 197. We suppose that ,  : Bn −→ Bn are time-reversal symmetrical, we take some μ ∈ Bn and we investigate the truth of ∀λ ∈ Bn , ∃ν ∈ Bn , (λ ◦  ν )(μ) = μ,

(30.4.1)

290

Boolean Systems

∀ν ∈ Bn , ∃λ ∈ Bn , ( ν ◦ λ )(μ) = μ,

(30.4.2)

which are to be compared with (30.1.3) and (30.1.4). For this, we see that the negation of (30.4.1): ∃λ ∈ Bn , ∀ν ∈ Bn , (λ ◦  ν )(μ) = μ is true in Example 154 for λ = (1, 1) and μ = (0, 1): ∀ν ∈ B2 , ( ν (0, 1)) = (1, 1) = (0, 1). The theorem that follows shows that for any μ ∈ Bn , (30.4.1), (30.4.2) take place under a weaker form. Theorem 230. For ,  : Bn −→ Bn and μ ∈ Bn , + μ−  = μ

(30.4.3)

∀λ ∈ Bn , (λ )−1 (μ) = ∅ =⇒ ∃ν ∈ Bn , (λ ◦  ν )(μ) = μ,

(30.4.4)

+ μ−  = μ

(30.4.5)

∀ν ∈ Bn , ( ν )−1 (μ) = ∅ =⇒ ∃λ ∈ Bn , ( ν ◦ λ )(μ) = μ.

(30.4.6)

implies

and

implies

Proof. (30.4.3)=⇒(30.4.4). For μ ∈ Bn , λ ∈ Bn arbitrary, we suppose that we have (λ )−1 (μ) + n

= ∅ and let μ ∈ (λ )−1 (μ) be arbitrary too. As μ ∈ μ−  = μ , some ν ∈ B exists with μ = ν λ ν λ  (μ). We have ( ◦  )(μ) =  (μ ) = μ, thus (30.4.4) is true. (30.4.5)=⇒(30.4.6). Similar. Example 158. We continue the example from Remark 197. We have taken there λ = (1, 1) giving in Example 154 that −1 (0, 1) = ∅. But we can take the value λ = (0, 1), for which ((0,1) )−1 (0, 1) = {(0, 0), (0, 1)} and then, for ν = (1, 1) we infer (λ ◦  ν )(0, 1) = (0,1) ((0, 1)) = (0,1) (0, 0) = (0, 1), the property that we were looking for. Theorem 231. If ,  : Bn → Bn are constant functions: ∃ξ ∈ Bn , ∀μ ∈ Bn , (μ) = ξ, (μ) = ξ , then ,  are time-reversal symmetrical.

Chapter 30 • Time-reversal symmetry

291

Proof. Let μ ∈ Bn arbitrary. We infer μ−  μ+ 

Theorem 11, page 13

=

(1.7.5)page 11

=

[(μ), μ] = [ξ , μ] = [μ, ξ ] = [μ, (μ)]

[μ, (μ)] = [μ, ξ ] = [ξ, μ] = [ξ , μ] = [(μ), μ]

(1.7.5)page 11

=

μ+ ,

Theorem 11, page 13

=

μ− .

Theorem 232. The following statements are equivalent: (a) ,  are time-reversal symmetrical, (b) ∗ ,  ∗ are time-reversal symmetrical. Proof. (a)=⇒(b) Let μ ∈ Bn arbitrary. The hypothesis states the truth of (30.1.1), (30.1.2), wherefrom + μ−  = μ ,

(30.4.7)

+ μ−  = μ .

(30.4.8)

We conclude, by using Theorem 12, page 13 that − μ− ∗ = μ 

(30.4.7)

=

+ μ+  = μ ∗ ,

− μ−  ∗ = μ

(30.4.8)

+ μ+  = μ ∗ .

=

The rest of the proof is similar. Theorem 233. We suppose that ,  : Bn → Bn are time-reversal symmetrical. For any μ ∈ Bn , (a) the sets of states which are reached from μ in k time units Ak (μ) ⊂ Bn , k ∈ N and the sets of states that reach μ in k time units Bk (μ) ⊂ Bn , k ∈ N, which were defined at Definitions 79, 80, page 112, satisfy ∀k ∈ N,  A k (μ) = Bk (μ),

(30.4.9)

Bk (μ) = A k (μ);

(30.4.10)

− + (μ) = O (μ), O

(30.4.11)

+ − O (μ) = O (μ).

(30.4.12)

(b) we have

Proof. (a) We prove (30.4.9) by induction on k ∈ N. The result is true for k = 0, when both terms are equal with {μ}, and we suppose that it is true for k arbitrary. We infer:    + − −  δ = δ = δ = Bk+1 (μ). A k+1 (μ) = δ∈A k (μ)

δ∈A k (μ)

δ∈Bk (μ)

292

Boolean Systems

The proof of (30.4.10) is similar. (b) We have − O (μ)



(12.2.12)page 115

=

Bk (μ)

(30.4.10)

=

k∈N + O (μ)

(12.2.11)page 115

=





A k (μ)

(12.2.11)page 115

=

+ O (μ),

Bk (μ)

(12.2.12)page 115

− O (μ).

k∈N

A k (μ)

k∈N

(30.4.9)

=



=

k∈N

We see that ∀μ ∈ Bn , ∀k

Remark 198. ∈ N, ((30.4.9) and (30.4.10)) implies that  and  are time-reversal symmetrical, i.e. we infer the equivalence of ∀μ ∈ Bn , ∀k ∈ N, ((30.4.9) and (30.4.10)) with the time-reversal symmetry of  and . Problem 19. Does the property ∀μ ∈ Bn , ((30.4.11) and (30.4.12)) imply the time-reversal symmetry of , ?

30.5 Morphisms vs. time-reversal symmetry Theorem 234. We consider the functions , , , ϒ : Bn −→ Bn . (a) If (h, h ) ∈ H om(γ , φ),  and  are time-reversal symmetrical and (g, g ) ∈ H om(ψ, υ), then ∀μ ∈ Bn , + g(h(μ−

)) ⊂ g(h(μ))ϒ ,

(30.5.1)

− g(h(μ+

)) ⊂ g(h(μ))ϒ ,

(30.5.2)

g(h(O − (μ))) ⊂ Oϒ+ (g(h(μ))),

(30.5.3)

g(h(O + (μ))) ⊂ Oϒ− (g(h(μ)));

(30.5.4)

in the special case when (h, h ) ∈ I so(γ , φ), (g, g ) ∈ I so(ψ, υ), the inclusions (30.5.1)–(30.5.4) become equalities. (b) We suppose that and  are time-reversal symmetrical, (h, h ) ∈ H om(φ, ψ), while  and ϒ are time-reversal symmetrical too. Then ∀μ ∈ Bn , − h(μ−

) ⊂ h(μ)ϒ ,

(30.5.5)

+ h(μ+

) ⊂ h(μ)ϒ ,

(30.5.6)

h(O − (μ)) ⊂ Oϒ− (h(μ)),

(30.5.7)

h(O + (μ)) ⊂ Oϒ− (h(μ));

(30.5.8)

in the special case when (h, h ) ∈ I so(φ, ψ), the inclusions (30.5.5)–(30.5.8) become equalities.

Chapter 30 • Time-reversal symmetry

293

Proof. We fix arbitrarily μ ∈ Bn and we use Theorem 80, page 92. (a) If (h, h ) ∈ H om(γ , φ), (g, g ) ∈ H om(ψ, υ), we have − + h(μ−

) ⊂ h(μ) = h(μ) , + − h(μ+

) ⊂ h(μ) = h(μ) ,

wherefrom − + + g(h(μ−

)) ⊂ g(h(μ) ) = g(h(μ) ) ⊂ g(h(μ))ϒ , + − − g(h(μ+

)) ⊂ g(h(μ) ) = g(h(μ) ) ⊂ g(h(μ))ϒ .

In the special case when (h, h ) ∈ I so(γ , φ), (g, g ) ∈ I so(ψ, υ), the inclusions are replaced by equalities. (b) If (h, h ) ∈ H om(φ, ψ), we can write + + − h(μ−

) = h(μ ) ⊂ h(μ) = h(μ)ϒ , − − + h(μ+

) = h(μ ) ⊂ h(μ) = h(μ)ϒ ,

and when (h, h ) ∈ I so(φ, ψ), the inclusions become equalities. Remark 199. If in Theorem 234 (a), in the special case when (h, h ) ∈ I so(γ , φ), (g, g ) ∈ I so(ψ, υ), we have g ◦ h = 1Bn , then the time-reversal symmetry of and ϒ follows. We have also the special case of this special case, when (h, h ), (g, g ) are the duality isomorphisms: then = ∗ is time-reversal symmetrical with ϒ =  ∗ , see Theorem 232, page 291.

30.6 Cartesian product Theorem 235. We consider the functions ,  : Bn −→ Bn , ϒ,  : Bm −→ Bm having the property that is the time-reversal symmetrical of , and ϒ is the time-reversal symmetrical of . Then × ϒ,  ×  : Bn+m −→ Bn+m are time-reversal symmetrical. Proof. We use Theorem 15, page 15 and let μ ∈ Bn , ν ∈ Bm arbitrary. We infer: − − + + + (μ, ν)− × = μ × ν = μ × νϒ = (μ, ν) ×ϒ , + + − − − (μ, ν)+ × = μ × ν = μ × νϒ = (μ, ν) ×ϒ .

31 Generator functions with one parameter In the following we prepare step by step the definition of the input systems. Since these systems are operated, preferably, with the input kept constant, until the state reaches a final value (the so called fundamental operating mode), we introduce for the moment, in Section 31.1, the generator functions with one parameter,  : Bn × Bm → Bn . This means that from now when writing (μ, ν), the first argument μ ∈ Bn will be considered variable, standing for the state, and the second argument ν ∈ Bm will be considered parameter, standing for the input. The iterates (k) and λ of  are defined in Section 31.2. The Cartesian product of generator functions with one parameter is introduced in Section 31.3. The successors and the predecessors of  are redefined in this new framework in Section 31.4. The state portrait family of  : Bn × Bm → Bn is the set G = {G(·,ν) |ν ∈ Bm } of state portraits, and this is the topic of Section 31.5. The bifurcations represent the situations when two parameters ν, ν  ∈ Bm exist, ν = ν  , such that (·, ν) and (·, ν  ) are not topologically equivalent. They are addressed in Section 31.6. The morphisms of generator functions with one parameter are introduced in Section 31.7.

31.1 Generator functions with one parameter Remark 200. Until now, the functions  : Bn → Bn , Bn  μ −→ (μ) ∈ Bn ,

(31.1.1)

called sometimes generator functions, or systems, had the argument μ ∈ Bn , considered as variable. We shall replace them from this moment with functions  : Bn × Bm → Bn , Bn × Bm  (μ, ν) −→ (μ, ν) ∈ Bn ,

(31.1.2)

that may be referred to as generator functions with one parameter, or input systems as well. These terminological issues will be clarified further in Chapter 33. In (31.1.2), the first argument μ ∈ Bn plays the role of the variable from (31.1.1), standing for the state, Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00037-4 Copyright © 2023 Elsevier Inc. All rights reserved.

295

296

Boolean Systems

and μ1 , ..., μn are called the state coordinates, while the second argument ν ∈ Bm is the parameter, standing for the input, and ν1 , ..., νm are called the input coordinates. Definition 118. The dual ∗ : Bn × Bm → Bn of  is defined as ∀μ ∈ Bn , ∀ν ∈ Bm , ∗ (μ, ν) = (μ, ν). Notation 39. For  : Bn × Bm → Bn and ν ∈ Bm , we denote with ν : Bn → Bn the function ∀μ ∈ Bn , ν (μ) = (μ, ν).

31.2 Iterates Definition 119. We consider  : Bn × Bm → Bn . The function (k) : Bn × Bm → Bn , k ∈ N that is called the k-th iterate of , is defined as: ∀μ ∈ Bn , ∀ν ∈ Bm , (k) (μ, ν) = (ν )(k) (μ),

(31.2.1)

where (ν )(k) is the one from Definition 13, page 8. Definition 120. For  : Bn ×Bm → Bn and λ ∈ Bn , we define the function λ : Bn ×Bm → Bn , called the λ-iterate of , by ∀μ ∈ Bn , ∀ν ∈ Bm , λ (μ, ν) = (ν )λ (μ),

(31.2.2)

see Definition 14, page 8 for the meaning of (ν )λ . Remark 201. In order to understand better these definitions, we fix μ ∈ Bn , ν ∈ Bm arbitrary. We get (0) (μ, ν) = (ν )(0) (μ) = μ, and for any k ∈ N, (k+1) (μ, ν) = (ν )(k+1) (μ) = ν ((ν )(k) (μ)) = ((k) (μ, ν), ν). On the other hand, for arbitrary λ ∈ Bn , i ∈ {1, ..., n}, we have  ( )i (μ, ν) = ((ν ) )i (μ) = λ

λ

μi , if λi = 0, = (ν )i (μ), if λi = 1

Remark 202. Eqs. (31.2.1), (31.2.2), interpreted as ((k) )ν = (ν )(k) , (λ )ν = (ν )λ



μi , if λi = 0, i (μ, ν), if λi = 1.

Chapter 31 • Generator functions with one parameter

297

since there μ continues to be variable and ν continues to be parameter, allow the notations (k) ν , λν for any of ((k) )ν , (ν )(k) , and any of (λ )ν , (ν )λ . Remark 203. Definitions 119 and 120 show how the iterations of the generator function  depending on the parameter ν ∈ Bm can be made: (k) iterates all its coordinates, these are the synchronous iterations, and λ iterates some coordinates only, these are the asynchronous iterations.

31.3 Cartesian product of functions Definition 121. Let the functions  : Bn+p × Bm+q → Bn+p ,  : Bn × Bm → Bn and  : Bp × Bq → Bp . If ∀μ ∈ Bn , ∀ν ∈ Bm , ∀δ ∈ Bp , ∀ξ ∈ Bq ,  i (μ, ν), if i ∈ {1, ..., n}, i ((μ, δ), (ν, ξ )) = i−n (δ, ξ ), if i ∈ {n + 1, ..., n + p}, then  is called the Cartesian product of  and . The usual notation is  =  × . Theorem 236. The following is true: ∀ν ∈ Bm , ∀ξ ∈ Bq , ( × )(ν,ξ ) = ν × ξ . Proof. Indeed, for any μ ∈ Bn , δ ∈ Bp , ν ∈ Bm , ξ ∈ Bq we infer ( × )(ν,ξ ) (μ, δ) = ( × )((μ, δ), (ν, ξ )) = ((μ, ν), (δ, ξ )) = (ν (μ), ξ (δ)) = (ν × ξ )(μ, δ). Theorem 237. For  : Bn × Bm → Bn ,  : Bp × Bq → Bp we have ( × )∗ = ∗ ×  ∗ , ( × )(k) = (k) ×  (k) , ( × )(λ,ζ ) = λ ×  ζ , where k ∈ N, λ ∈ Bn , ζ ∈ Bp . Proof. We prove the third assertion: ∀μ ∈ Bn , ∀ν ∈ Bm , ∀δ ∈ Bp , ∀ξ ∈ Bq , ( × )(λ,ζ ) ((μ, δ), (ν, ξ ))

(31.2.2)

= ((ν )λ (μ), (ξ )ζ (δ))

=

(( × )(ν,ξ ) )(λ,ζ ) (μ, δ)

(31.2.2)

=

Theorem 236

=

(ν × ξ )(λ,ζ ) (μ, δ)

(λ (μ, ν),  ζ (δ, ξ )) = (λ ×  ζ )((μ, δ), (ν, ξ )).

Remark 204. Theorem 236 shows the existing compatibility between the Definition 121 of  =  ×  and Definition 15, page 10, while Theorem 237 is to be compared with Theorem 8, page 10.

298

Boolean Systems

31.4 Successors and predecessors Definition 122. Let  : Bn × Bm → Bn . We have the following sets of successors and predecessors: ∀μ ∈ Bn , ∀ν ∈ Bm , λ n μ+ ν = {ν (μ)|λ ∈ B }, + O (μ) = {(λν ◦ ... ◦ γν )(μ)|λ ∈ Bn , ..., γ ∈ Bn }, ν n n λ μ− ν = {δ|δ ∈ B , ∃λ ∈ B , ν (δ) = μ}, − O (μ) = {δ|δ ∈ Bn , ∃λ ∈ Bn , ..., ∃γ ∈ Bn , (λν ◦ ... ◦ γν )(δ) = μ}. ν

Remark 205. The properties of these sets of successors and predecessors of μ depending on ν are the expected ones, for example Theorem 13, page 14 becomes Theorem 238. The functions ,  : Bn × Bm −→ Bn are given. We have ∀μ ∈ Bn , ∀ν ∈ + Bm , μ+ ν = μν if and only if  = . Proof. If. Let μ ∈ Bn , ν ∈ Bm arbitrary. We can write: + μ+ ν = [μ, (μ, ν)] = [μ, (μ, ν)] = μν . + Only if. In case that ∀μ ∈ Bn , ∀ν ∈ Bm , μ+ ν = μν we can apply Theorem 3 (e), page 4 and get (μ, ν) = (μ, ν).

31.5 State portrait families Notation 40. If the function H : Bn × Bm → P ({1, ..., n}) is given, we denote with Hν : Bn → P ({1, ..., n}), ν ∈ Bm , the function ∀μ ∈ Bn , Hν (μ) = H (μ, ν). Definition 123. A state portrait family (of dimension n × m) is a set GH = {(Bn , E Hν )|ν ∈ Bm }, where (Bn , E Hν ) are state portraits (of dimension n), ν ∈ Bm , defined as E Hν = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ Hν (μ)} i∈A

(31.5.1)

by means of the function H : Bn × Bm → P ({1, ..., n}). Definition 124. The state portrait family of  : Bn × Bm → Bn is defined like this: G = {(Bn , Eν )|ν ∈ Bm }, where Eν = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  ν (μ)}. i∈A

(31.5.2)

Notation 41. The set of the state portrait families of dimension n × m is denoted with Spn,m .

Chapter 31 • Generator functions with one parameter

299

Example 159. The function  : B2 × B −→ B2 , ∀μ ∈ B2 , ∀ν ∈ B, (μ, ν) = (μ1 μ2 , ν(μ1 ∪ μ2 )) defines 0 , 1 : B2 −→ B2 by ∀μ ∈ B2 , 0 (μ) = (μ1 μ2 , 0), 1 (μ) = (μ1 μ2 , μ1 ∪ μ2 ). The state portrait family of , consisting in the state portraits of 0 , 1 , was drawn below. (0, 0)  6 (1, 0) 

(0, 1)

(0, 0) 6

(0, 1) -

(1, 1)

(1, 0)

- (1, 1)

ν=0

ν=1

Remark 206. The state portrait families make us think of evolution under constant ν, where ν is the parameter. Notation 42.  : Bn × Bm → Bn and H : Bn × Bm → P ({1, ..., n}) are given. We denote with H : Bn × Bm → Bn and H : Bn × Bm → P ({1, ..., n}) the functions ∀μ ∈ Bn , ∀ν ∈ Bm , H (μ, ν) = μ ⊕



εi ,

i∈H (μ,ν)

H (μ, ν) = μ  (μ, ν). Remark 207. The relations between Definitions 123, 124 are G = {(Bn , Eν )|ν ∈ Bm } = {(Bn , E (H )ν )|ν ∈ Bm } = GH , GH = {(Bn , E Hν )|ν ∈ Bm } = {(Bn , E(H )ν )|ν ∈ Bm } = GH , analogously with Remark 24, page 33. Notation 43. We denote with Fn,m the set of the functions with a parameter  : Bn × Bm → Bn . Theorem 239. We define the functions U : Fn,m → Spn,m , ∀ ∈ Fn,m , U () = G ,

300

Boolean Systems

and V : Spn,m → Fn,m , ∀GH ∈ Spn,m , V (GH ) = H . The following statements hold: (a) For any  : Bn × Bm → Bn and H : Bn × Bm → P ({1, ..., n}), HH = H, H = , (b) U ◦ V = 1Spn,m and V ◦ U = 1Fn,m . Proof. The proof coincides formally with the proof of Theorem 33, page 33.

31.6 Bifurcations Definition 125. Let  : Bn × Bm → Bn . The topological equivalence (Definition 22, page 23) of the Bn −→ Bn functions defines the equivalence ≡ on Bm like this: ∀ν ∈ Bm , ∀ν  ∈ Bm , ν ≡ ν  ⇐⇒ I so(ν , ν  ) = ∅. The equivalence class of ν modulo ≡, denoted [ν], is called a stratum (of ). The set of the strata is called the parametric portrait of . Definition 126. If ∃ν ∈ Bm , ∃ν  ∈ Bm such that not ν ≡ ν  , we say that a bifurcation of  exists, or that  has a bifurcation (resulted at the variation of the parameter ν). Definition 127. We suppose that  has a bifurcation and let the strata [ν], ..., [ν  ] be its parametric portrait. The sequence of state portraits of ν , ..., ν  is called the bifurcation diagram of . Example 160. The function  : B2 × B2 −→ B2 , ∀μ ∈ B2 , ∀ν ∈ B2 , (μ1 , μ2 , ν1 , ν2 ) = (ν1 ν2 μ1 ∪ ν1 ν2 (μ1 ∪ μ2 ) ∪ ν1 ν2 μ1 ∪ ν1 ν2 μ1 μ2 , ν1 ν2 (μ1 ∪ μ2 ) ∪ ν1 ν2 μ2 ∪ ν1 ν2 μ1 μ2 ∪ ν1 ν2 μ2 ) gives for the four possible values of the parameter ν ∈ B2 the generator functions: (0,0) (μ1 , μ2 ) = (μ1 , μ1 ∪ μ2 ), (0,1) (μ1 , μ2 ) = (μ1 ∪ μ2 , μ2 ), (1,1) (μ1 , μ2 ) = (μ1 , μ1 μ2 ),

Chapter 31 • Generator functions with one parameter

301

(1,0) (μ1 , μ2 ) = (μ1 μ2 , μ2 ), whose state portraits were drawn below. (0, 0)

(1, 0)

(0, 0)

(1, 0)

? (0, 1)

(1, 1)

(0, 1)

- (1, 1)

ν = (0, 0)

ν = (0, 1)

(0, 0)

(1, 0) 6

(0, 1)

(1, 1)

(0, 0) 

(1, 0)

(0, 1)

(1, 1)

ν = (1, 1)

ν = (1, 0)

Function  has no bifurcations, since (0,0) , ..., (1,1) are topologically equivalent. Example 161. Let the function  : B2 × B −→ B2 defined by ∀μ ∈ B2 , ∀ν ∈ B, (μ1 , μ2 , ν) = (μ1 ∪ νμ2 , μ2 ). The state portrait family of , consisting in the state portraits of 0 (μ1 , μ2 ) = (μ1 ∪ μ2 , μ2 ), 1 (μ1 , μ2 ) = (μ1 , μ2 ) has been drawn below. (0, 0)

(1, 0)

(0, 0)

(1, 0)

(0, 1)

- (1, 1)

(0, 1)

(1, 1)

ν=0

ν=1

Since 0 , 1 are not topologically equivalent, we get that this is the bifurcation diagram of .

302

Boolean Systems

Example 162. The function  : B2 × B −→ B2 , ∀μ ∈ B2 , ∀ν ∈ B, (μ1 , μ2 , ν) = (μ1 , ν μ2 ∪ ν) (0, 0) 6

(1, 0) 6

(0, 0)

(1, 0)

? (0, 1)

? (1, 1)

? (0, 1)

? (1, 1)

ν=0

ν=1

has the property that G = {(θ (0,0) , 1B2 ), (θ (1,0) , 1B2 )} is a group with ∀(h, h ) ∈ G, ∀μ ∈ B2 , ∀ν ∈ B, ∀δ ∈ B2 , 

h (δ) (h(μ), ν) = h(δ (μ, ν)), i.e. ∀ν ∈ B, G ⊂ Aut (ν ). This is the example of a bifurcation with symmetry.

31.7 Morphisms Definition 128. A morphism from  : Bn × Bm → Bn to  : Bp × Bq → Bp , denoted (h, h , h ) :  → , is given by three functions h, h : Bn → Bp , h : Bm → Bq such that for any λ ∈ Bn , the diagram λ - n B

Bn × Bm h × h

h  (λ) ? ? h  - Bp Bp × Bq

is commutative or equivalently, ∀ν ∈ Bm , ∀λ ∈ Bn , the diagram Bn h ? Bp

λν - n B h (λ) h (ν)

h ? - Bp

commutes. This means that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀λ ∈ Bn , 

h(λ (μ, ν)) =  h (λ) (h(μ), h (ν)). Notation 44. We denote by H om(, ) the set of the morphisms from  to . Example 163. For  : Bn × Bm → Bn , we have the morphism (1Bn , 1Bn , 1Bm ) :  −→ , which is usually denoted 1 .

Chapter 31 • Generator functions with one parameter

303

Example 164. The functions  : Bn × Bm → Bn ,  : Bp × Bq → Bp are given and we define π, π  : Bn+p → Bn , π  : Bm+q → Bm by π(μ, δ) = μ, π  (λ, ρ) = λ, π  (ν, ξ ) = ν, where μ, λ ∈ Bn , ν ∈ Bm , δ, ρ ∈ Bp , ξ ∈ Bq . We have π(( × )(λ,ρ) ((μ, δ), (ν, ξ ))) = π((λ ×  ρ )((μ, δ), (ν, ξ ))) = π(λ (μ, ν),  ρ (δ, ξ )) 



= λ (μ, ν) = π (λ,ρ) (π(μ, δ), π  (ν, ξ )) = π (λ,ρ) ((π × π  )((μ, δ), (ν, ξ ))), i.e. the diagram Bn+p × Bm+q

( × )(λ,ρ) -

π × π  ? Bn × Bm

π  (λ,ρ)



Bn+p

π ? - Bn

is commutative and (π, π  , π  ) :  ×  →  is a morphism. Remark 208. The composition of the morphisms (h, h , h ) :  → , (f, f  , f  ) :  →  is (f ◦ h, f  ◦ h , f  ◦ h ) :  → . The isomorphisms (h, h , h ) :  →  are defined by the requests that h, h , h are bijections and their set is denoted by I so(, ). In case that I so(, ) = ∅, the generator functions ,  are said to be topologically equivalent (or conjugated). Example 165. Given  : Bn × Bm → Bn and ν ∈ Bn , we have the morphism (1Bn , h , 1Bm ) : ν → , where h : Bn → Bn is defined by ∀λ ∈ Bn , h (λ) = λν, see Theorem 27, page 28. Example 166. (See also Theorem 24, page 24.) The function  : Bn × Bm → Bn and the bijections s : {1, ..., n} → {1, ..., n}, t : {1, ..., m} → {1, ..., m} are considered. We define the bijections js : Bn → Bn , jt : Bm → Bm by ∀μ ∈ Bn , ∀ν ∈ Bm , js (μ1 , ..., μn ) = (μs(1) , ..., μs(n) ), jt (ν1 , ..., νm ) = (νt (1) , ..., νt (m) ). We get the existence of  : Bn × Bm → Bn ,  = js ◦  ◦ (js × jt )−1 = js ◦  ◦ (js −1 × jt −1 ), such that (js , js , jt ) ∈ I so(, ).  coincides with  modulo the order of its coordinates. Theorem 240. Given ,  : Bn × Bm −→ Bn , the following property holds:  is the dual of  if and only if (θ δ , 1Bn , θ ξ ) ∈ I so(, ), where δ = (1, ..., 1) ∈ Bn , ξ = (1, ..., 1) ∈ Bm . Proof. The if part of the proof was made in the parameterless case for Theorem 25, page 26.

304

Boolean Systems

Only if. Let μ ∈ Bn , ν ∈ Bm and λ ∈ Bn arbitrary, fixed. The hypothesis states that i.e.

 ∗ (μ, ν) = (μ, ν),

( λ )∗ (μ, ν) = ( ∗ )λ (μ, ν) = λ (μ, ν). We infer  λ (μ, ν) = λ (μ, ν) = (θ δ ◦ λ ◦ (θ δ × θ ξ ))(μ, ν). As far as (θ δ × θ ξ )−1 = (θ δ × θ ξ ), this gives  λ ◦ (θ δ × θ ξ ) = θ δ ◦ λ , i.e. (θ δ , 1Bn , θ ξ ) ∈ I so(, ).

32 Input flows and equations of evolution The input flows are the flows which are generated by the functions  : Bn × Bm → Bn and they are defined in Section 32.1. Theorems that characterize the causality and the composition properties of the input flows are given in Section 32.2. The equations of evolution of the  : Bn × Bm → Bn generator functions are addressed in Section 32.3. The morphisms of input flows are presented in Section 32.4.

32.1 Input flows Definition 129. We consider the function  : Bn × Bm → Bn . The (input) flow (or evolution function, or state transition function, or next state function) φ : n × Bn × S (m) × N → Bn is defined by ∀α ∈ n , ∀μ ∈ Bn , ∀u ∈ S (m) , ∀k ∈ N,  φ (μ, u, k) = α

μ, if k = 0, − 1), u(k − 1)), if k ≥ 1.

k−1 α (φ α (μ, u, k

Function  is called the generator function of φ, and we use to say that φ is generated by . The signal x ∈ S (n) given by x(k) = φ α (μ, u, k) is called state (function); k is the present time, 0 is the initial time, μ is the initial (value of the) state, α is the computation function and u is the input (or the control). Example 167. The function  : Bn × Bm → Bn defined by ∀μ ∈ Bn , ∀ν ∈ Bm , (μ, ν) = μ fulfills the property (which may be proved by induction on k) that ∀α ∈ n , ∀μ ∈ Bn , ∀u ∈ S (m) , ∀k ∈ N, φ α (μ, u, k) = μ. Notation 45. We denote with F ln,m the set of the flows φ of the  ∈ Fn,m generator functions: F ln,m = {φ| ∈ Fn,m }. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00038-6 Copyright © 2023 Elsevier Inc. All rights reserved.

305

306

Boolean Systems

Theorem 241. We define the functions U : Fn,m → F ln,m , ∀ ∈ Fn,m , U () = φ, and V : F ln,m → Fn,m , ∀φ ∈ F ln,m , ∀μ ∈ Bn , ∀ν ∈ Bm , V (φ)(μ, ν) = φ α (μ, u, 1), where α ∈ n is arbitrary, such that α 0 = (1, ..., 1), and u ∈ S (m) is arbitrary such that u(0) = ν. (a) ∀φ ∈ F ln,m , V (φ) = , (b) U ◦ V = 1F ln,m and V ◦ U = 1Fn,m . Proof. The theorem is similar with Theorem 58, page 61. Notation 46. For any α ∈ n , μ ∈ Bn and u ∈ S (m) , we denote O α (μ, u) = {φ α (μ, u, k)|k ∈ N}, −

O α (μ, u) = {δ|δ ∈ Bn , ∃k ∈ N, φ α (δ, u, k) = μ}.

32.2 Causality and composition Theorem 242. (Causality, or non-anticipation) The function  : Bn × Bm → Bn is given, and let α ∈ n , β ∈ n , u, v ∈ S (m) , μ ∈ Bn , k  ≥ 1 arbitrary. If ∀k ∈ {0, ..., k  − 1}, αk = β k ,

(32.2.1)

u(k) = v(k),

(32.2.2)

φ α (μ, u, k) = φ β (μ, v, k).

(32.2.3)

then ∀k ∈ {0, ..., k  },

Proof. At k = 0, the statement of the theorem is true since both terms of (32.2.3) are equal with μ. We suppose that (32.2.3) is true for an arbitrary k ∈ {0, ..., k  − 1} and we infer k

φ α (μ, u, k + 1) = α (φ α (μ, u, k), u(k)) (32.2.2)

=

k

α (φ β (μ, v, k), v(k))

(32.2.1)

=

(32.2.3)

=

k

α (φ β (μ, v, k), u(k))

k

β (φ β (μ, v, k), v(k)) = φ β (μ, v, k + 1).

Theorem 243. (Composition) ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , ∀u ∈ S (m) , ∀k  ∈ N, φ α (μ, u, k  ) = μ =⇒ ∀k ∈ N, φ α (μ, u, k + k  ) = φ σ

k  (α)



(μ , σ k (u), k).

Chapter 32 • Input flows and equations of evolution

307

Proof. The hypothesis states that φ α (μ, u, k  ) = μ and we make the proof by induction on k ∈ N. For k = 0 the equality is true, thus we suppose that it is true for k. We infer, since    α k+k = σ k (α)k and u(k + k  ) = σ k (u)(k): φ α (μ, u, k + k  + 1) = α = σ

k  (α)k

(φ σ

k  (α)



k+k 

(φ α (μ, u, k + k  ), u(k + k  ))



(μ , σ k (u), k), σ k (u)(k)) = φ σ

k  (α)



(μ , σ k (u), k + 1).

Theorem 244. The function  : Bn × Bm −→ Bn is given.  (a) For p ≥ 1 and α 0 , ..., α p ∈ n , μ0 , ..., μp ∈ Bn , u0 , ..., up ∈ S (m) , k0 , ..., kp−1 ≥ 1 such that φ α (μ0 , u0 , k0 ) = μ1 , 0

... φα

p−1

 (μp−1 , up−1 , kp−1 ) = μp ,

we define γ ∈ n and v ∈ S (m) in the following way: ⎧ ⎪ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪ ⎪ 1,k−k0 ⎪ α , if k ∈ {k0 , ..., k0 + k1 − 1}, ⎪ ⎨ ... γk =  ⎪ p−1,k−k0 −...−kp−2    ⎪ ⎪ , if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 − 1}, ⎪ α ⎪   ⎪ p,k−k0 −...−kp−1 ⎩   α , if k ≥ k + ... + k , 0

v(k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

p−1

u0 (k), if k ∈ {0, ..., k0 − 1}, − k0 ), if k ∈ {k0 , ..., k0 + k1 − 1}, ...    up−1 (k − k0 − ... − kp−2 ), if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 − 1},     p u (k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 .

We have

φ γ (μ0 , v, k) =

u1 (k

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

φ α (μ0 , u0 , k), if k ∈ {0, ..., k0 }, 1 α φ (μ1 , u1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p−1    α p−1 p−1 (μ ,u , k − k0 − ... − kp−2 ), if k ∈ {k0 + ... + kp−2 , φ   ..., k0 + ... + kp−1 }, p   φ α (μp , up , k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 . 0

(b) We consider α 0 , ..., α p , ... ∈ n , μ0 , ..., μp , ... ∈ Bn , u0 , ..., up , ... ∈ S (m) , k0 , ..., kp , ... ≥ 1 with the property that φ α (μ0 , u0 , k0 ) = μ1 , 0

(32.2.4)

308

Boolean Systems

... φ α (μp , up , kp ) = μp+1 , p

(32.2.5)

... For the computation function γ ∈ n and v ∈ S (m) defined as: ⎧ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪  ⎪ 1,k−k ⎪ 0 , if k ∈ {k  , ..., k  + k  − 1}, ⎪ α ⎨ 0 0 1 ... γk =  ⎪ p,k−k0 −...−kp−1 ⎪  ⎪ , if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, α ⎪ ⎪ ⎩ ...

v(k) =

we have

φ γ (μ0 , v, k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u0 (k), if k ∈ {0, ..., k0 − 1}, − k0 ), if k ∈ {k0 , ..., k0 + k1 − 1}, ...   ), if k ∈ {k0 + ... + kp−1 , ..., up (k − k0 − ... − kp−1   k0 + ... + kp − 1}, ...

(32.2.6)

u1 (k

(32.2.7)

φ α (μ0 , u0 , k), if k ∈ {0, ..., k0 }, ... p   ), if k ∈ {k0 + ... + kp−1 , ..., φ α (μp , up , k − k0 − ... − kp−1   k0 + ... + kp }, ... 0

Proof. (b) We make the proof by induction on p ≥ 0. For p = 0, we have ∀k ∈ {0, ..., k0 − 1}, γk v(k)

(32.2.6)

α 0,k ,

(32.2.7)

u0 (k),

=

=

and Theorem 242 implies ∀k ∈ {0, ..., k0 }, 0

φ γ (μ0 , v, k) = φ α (μ0 , u0 , k), wherefrom φ γ (μ0 , v, k0 ) = φ α (μ0 , u0 , k0 ) 0

(32.2.4)

=

μ1 .

The hypothesis of the induction is, for p ≥ 1:  ) = φα φ γ (μ0 , v, k0 + ... + kp−1

p−1

 (μp−1 , up−1 , kp−1 ) = μp .

(32.2.8)

Chapter 32 • Input flows and equations of evolution

309

We notice that ∀k ∈ {0, ..., kp − 1}, σ σ

 k0 +...+kp−1

 k0 +...+kp−1

(γ )k

(v)(k)

(32.2.6)

α p,k ,

(32.2.7)

up (k).

=

=

(32.2.9) (32.2.10)

We have ∀k ∈ {0, ..., kp }:  φ γ (μ0 , v, k + k0 + ... + kp−1 ) Theorem 243

=

φσ

 k0 +...+kp−1 (γ )

(32.2.8)

=

φ

 (φ γ (μ0 , v, k0 + ... + kp−1 ), σ

 k  +...+kp−1 σ 0 (γ )

(μp , σ

(32.2.9), (32.2.10), Theorem 242

=

 k0 +...+kp−1

 k0 +...+kp−1

(v), k)

(v), k)

p

φ α (μp , up , k),

therefore  φ γ (μ0 , v, k0 + ... + kp−1 + kp ) = φ α (μp , up , kp ) p

(32.2.5)

=

μp+1 .

Remark 209. Theorem 244 shows what happens in the case that α 0 , ..., α p , ... and u0 , ..., up , ... are ‘concatenated’.

32.3 Equations of evolution Remark 210. We consider  : Bn × Bm −→ Bn and the equations ∀k ∈ N, x(k) = φ α (μ, u, k), k

y(k + 1) = α (y(k), u(k)),

(32.3.1)

⎧ ⎪ ⎨ z1 (k + 1) = 1 (z(k), u(k))α1k ∪ z1 (k)α1k , ... ⎪ ⎩ zn (k + 1) = n (z(k), u(k))αnk ∪ zn (k)αnk ,

(32.3.2)

see Theorem 65, page 67. For any α ∈ n , μ ∈ Bn , and u ∈ S (m) , we suppose that y(0) = z(0) = μ. Then x = y = z. Definition 130. Eqs. (32.3.1), (32.3.2) are called, each of them, equation of evolution of  (or of φ) and their notation is Eq . Notation 47. The set of the equations of evolution of the functions  : Bn × Bm → Bn has the notation Eqn,m : Eqn,m = {Eq | ∈ Fn,m }.

310

Boolean Systems

Theorem 245. We define the functions U : Fn,m → Eqn,m , ∀ ∈ Fn,m , U () = Eq , and V : Eqn,m → Fn,m , ∀Eq ∈ Eqn,m , ∀μ ∈ Bn , ∀ν ∈ Bm , V (Eq )(μ, ν) = y(1).

(32.3.3)

In the last equation, y is the solution of (32.3.1) with the initial value μ, α ∈ n arbitrary, such that α 0 = (1, .., 1) and u ∈ S (m) arbitrary such that u(0) = ν. We have: (a) ∀Eq ∈ Eqn,m , V (Eq ) = , (b) U ◦ V = 1Eqn,m and V ◦ U = 1Fn,m . Proof. The theorem is similar with Theorem 66, page 67.

32.4 Morphisms Definition 131. The functions  : Bn × Bm → Bn ,  : Bp × Bq → Bp , h : Bn → Bp , h : n → p , h : S (m) → S (q) are given. We say that the triple (h, h , h ) defines a morphism from φ to ψ, denoted (h, h , h ) : φ → ψ, if ∀α ∈ n , ∀μ ∈ Bn , ∀u ∈ S (m) , ∀k ∈ N, 

h(φ α (μ, u, k)) = ψ h (α) (h(μ), h (u), k).

(32.4.1)

Notation 48. We denote with H om(φ, ψ) the set of morphisms from φ to ψ.  Example 168. For the function  : Bn × Bm → Bn , the triple (1Bn , 1 Bn , 1 Bm ) : φ → φ is a n (m) morphism, since ∀α ∈ n , ∀μ ∈ B , ∀u ∈ S , ∀k ∈ N,  1Bn (φ α (μ, u, k)) = φ α (μ, u, k) = φ 1Bn (α) (1Bn (μ), 1 Bm (u), k).

Example 169. Let  : Bn × Bm → Bn arbitrary, and  : Bp × Bq → Bp defined by ∀δ ∈ Bp , ∀ξ ∈ Bq , (δ, ξ ) = δ. If h : Bn → Bp is the constant function: ∃δ  ∈ Bp , ∀μ ∈ Bn , h(μ) = δ  and h : n → p , h : S (m) → S (q) are arbitrary, then (h, h , h ) : φ → ψ is a morphism: ∀α ∈ n , ∀μ ∈ Bn , ∀u ∈ S (m) , ∀k ∈ N, h(φ α (μ, u, k)) = δ 

Example 167, page 305

=



ψ h (α) (δ  , h (u), k)



= ψ h (α) (h(μ), h (u), k). Remark 211. The composition of the morphisms (h, h , h ) : φ → ψ, (f, f  , f  ) : ψ → υ is the morphism (f, f  , f  ) ◦ (h, h , h ) : φ → υ, (f, f  , f  ) ◦ (h, h , h ) = (f ◦ h, f  ◦ h , f  ◦ h ).

Chapter 32 • Input flows and equations of evolution

311

The isomorphisms (h, h , h ) : φ → ψ are the morphisms for which h, h , h are bijections, and their set is denoted I so(φ, ψ). When I so(φ, ψ) = ∅, the flows φ, ψ are said to be topologically equivalent, or conjugated. The isomorphisms (h, h , h ) : φ → φ are called automorphisms, and the set of the automorphisms of φ is denoted by Aut (φ). Theorem 246. We consider the functions  : Bn × Bm → Bn ,  : Bp × Bq → Bp . (a) If (h, h , h ) ∈ H om(φ, ψ) and g  ∈ n,p , g  : Bm → Bq exist, with the property that ∀α ∈ n , h (α) = g (α),

(32.4.2)

∀u ∈ S (m) , h (u) = g (u),

(32.4.3)

then (h, g  , g  ) ∈ H om(, ).

 , h ) ∈ (b) If h : Bn → Bp , h ∈ n,p , h : Bm → Bq fulfill (h, h , h ) ∈ H om(, ), then (h, h H om(φ, ψ).

Proof. (a) We notice first that writing (32.4.2) is possible since g  ∈ n,p . We take μ ∈ Bn , ν ∈ Bm , λ ∈ Bn arbitrary, and we take also α ∈ n , u ∈ S (m) arbitrary, such that α 0 = λ, u(0) = ν. We have hyp



h(λ (μ, ν)) = h(α (μ, u(0))) = h(φ α (μ, u, 1)) = ψ h (α) (h(μ), h (u), 1) 0

(32.4.2),(32.4.3)

=

 0   ψ g (α) (h(μ), g (u), 1) =  g (α ) (h(μ), g  (u(0))) =  g (λ) (h(μ), g  (ν)).

(b) We take α ∈ n , μ ∈ Bn and u ∈ S (m) arbitrary, and we must prove that ∀k ∈ N,  h(φ α (μ, u, k)) = ψ h (α) (h(μ), h (u), k)

(32.4.4)

is true. We note that (32.4.4) makes sense because h ∈ n,p . The proof is made by induction on k ∈ N. For k = 0, Eq. (32.4.4) is true with both terms equal with h(μ), so that we can suppose now its truth for k. We get: 

h(φ α (μ, u, k + 1)) = h(α (φ α (μ, u, k), u(k))) =  h (α ) (h(φ α (μ, u, k)), h (u(k))) k

(32.4.4)

=





k



 h (α ) (ψ h (α) (h(μ), h (u), k), h (u(k))) = ψ h (α) (h(μ), h (u), k + 1). k

Definition 132. If (h, h , h ) ∈ H om(, ) and h ∈ n,p , we say that the morphism (h, h , h ) of generator functions with one parameter induces the morphism of input flows  , h ) ∈ H om(φ, ψ). (h, h Example 170. The morphism (1Bn , 1Bn , 1Bm ) :  −→  induces the morphism (1Bn , 1 Bn ,  1Bm ) : φ −→ φ, see Example 168, page 310. Remark 212. Morphisms of generator functions exist which cannot induce morphisms of flows, and also morphisms of flows exist that cannot define morphisms of generator functions.

33 Input systems We summarize in the first section some previous constructions and results, allowing us to identify the generator functions with one parameter , the state portrait families G , the input flows φ and the equations of evolution Eq , like in the case of the autonomous systems. Any of them is defined in Section 33.2 as input system. The input subsystems are introduced in Section 33.3, and Section 33.4 deals with the state space decomposition of the input systems. This refers to the situation when, instead of analyzing the whole system, we can analyze systems with a smaller dimension. The Cartesian product of input systems is introduced in Section 33.5. We indicate in Section 33.6 three equivalent possibilities of addressing autonomy: (a) the system has no input, (b) the system has input, but the generator function does not depend on it, (c) the system has input, which is constant.

33.1 Several equivalent perspectives Remark 213. We have used the notation Fn,m for the set of the  : Bn × Bm → Bn functions, and recall that ∀μ ∈ Bn , ∀ν ∈ Bm , ν (μ) = (μ, ν). We have also denoted with Spn,m the set of the state portrait families G of dimension n × m. In addition, F ln,m is the set of the flows φ of the generator functions , and the set of the equations of evolution Eq of the functions  has the notation Eqn,m . The following functions are defined: Fn,m

Fn,m U

6 V

U

? Spn,m

6 V

? F ln,m

Fn,m 6 V  ? Eqn,m

U 

(i) U : Fn,m → Spn,m , ∀ ∈ Fn,m , U () = G , where G = {(Bn , Eν )|ν ∈ Bm } is the family of state portraits given by Eν = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ μ  ν (μ)}, i∈A

V : Spn,m → Fn,m , ∀GH ∈ Spn,m , V (GH ) = H , Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00039-8 Copyright © 2023 Elsevier Inc. All rights reserved.

313

314

Boolean Systems

where H : Bn × Bm → P ({1, ..., n}), GH = {(Bn , E Hν )|ν ∈ Bm } is the family with E Hν = {(μ, μ ⊕  ε i )|μ ∈ Bn , ∅ = A ⊂ Hν (μ)} i∈A

and ∀μ ∈ Bn , ∀ν ∈ Bm , H (μ, ν) = μ ⊕



εi .

i∈Hν (μ)

We have proved at Theorem 239, page 299, that U ◦ V = 1Spn,m , V ◦ U = 1Fn,m . (ii) U  : Fn,m → F ln,m , ∀ ∈ Fn,m , U  () = φ, V  : F ln,m → Fn,m , ∀φ ∈ F ln,m , V  (φ) : Bn × Bm −→ Bn is given by ∀μ ∈ Bn , ∀ν ∈ Bm , V  (φ)(μ, ν) = φ α (μ, u, 1), where α ∈ n is arbitrary with α 0 = (1, ..., 1) ∈ Bn and u ∈ S (m) is arbitrary with u(0) = ν. Theorem 241, page 306 states that U  ◦ V  = 1F ln,m , V  ◦ U  = 1Fn,m . (iii) U  : Fn,m → Eqn,m is defined by: ∀ ∈ Fn,m , U  () = Eq , V  : Eqn,m → Fn,m , ∀Eq ∈ Eqn,m , V  (Eq) : Bn × Bm −→ Bn is given by ∀μ ∈ Bn , ∀ν ∈ Bm , V  (Eq)(μ, ν) = x(1), where x is the solution of Eq with the initial value x(0) = μ, α is arbitrary with α 0 = (1, ..., 1) ∈ Bn and u ∈ S (m) is arbitrary with u(0) = ν. Then Theorem 245, page 310 states that U  ◦ V  = 1Eqn,m , V  ◦ U  = 1Fn,m hold. We conclude that we can identify the elements of Fn,m , Spn,m , F ln,m , Eqn,m .

33.2 Definition Definition 133. An n-dimensional Boolean, universal, regular, asynchronous, nondeterministic, non-initialized, discrete time input system (or network) with unbounded delays, shortly a system, is any of (a) a function  : Bn × Bm −→ Bn , (b) a family G = {(Bn , Eν )|ν ∈ Bm }, (c) a flow φ : n × Bn × S (m) × N −→ Bn , (d) an equation Eq . Function  is the generator function of the system, G is its state portrait family, φ is its flow and Eq is its equation of evolution; signal x(k) = φ α (μ, u, k) is the state, α ∈ n is the computation function (of ), μ ∈ Bn is the initial (value of the) state, u ∈ S (m) is the input (or control) and k ∈ N is the time (instant). Remark 214. This definition is to be compared with Definition 62, page 72, where the word ‘autonomous’ was replaced with the word ‘input’. A synonym for ‘input’ (system) is ‘control’ (system).

Chapter 33 • Input systems

315

Remark 215. If we change the unbounded delays to bounded delays, or fixed delays, we reL place in the previous framework n with ≤L n or n , see Notations 21, 22, page 57 (or with other spaces of progressive computation functions that fit the intuition of giving bounded, respectively fixed delays).

33.3 Subsystem Lemma 4. Let  : Bn × Bm → Bn , : Bn1 × Bm → Bn1 , with 0 < n1 < n. We suppose that ∀μ ∈ Bn1 , ∀δ ∈ Bn−n1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n1 }, i ((μ, δ), ν) = i (μ, ν).

(33.3.1)

Then for any μ, λ ∈ Bn1 , δ, ξ ∈ Bn−n1 , ν ∈ Bm and i ∈ {1, ..., n1 }, we get (λ,ξ )

i

((μ, δ), ν) = iλ (μ, ν).

(33.3.2)

Proof. Indeed, we can write that 

μi , if λi = 0, i ((μ, δ), ν), if λi = 1

(λ,ξ ) i ((μ, δ), ν) = (33.3.1)



=

μi , if λi = 0, = iλ (μ, ν). i (μ, ν), if λi = 1

Theorem 247. We suppose that  and fulfill the hypothesis of Lemma 4. Then for any μ ∈ Bn1 , δ ∈ Bn−n1 , α ∈ n1 , β ∈ n−n1 , u ∈ S (m) , k ∈ N and i ∈ {1, ..., n1 }, we have (α,β)

φi

((μ, δ), u, k) = ψiα (μ, u, k).

(33.3.3)

Proof. We take μ ∈ Bn1 , δ ∈ Bn−n1 , α ∈ n1 , β ∈ n−n1 and u ∈ S (m) arbitrary, fixed. We prove that ∀i ∈ {1, ..., n1 }, (33.3.3) is true by induction on k ≥ 0. For k = 0 both terms of the equation are equal with μi , thus we suppose that ∀i ∈ {1, ..., n1 }, (33.3.3) is true for arbitrary k > 0 and we prove it for k + 1. We denote with ζ ∈ Bn−n1 the last n − n1 coordinates of φ (α,β) ((μ, δ), u, k): φ (α,β) ((μ, δ), u, k) = (ψ α (μ, u, k), ζ ),

(33.3.4)

thus ∀i ∈ {1, ..., n1 }, (α,β)

φi

(α,β)k

((μ, δ), u, k + 1) = i (33.3.4)

=

Lemma 4

=

(α k ,β k )

i

(φ (α,β) ((μ, δ), u, k), u(k))

((ψ α (μ, u, k), ζ ), u(k))

k

iα (ψ α (μ, u, k), u(k)) = ψiα (μ, u, k + 1).

Definition 134. If  and fulfill the property that ∀μ ∈ Bn1 , ∀δ ∈ Bn−n1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n1 }, (33.3.1) is true, the system generated by is said to be a subsystem of the system generated by  and we denote ⊂ . By definition,  is a subsystem of itself.

316

Boolean Systems

Remark 216. Working with the first n1 coordinates of the state does not restrict the generality, see Example 166, page 303. Remark 217. The possibility exists that the subsystem ⊂  does not depend on all of ν1 , ..., νm . We define ‘ does not depend on νi ’, i ∈ {1, ..., m} in the following way1 : ∀μ ∈ Bn1 , ∀ν ∈ Bm , (μ1 , ..., μn1 , ν1 , ..., νi , ..., νm ) = (μ1 , ..., μn1 , ν1 , ..., νi , ..., νm ). Remark 218. The fact that is a subsystem of  is characterized by the commutativity of the diagram Bn × Bm

(λ,ξ ) - n B

π × 1Bm ? Bn1 × Bm

π ?

λ - n1 B

where λ ∈ Bn1 , ξ ∈ Bn−n1 and π : Bn → Bn1 is defined as ∀μ ∈ Bn1 , ∀δ ∈ Bn−n1 , π(μ, δ) = μ. In other words, (π, π, 1Bm ) :  → and (π,  π , 1 Bm ) : φ → ψ are morphisms.

33.4 State space decomposition Definition 135. The function  : Bn × Bm → Bn is given, with the property that the numbers 0 = n0 < n1 < n2 < ... < np = n and the functions 1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 , 2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ..., p : Bnp −np−1 ×Bnp−1 +m → Bnp −np−1 exist such that ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, ⎧ 1 (μ1 , ν), if i ∈ {n0 + 1, .., n1 }, ⎪ ⎪ ⎪ 2 i−n02 ⎪ 1 ⎪ ⎨ i−n1 (μ , (μ , ν)), if i ∈ {n1 + 1, ..., n2 }, ... i ((μ1 , ..., μp ), ν) = (33.4.1) ⎪ p p , (μ1 , ..., μp−1 , ν)), ⎪ ⎪ (μ ⎪ i−np−1 ⎪ ⎩ if i ∈ {np−1 + 1, ..., np }. Then 1 , 2 , ..., p are called a state space decomposition of . Or, equivalently,  : Bn1 × Bm−1 → Bn1 exists such that (μ1 , ..., μn1 , ν1 , ..., νi , ..., νm ) =  (μ1 , ..., μn1 , ν1 , ..., νi , ..., νm ), where νi indicates that νi is missing. 1

Chapter 33 • Input systems

317

Remark 219. The existence of 1 , 2 , ..., p has the meaning of showing that a decomposition μ = (μ1 , μ2 , ..., μp ) of Bn = Bn1 −n0 × Bn2 −n1 × ... × Bnp −np−1 exists, such that i , i ∈ {n0 + 1, ..., n1 } do not depend on μ2 , μ3 , ..., μp , i , i ∈ {n1 + 1, ..., n2 } do not depend on μ3 , ..., μp , ... i , i ∈ {np−1 + 1, ..., np } depend on all of μ1 , μ2 , ..., μp . Remark 220. Note in Definition 135 the way that the state coordinates μ1 of  become input coordinates of 2 , ..., the state coordinates μ1 , ..., μp−1 of  become input coordinates of p . Remark 221. The syntagm ‘state space decomposition’ has previously occurred at Problem 4, page 77, stated for the autonomous systems. It was a parallel state space decomposition there, and it is a cascading state space decomposition here. The autonomous systems have no cascading state space decomposition in autonomous systems, but in the case of the input systems, the parallel state space decomposition in input systems can be considered, see the next section. Theorem 248. Let the function  : Bn × Bm → Bn , the numbers 0 = n0 < n1 < n2 < ... < np = n, and we suppose that 1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 , 2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ..., p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 exist with the property that ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, (33.4.1) is true. For arbitrary α 1 ∈ n1 −n0 , α 2 ∈ n2 −n1 , ..., α p ∈ np −np−1 , μ1 ∈ Bn1 −n0 , μ2 ∈ Bn2 −n1 , ..., μp ∈ Bnp −np−1 and u ∈ S (m) , we denote with v 1 ∈ S (n0 +m) , v 2 ∈ S (n1 +m) , ..., v p ∈ S (np−1 +m) the functions2 ∀k ∈ N, v 1 (k) = u(k), 1

v 2 (k) = ((ψ 1 )α (μ1 , v 1 , k), u(k)), ... 1

v p (k) = ((ψ 1 )α (μ1 , v 1 , k), ..., (ψ p−1 )α

p−1

(μp−1 , v p−1 , k), u(k)).

We have ∀k ∈ N, φ (α 1

1 ,α 2 ,...,α p )

((μ1 , μ2 , ..., μp ), u, k) 2

p

= ((ψ 1 )α (μ1 , v 1 , k), (ψ 2 )α (μ2 , v 2 , k), ..., (ψ p )α (μp , v p , k)). 2

See in (33.4.1) the second argument of 1 , 2 , ..., p .

318

Boolean Systems

Proof. We prove that the property is true by induction on k ≥ 0 and let α 1 ∈ n1 −n0 , α 2 ∈ n2 −n1 , ..., α p ∈ np −np−1 , μ1 ∈ Bn1 −n0 , μ2 ∈ Bn2 −n1 , ..., μp ∈ Bnp −np−1 , u ∈ S (m) arbitrary, fixed. For k = 0 we have φ (α

1 ,α 2 ,...,α p )

((μ1 , μ2 , ..., μp ), u, 0) = (μ1 , μ2 , ..., μp )

1

2

p

= ((ψ 1 )α (μ1 , v 1 , 0), (ψ 2 )α (μ2 , v 2 , 0), ..., (ψ p )α (μp , v p , 0)). We suppose that the statement of the theorem is true for an arbitrary k and we prove it for k + 1 in the following way: ∀i ∈ {1, ..., n}, (α 1 ,α 2 ,...,α p )

φi

(α 1 ,α 2 ,...,α p )k

= i hyp

= i(α

1,k ,α 2,k ,...,α p,k )

Lemma 4

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

(φ (α

((μ1 , μ2 , ..., μp ), u, k + 1)

1 ,α 2 ,...,α p )

1

((μ1 , μ2 , ..., μp ), u, k), u(k)) 2

p

(((ψ 1 )α (μ1 , v 1 , k), (ψ 2 )α (μ2 , v 2 , k), ..., (ψ p )α (μp , v p , k)), u(k)) 1,k

1

α 2,k

2

1 ( i−n )α ((ψ 1 )α (μ1 , v 1 , k), v 1 (k)), if i ∈ {n0 + 1, ..., n1 }, 0

((ψ 2 )α (μ2 , v 2 , k), v 2 (k)), if i ∈ {n1 + 1, ..., n2 }, ⎪ ... ⎪ ⎪ p,k p ⎪ α p α p p ⎩ ( p ((ψ ) (μ , v , k), v p (k)), if i ∈ {np−1 + 1, ..., np } i−np−1 ) ⎧ 1 1 (ψi−n )α (μ1 , v 1 , k + 1), if i ∈ {n0 + 1, ..., n1 }, ⎪ ⎪ 0 ⎪ 2 ⎨ 2 (ψi−n )α (μ2 , v 2 , k + 1), if i ∈ {n1 + 1, ..., n2 }, 1 = ⎪ ... ⎪ ⎪ p p ⎩ α p p (ψi−np−1 ) (μ , v , k + 1), if i ∈ {np−1 + 1, ..., np }. 2 ( i−n ) 1

Remark 222. In literature, the cascading state space decomposition of  : Bn × Bm → Bn refers to the existence of the numbers 0 = n0 < n1 < n2 < ... < np = n and of the functions 1 : Bn1 × Bm → Bn1 −n0 , 2 : Bn2 × Bm → Bn2 −n1 , ..., p : Bnp × Bm → Bnp −np−1 such that ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, ⎧ 1i−n0 (μ1 , ν), if i ∈ {n0 + 1, .., n1 }, ⎪ ⎪ ⎪ ⎪ 2 1 2 ⎪ ⎨ i−n1 ((μ , μ ), ν), if i ∈ {n1 + 1, ..., n2 }, 1 p ... i ((μ , ..., μ ), ν) = ⎪ p 1 ⎪ ⎪ i−np−1 ((μ , ..., μp−1 , μp ), ν), ⎪ ⎪ ⎩ if i ∈ {np−1 + 1, ..., np }, to be compared with Definition 135.

Chapter 33 • Input systems

319

33.5 Cartesian product Theorem 249. Let  : Bn × Bm → Bn , : Bp × Bq → Bp and  : Bn+p × Bm+q → Bn+p ,  =  × , see Definition 121, page 297. Then ∀α ∈ n , ∀β ∈ p , ∀μ ∈ Bn , ∀ν ∈ Bp , ∀u ∈ S (m) , ∀v ∈ S (q) , ∀k ∈ N, γ (α,β) ((μ, ν), (u, v), k) = (φ α (μ, u, k), ψ β (ν, v, k))

(33.5.1)

holds. Proof. Let us take α ∈ n , β ∈ p , μ ∈ Bn , ν ∈ Bp , u ∈ S (m) , v ∈ S (q) arbitrary and we prove (33.5.1) by induction on k ≥ 0. For k = 0, the equation is true with both terms equal with (μ, ν), thus we suppose its truth for arbitrary k > 0. We infer: k

γ (α,β) ((μ, ν), (u, v), k + 1) =  (α,β) (γ (α,β) ((μ, ν), (u, v), k), (u(k), v(k))) (33.5.1)

=

( × )(α k

k ,β k )

((φ α (μ, u, k), ψ β (ν, v, k)), (u(k), v(k)))

k

= (α × β )((φ α (μ, u, k), ψ β (ν, v, k)), (u(k), v(k))) k

k

= (α (φ α (μ, u, k), u(k)), β (ψ β (ν, v, k), v(k))) = (φ α (μ, u, k + 1), ψ β (ν, v, k + 1)). Problem 20. We consider the system  : Bn × Bm → Bn for which the following problem is   formulated: to be found : Bn × Bm → Bn isomorphic with  such that 1 : Bn1 × Bm1 →     Bn1 , ..., p : Bnp × Bmp → Bnp exist with = 1 × ... × p , where n1 + ... + np = n and m1 + ... + mp = m hold. Remark 223. Note that the parallel state space decomposition from Problem 20 is a special case of cascading state space decomposition from Definition 135.

33.6 Autonomy Theorem 250. The functions  : Bn × Bm −→ Bn , : Bn −→ Bn are given and let α ∈ n , μ ∈ Bn , u ∈ S (m) . Either of ∀μ ∈ Bn , ∀ν ∈ Bm , (μ , ν) = (μ ),

(33.6.1)

∃ν ∈ Bm , ∀k ∈ N, u(k) = ν and ∀μ ∈ Bn , (μ , ν) = (μ )

(33.6.2)

φ α (μ, u, k) = ψ α (μ, k).

(33.6.3)

implies ∀k ∈ N,

320

Boolean Systems

Proof. (33.6.1)=⇒(33.6.3) We prove first that for arbitrary μ ∈ Bn , ν ∈ Bm , λ ∈ Bn we have λ (μ , ν) = λ (μ ).

(33.6.4)

This follows from the fact that ∀i ∈ {1, ..., n},   μi , if λi = 0, μi , if λi = 0, (33.6.1) λ  = iλ (μ ). i (μ , ν) = =  i (μ , ν), if λi = 1 i (μ ), if λi = 1 Let now α ∈ n , μ ∈ Bn , u ∈ S (m) arbitrary. We make the proof by induction on k ∈ N. For k = 0, (33.6.3) is true with both terms equal with μ. We suppose that it is true for arbitrary k. Then k

φ α (μ, u, k + 1) = α (φ α (μ, u, k), u(k)) (33.6.4)

=

(33.6.3)

=

k

α (ψ α (μ, k), u(k))

k

α (ψ α (μ, k)) = ψ α (μ, k + 1).

Remark 224. The previous theorem gives autonomy the following interpretations: (a) there is no input ( is in this situation); (b) an input exists, but the generator function  does not depend on it (property (33.6.1)); (c) an input exists, but it is constant (property (33.6.2)). In our approach, autonomy was studied by working with functions : Bn −→ Bn , case (a). We can think however that the evolution takes place under constant input, case (c), until the final value of the state is reached, and at that moment two possibilities exist: either (c.1) the input function, that has reached its final value also, continues to be constant, or (c.2) the input function switches to a new value and a new evolution under constant input begins. This is called the fundamental mode of operation of the asynchronous systems.

34 The fundamental (operating) mode In Section 34.1 we show that the upper boundedness of the delays with which the coordinates of  are computed does not insure the existence of an upper bound of the transfer time from one value of the state to another. This remark is followed by several other similar remarks in Section 34.2, trying to state some common sense requests concerning the work of the input systems. Section 34.3, dedicated to the fundamental (operating) mode of the asynchronous systems, gathers to some extent such requests. The fundamental mode consists in keeping the input constant until the final value of the state is reached, and possibly changing it to a new value afterwards.

34.1 An introductory remark Example 171. We consider the autonomous system with the following state portrait (0, 0) 

(1, 0) 

- (0, 1)

? (1, 1) and let the sequence α 0 , ..., α p , ... ∈ 2 of computation functions α p = (0, 1), (1, 1), (0, 1), (1, 1), ..., (0, 1), (1, 1), (1, 1), ..., 0

1

2

3

2p

2p+1

2p+2

p ∈ N, where the terms of rank ≥ 2p + 3 are arbitrary such that α p ∈ ≤1 2 : p,k 

∀i ∈ {1, 2}, ∀k ∈ N, ∃k  ∈ {k, k + 1}, αi The state x(k) = φ

αp

= 1.

⎧ ⎨ (0, 0), if k ∈ {0, 2, ..., 2p + 2}, ((0, 0), k) = (0, 1), if k ∈ {1, 3, ..., 2p + 1}, ⎩ (1, 1), if k ≥ 2p + 3

has the property that the transfer from (0, 0) to (1, 1) takes place in unbounded time, in the following sense: ∀L ∈ N, ∃p ∈ N, 2p + 3 > L and ∀k ∈ {0, ..., 2p + 2}, x(k) = (1, 1) and x(2p + 3) = (1, 1) Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00040-4 Copyright © 2023 Elsevier Inc. All rights reserved.

321

322

Boolean Systems

in spite of the fact that the durations of computation of 1 , 2 are superiorly bounded by 2. Remark 225. The previous example shows that asking the existence of an upper bound for the delays with which the coordinates of  are computed insures the existence of no upper bound of the transfer time of the state from one value to another. This is the reason why, even if in this chapter we shall ask that the input is kept constant until the transfer between rest positions is surely completed, the computation functions α ∈ n will not be restricted to α ∈ ≤L n , L ≥ 1: what ‘surely’ means needs further investigations.

34.2 Looking for common sense requests Remark 226. The function  : Bn × Bm → Bn is given. Let α, β ∈ n , μ, μ ∈ Bn , u, u ∈ S (m) and we suppose that for some k  ≥ 1 we have: φ α (μ, u, k  ) = μ .

(34.2.1)

(a) If we define γ ∈ n , v ∈ S (m) by  k α , if k ∈ {0, ..., k  − 1}, γk =  β k−k , if k ≥ k  ,  v(k) =

u(k), if k ∈ {0, ..., k  − 1}, u (k − k  ), if k ≥ k  ,

then Theorem 244 (a), page 307 implies for p = 1, that  α φ (μ, u, k), if k ∈ {0, ..., k  }, γ φ (μ, v, k) = φ β (μ , u , k − k  ), if k ≥ k  . This is the way that a system ‘concatenates’ the computation functions α, β and the inputs u, u . (b) In addition to (34.2.1) we ask that ∀k ≥ k  , φ α (μ, u, k) = μ , and for an arbitrary k1 ≥ k  we define γ ∈ n , v ∈ S (m) in the following way:  k α , if k ∈ {0, ..., k1 − 1}, k γ = β k−k1 , if k ≥ k1 ,  v(k) = Then:

u(k), if k ∈ {0, ..., k1 − 1}, u (k − k1 ), if k ≥ k1 . 

φ (μ, v, k) = γ

φ α (μ, u, k), if k ∈ {0, ..., k1 }, φ β (μ , u , k − k1 ), if k ≥ k1 ,

(34.2.2)

Chapter 34 • The fundamental (operating) mode

323

to be compared with (a). The request (34.2.2) is that the intermediate point μ is a rest position: its meaning is that, in conditions of uncertainty, we can wait there as long as necessary until the transfer from μ to μ was surely completed and switch from u to u afterwards only. Remark 227. It is convenient in practice to ask the fulfillment of some other expectations too. (a) μ is reached in (34.2.1) under constant input: ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν; the advantage that results is that we can use the study of the autonomous systems (i.e. systems without input), which are equivalent with the systems with constant input, see Theorem 250, page 319. In this case the generator function is ν : Bn → Bn . Moreover, keeping the input constant brings no additional uncertainties. (b) all the computation functions insure the transfer from μ to μ , ∀α ∈ n , μ ∈ O α (μ, u),

(34.2.3)

i.e. in (34.2.1) μ, μ , u are fixed and α, k  are variable. This property, called the technical condition of proper operation, is desirable because the states x(k) = φ α (μ, u, k) have unknown computation functions, and we must consider all the possibilities. (c) similar with (b), but (34.2.3) is replaced by  α ∀α ∈ ≤L n , μ ∈ O (μ, u),

(34.2.4)

the delays that occur in the computation of 1 , ..., n are superiorly bounded, α ∈ ≤L n , for some L ≥ 1. This request is natural, but it implies no boundedness of the transfer time of the state from the value μ to the value μ , as we have seen at Example 171. (d) ∃kL ∈ N making (34.2.1) true for some functions α such that k  ≤ kL , the duration of the transfer from μ to μ , under u, for any α ∈  ⊂ n , is bounded by a constant kL . The subset  of n is to be conveniently defined. (e) the system remains in (34.2.1) in the rest position μ at least until the longest transfer time kL was reached k1 ≥ kL , see Remark 226 (b) for the meaning of k1 .

34.3 The fundamental (operating) mode Theorem 251. We consider the function  : Bn × Bm → Bn .  (a) For p ≥ 1 and α 0 , ..., α p ∈ n , μ0 , ..., μp ∈ Bn , u0 , ..., up ∈ S (m) , k0 , ..., kp−1 ≥ 1 that satisfy ∀k ≥ k0 , φ α (μ0 , u0 , k) = μ1 , 0

(34.3.1)

324

Boolean Systems

...  , φα ∀k ≥ kp−1

p−1

(μp−1 , up−1 , k) = μp ,

(34.3.2)

 arbitrary and we define γ ∈ n , v ∈ S (m) by we fix k0 ≥ k0 , ..., kp−1 ≥ kp−1 ⎧ ⎪ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪ ⎪ 1,k−k 0 , if k ∈ {k , ..., k + k − 1}, ⎪ α ⎨ 0 0 1 k γ = ... ⎪ ⎪ ⎪ α p−1,k−k0 −...−kp−2 , if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 − 1}, ⎪ ⎪ ⎩ α p,k−k0 −...−kp−1 , if k ≥ k0 + ... + kp−1 ,

v(k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u0 (k), if k ∈ {0, ..., k0 − 1}, u1 (k − k0 ), if k ∈ {k0 , ..., k0 + k1 − 1}, ... up−1 (k − k0 − ... − kp−2 ), if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 − 1}, up (k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 .

We have

φ γ (μ0 , v, k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

φ α (μ0 , u0 , k), if k ∈ {0, ..., k0 }, 1 φ α (μ1 , u1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p−1 φ α (μp−1 , up−1 , k − k0 − ... − kp−2 ), if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 }, p φ α (μp , up , k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 .

(b) We take α 0 , ..., α p , ... ∈ n , μ0 , ..., μp , ... ∈ Bn , u0 , ..., up , ... ∈ S (m) , k0 , ..., kp , ... ≥ 1 with ∀k ≥ k0 , φ α (μ0 , u0 , k) = μ1 , 0

(34.3.3)

... ∀k ≥ kp , φ α (μp , up , k) = μp+1 , p

(34.3.4)

... For k0 ≥ k0 , ..., kp v ∈ S (m) as:

γk =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

≥ kp , ... arbitrary, we define the computation function γ

∈ n and the input

α 0,k , if k ∈ {0, ..., k0 − 1}, if k ∈ {k0 , ..., k0 + k1 − 1}, ... α p,k−k0 −...−kp−1 , if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, ... α 1,k−k0 ,

(34.3.5)

Chapter 34 • The fundamental (operating) mode

v(k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u0 (k), if k ∈ {0, ..., k0 − 1}, − k0 ), if k ∈ {k0 , ..., k0 + k1 − 1}, ... up (k − k0 − ... − kp−1 ), if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, ...

325

u1 (k

(34.3.6)

We have

φ γ (μ0 , v, k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

φ α (μ0 , u0 , k), if k ∈ {0, ..., k0 }, ... p α p p φ (μ , u , k − k0 − ... − kp−1 ), if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp }, ...

Proof. Item (a) is a special case of Theorem 244 (a), page 307, when additional requests of stability (34.3.1), ..., (34.3.2) are fulfilled, and item (b) is a special case of Theorem 244 (b), when additional requests of stability (34.3.3), ..., (34.3.4), ... are true. Remark 228. In Theorem 251, item (a) is a special case of item (b), when μp = μp+1 = ..., ∀k ≥ kp , α p,k = α p+1,k−kp , up (k) = up+1 (k − kp ), ∀k ≥ kp+1 , α p+1,k = α p+2,k−kp+1 , up+1 (k) = up+2 (k − kp+1 ), ...  ≥ 1, ... are true, for arbitrary kp ≥ kp ≥ 1, kp+1 ≥ kp+1

Definition 136. In the circumstances of the hypothesis of Theorem 251 (b), which are expressed by the requests (34.3.3), ..., (34.3.4), ... to which we add the constancy of the inputs: ν 0 , ..., ν p , ... ∈ Bm exist with ∀k ∈ N, u0 (k) = ν 0 , ...

(34.3.7)

326

Boolean Systems

∀k ∈ N, up (k) = ν p ,

(34.3.8)

... and the definitions (34.3.5), (34.3.6) of γ , v, the triple (γ , μ0 , v) is called a fundamental (operating) mode of , or of φ. When x(k) = φ γ (μ0 , v, k), k ∈ N, and (γ , μ0 , v) is a fundamental mode, we say that the system  (or φ) runs in the fundamental mode. Definition 137. If μ0 , ..., μp , ... ∈ Bn are given and α 0 , ..., α p , ... ∈ n , k0 , ..., kp , ... ∈ N, ν 0 , ..., ν p , ... ∈ Bm exist such that (γ , μ0 , v) is a fundamental mode, we say that  (or φ) has (accepts) a fundamental operating mode along μ0 , ..., μp , ... ∈ Bn . Remark 229. We conclude that the fundamental operating mode of  consists in successive evolutions under constant input, see (34.3.7), ..., (34.3.8), ... which provide stability, see (34.3.3), ..., (34.3.4), ... The concept of stability is understood in the autonomous sense. Example 172. We give the following example of the function  : B × B → B, ∀(μ, ν) ∈ B × B, (μ, ν) = ν. 0

1 ν=0

- 1

0 ν=1

≤p ,

We ask that α ∈ where p ≥ 1 i.e. α can have no more than p successive 0 s. For a sequence 0 = i0 < i1 < i2 < ... we ask that the input v fulfills  0, if k ∈ {i0 , i0 + 1, ..., i1 − 1} ∨ {i2 , i2 + 1, ..., i3 − 1} ∨ ..., v(k) = 1, if k ∈ {i1 , i1 + 1, ..., i2 − 1} ∨ {i3 , i3 + 1, ..., i4 − 1} ∨ ... If i1 − i0 ≥ p + 1, i2 − i1 ≥ p + 1, i3 − i2 ≥ p + 1, ... then (α, 0, v) is a fundamental mode of . This produces the situation when each 0 at the input v(i2k ) = 0 makes the state be 0, x(i2k+1 ) = 0 and each 1 at the input v(i2k+1 ) = 1 makes the state be 1, x(i2k+2 ) = 1, k ∈ N.

35 Combinational systems with one level The combinational systems are the systems without feedback. Of them, the combinational systems with one level, shortly called 1-combinational systems, are defined as systems  : Bn × Bm → Bn that do not depend on the state coordinates: χ : Bm → Bn exists, such that ∀μ ∈ Bn , ∀ν ∈ Bm , χ (ν) = (μ, ν). Then χ is called the input-output function of . If the input is constant, ∀k ∈ N, u(k) = ν, the system behaves like an autonomous system with constant generator function. These systems are introduced in Section 35.1 and in Section 35.2 we give examples. An important property of the combinational systems is stability, in particular the states of the 1-combinational systems with a constant input ν converge coordinate-wise monotonically towards the final value χ (ν), as shown in Section 35.3. The Cartesian product of 1-combinational systems and the predecessors/successors of such systems are addressed in Sections 35.4 and 35.5. The isomorphisms (h, h , h ) ∈ I so(, ) bring, in certain circumstances, the 1-combinational system  in the 1-combinational system . This, together with the special case when the isomorphisms are translations, is treated in Sections 35.6 and 35.7. A property of invariance when  is 1-combinational is given in Section 35.8. In Section 35.9 we characterize the subsystems of a 1-combinational system.

35.1 Definition Lemma 5. Let the function  : Bn × Bm → Bn . (a) The following statements are equivalent: (a.1) ∀μ ∈ Bn , ∀μ ∈ Bn , ∀ν ∈ Bm , (μ, ν) = (μ , ν), (a.2) the function χ : Bm → Bn exists, such that ∀μ ∈ Bn , ∀ν ∈ Bm , χ (ν) = (μ, ν). (b) If one of (a.1), (a.2) is true, then χ from (a.2) is unique. Proof. (b) We suppose that (a.2) is true and we suppose also against all reason the existence of h : Bm → Bn , different from χ , with the property that ∀μ ∈ Bn , ∀ν ∈ Bm , h(ν) = (μ, ν), Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00041-6 Copyright © 2023 Elsevier Inc. All rights reserved.

327

328

Boolean Systems

therefore ν  ∈ Bm exists with χ (ν  ) = h(ν  ). We get the contradiction ∀μ ∈ Bn , χ (ν  ) = (μ, ν  ) = h(ν  ). Definition 138. The function  that fulfills (a.1) is called 1-combinational1 or combinational with one level, and χ defined by (a.2) is said to be the input-output function of . Remark 230. The 1-combinational functions (or systems) are those for which (μ, ν) does not depend on μ, thus if ν is fixed, we have the autonomous system ν : Bn → Bn with a constant generator function. The situation was briefly mentioned in the section ‘Flows with constant generator functions’, page 68. Remark 231. Let  : Bn × Bm → Bn 1-combinational and ν ∈ Bm arbitrary, fixed. The possibility of working with ν : Bn → Bn and asking that ∀k ∈ N, u(k) = ν allows referring to stability, predecessors and successors, invariance, basins of attraction, attractors, i.e. topics which were defined for the autonomous systems. We shall use this possibility.

35.2 Examples Example 173. We have the 1-combinational function  : B × B2 → B which is defined by ∀μ ∈ B, ∀(ν1 , ν2 ) ∈ B2 , (μ, (ν1 , ν2 )) = ν1 ν2 (the AND logical gate). Its state portrait family presented under a condensed form as 0

1

- 1

0

ν1 ν2 = 0

ν1 ν2 = 1

recalls Example 172, page 326. When (ν1 , ν2 ) is fixed, (ν1 ,ν2 ) is constant. Example 174. The 1-combinational function  : B2 × B → B2 defined like this: ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ((μ1 , μ2 ), ν) = (ν, ν) has the following state portrait family: (0, 0) 

(0, 1)

? (1, 0) 

? (1, 1)

ν=0 1

Or combinatorial.

(0, 0) 6

- (0, 1) - 6

(1, 0)

- (1, 1) ν=1

Chapter 35 • Combinational systems with one level

329

35.3 Stability Theorem 252. Let the 1-combinational function  : Bn × Bm → Bn and we ask that u ∈ S (m) is a constant input: ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν. Then ∀α ∈ n , ∀μ ∈ Bn , ∀k ∈ N, ∀i ∈ {1, ..., n}, φiα (μ, u, k) =

⎧ ⎨

μi , if k = 0, μi , if k ≥ 1 and αi0 ∪ ... ∪ αik−1 = 0, ⎩ χ,i (ν), if k ≥ 1 and αi0 ∪ ... ∪ αik−1 = 1,

(35.3.1)

in particular φ α (μ, u, ·) → χ (ν) coordinate-wise monotonically. Proof. The property results from Theorem 67, page 68, where  : Bn → Bn and ∃μ ∈ Bn , ∀μ ∈ Bn , (μ) = μ are replaced by  : Bn × Bm → Bn and ∀ν ∈ Bm , ∃μ ∈ Bn , i.e. μ = χ (ν), such that ∀μ ∈ Bn , (μ, ν) = χ (ν). Remark 232. The property of global asymptotic stability, Definition 116, page 277, was formulated for the autonomous systems  : Bn → Bn . When we have  : Bn × Bm → Bn and we ask the constancy of the input ∀k ∈ N, u(k) = ν, we can replace in the definition of the global asymptotic stability  with ν . Remark 233. The stability of the 1-combinational systems ∀ν ∈ Bm , ∃μ ∈ Bn , namely μ = χ (ν), such that ∀μ ∈ Bn , ∀α ∈ n , lim φ α (μ , u, k) = μ, k→∞

see (29.1.8), page 276, shows the advantage of keeping the input constant long enough so that the system stabilizes, and also the connection with the fundamental operating mode. Remark 234. Similarly with the previous reasoning, dependence on the initial conditions, Definition 82, page 119, was formulated for the autonomous systems  : Bn → Bn . The hypothesis of Theorem 252 of constancy of the input allows considering this property by the replacement of  with ν . We infer from the theorem the satisfaction of the following property of independence on the initial conditions: ∀ν ∈ Bm , ∀α ∈ n , ∀μ ∈ Bn , ∀μ ∈ Bn , ∃k ∈ N, φ α (μ, u, k) = φ α (μ , u, k), representing the negation of (13.1.1)page 119 . This means that, under constant input, all the dependence on the initial conditions properties (13.1.1)page 119 –(13.1.6)page 120 are false.

330

Boolean Systems

35.4 Cartesian product Theorem 253. We consider the 1-combinational functions  : Bn × Bm → Bn ,  : Bp × Bq → Bp . (a) The Cartesian product  ×  is 1-combinational, (b) the input-output function χ× : Bm+q → Bn+p of  ×  fulfills χ× = χ × χ . Proof. (a) The Cartesian product (Definition 121, page 297)  ×  : Bn+p × Bm+q → Bn+p satisfies ∀μ ∈ Bn , ∀μ ∈ Bn , ∀δ ∈ Bp , ∀δ  ∈ Bp , ∀ν ∈ Bm , ∀ρ ∈ Bq , ( × )((μ, δ), (ν, ρ)) = ((μ, ν), (δ, ρ)) = ((μ , ν), (δ  , ρ)) = ( × )((μ , δ  ), (ν, ρ)), thus  ×  is 1-combinational. (b) With arbitrary μ ∈ Bn , δ ∈ Bp , ν ∈ Bm , ρ ∈ Bq we can write χ× (ν, ρ) = ( × )((μ, δ), (ν, ρ)) = ((μ, ν), (δ, ρ)) = (χ (ν), χ (ρ)) = (χ × χ )(ν, ρ).

35.5 Predecessors and successors Notation 49. The system  : Bn × Bm → Bn and μ ∈ Bn , u ∈ S (m) are given. We denote  O α (μ, u), O + (μ, u) = α∈n

O − (μ, u) =



−

O α (μ, u),

α∈n

see Notation 46, page 306. Theorem 254. The system  : Bn × Bm → Bn and μ ∈ Bn are given. We ask that the input u ∈ S (m) is constant: ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν. (a) If  is 1-combinational, then O + (μ, u) = [μ, χ (ν)] = μ+ ν , O − (μ, u) = [χ (ν), μ] = μ− ν .

Chapter 35 • Combinational systems with one level

331

(b) In the special case when at (a) we have μ = χ (ν), we infer O + (χ (ν), u) = {χ (ν)}, i.e. ∀α ∈ n , ∀k ∈ N, φ α (χ (ν), u, k) = χ (ν). Proof. (a) We prove that O + (μ, u) ⊂ [μ, χ (ν)] and let λ ∈ O + (μ, u) arbitrary. This means that α ∈ n and k ∈ N exist such that λ = φ α (μ, u, k). From Theorem 252, page 329 we know that ∀i ∈ {1, ..., n}, λi takes one of the values μi and i (μ, ν) = χ,i (ν), thus A ⊂ μ  χ (ν) exists with λ = μ ⊕ ε i . We have ini∈A

ferred that λ ∈ [μ, χ (ν)]. The inclusion + [μ, χ (ν)] = μ+ ν ⊂ O (μ, u)

is obvious. We prove O − (μ, u) ⊂ [χ (ν), μ] and we take λ ∈ O − (μ, u) arbitrary, i.e. α ∈ n and k  ∈ N exist such that μ = φ α (λ, u, k  ).

(35.5.1)

We suppose against all reason that λ ∈ / [χ (ν), μ], i.e. not(∀i ∈ {1, ..., n}, λi = χ,i (ν) or λi = μi ) and this means that ∃i ∈ {1, ..., n}, λi = χ,i (ν) = μι .

(35.5.2)

As φiα (λ, u, k) → χ,i (ν) converges monotonically, φiα (λ, u, k) can take the values λi and χ,i (ν) i.e. it is constant, thus (35.5.2) implies ∀k ∈ N, φiα (λ, u, k) = λi = χ,i (ν) = μi . Statements (35.5.1) and (35.5.3) are contradictory. The inclusion [χ (ν), μ]

Theorem 11, page 13

=

− μ− ν ⊂ O (μ, u)

takes place too. (b) We take in consideration that ∀α ∈ n , ∀k ∈ N, φ α (χ (ν), u, k) ∈ O + (χ (ν), u) = [χ (ν), χ (ν)] = {χ (ν)}.

(35.5.3)

332

Boolean Systems

35.6 Isomorphisms Theorem 255. The functions ,  : Bn × Bm −→ Bn are given. We suppose that  is 1combinational, and an isomorphism (h, h , h ) ∈ I so(, ) exists with the property that h ∈ n . Then (a)  is 1-combinational, (b) the input-output function of  is χ : Bm −→ Bn , ∀ν ∈ Bm , χ (ν) = h(χ (h−1 (ν))). Proof. The fact that h ∈ n implies, from Theorem 56, page 57, that h (1, ..., 1) = (1, ..., 1). This shows that ∀μ ∈ Bn , ∀λ ∈ Bn , ∀ν ∈ Bm , 

 h (λ) (h(μ), h (ν)) = h(λ (μ, ν)) implies, for λ = (1, ..., 1): (h(μ), h (ν)) = h((μ, ν)), thus (h(μ), h (ν)) = h(χ (ν)). As h, h are bijections,  does not depend on the first argument, therefore χ exists with (h(μ), h (ν)) = χ (h (ν)), and statement (b) is also true.

35.7 Symmetry relative to translations Example 175. The 1-combinational function  : B2 × B → B2 from Example 174, page 328 defined by ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ((μ1 , μ2 ), ν) = (ν, ν) and  : B2 × B → B2 , ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ((μ1 , μ2 ), ν) = (ν, ν)

Chapter 35 • Combinational systems with one level

333

have the property that (θ (1,0) , 1B2 , h ) :  →  is an isomorphism, where h (ν) = ν. We can compare the state portrait family of  from page 328 with the state portrait family of  (1, 0) 

(1, 1)

? (0, 0) 

? (0, 1)

(1, 0) 6

- (1, 1) - 6

(0, 0)

- (0, 1)

ν=1

ν=0

that was drawn in the order ν = 1 first, ν = 0 last, due to h . Example 176. The function  : B2 × B → B2 , ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ((μ1 , μ2 ), ν) = (ν, ν) is 1-combinational. We keep the meaning of  from Example 175 and we see that ∀ν ∈ B, ν , ν are time-reversal symmetrical, see Theorem 231, page 290. On the other hand we notice the existence of the isomorphism (θ (1,1) , 1B2 , 1B2 ) :  → , i.e. for any μ, λ ∈ B2 and any ν ∈ B, we have θ (1,1) ( λ (μ, ν)) = (λ1 μ1 ∪ λ1 ν, λ2 μ2 ∪ λ2 ν) = ((λ1 ∪ μ1 )(λ1 ∪ ν), (λ2 ∪ μ2 )(λ2 ∪ ν)) = (λ1 ν ∪ λ1 μ1 ∪ μ1 ν, λ2 ν ∪ λ2 μ2 ∪ μ2 ν) = (λ1 μ1 ∪ λ1 ν, λ2 μ2 ∪ λ2 ν) =  λ (μ, ν) =  λ (θ (1,1) (μ), ν).

35.8 Invariance Theorem 256. Let  : Bn × Bm → Bn be a 1-combinational system. For any ν ∈ Bm such that ∀k ∈ N, u(k) = ν, the following 3-invariance property is true: ∀μ ∈ Bn , ∀ξ ∈ O + (μ, u), ∀α ∈ n , O α (ξ, u) ⊂ O + (μ, u). Proof. We can use Theorem 128, page 160, and on the other hand, we could have omitted this theorem due to Theorem 265 to follow, but here is a straight proof. We take an arbitrary μ ∈ Bn . We have from Theorem 254, page 330 that O + (μ, u) = [μ, χ (ν)] and let ξ ∈ [μ, χ (ν)] arbitrary. For any α ∈ n , we can write O α (ξ, u) ⊂ O + (ξ, u) = [ξ, χ (ν)] ⊂ [μ, χ (ν)] = O + (μ, u).

334

Boolean Systems

35.9 Subsystem Theorem 257. We consider the 1-combinational system  : Bn × Bm → Bn and let  : Bn1 × Bm → Bn1 , where 0 < n1 ≤ n. The following statements are equivalent: (a)  is a subsystem of , (b) ∀(μ1 , ..., μn1 ) ∈ Bn1 , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {1, ..., n1 }, i (μ1 , ..., μn1 , ν1 , ..., νm ) = χ,i (ν1 , ..., νm ). Proof. (b)=⇒(a) For arbitrary (μ1 , ..., μn1 , μn1 +1 , ..., μn ) ∈ Bn , (ν1 , ..., νm ) ∈ Bm , we have ∀i ∈ {1, ..., n1 }, i (μ1 , ..., μn1 , ν1 , ..., νm ) = χ,i (ν1 , ..., νm ) = i (μ1 , ..., μn1 , μn1 +1 , ..., μn , ν1 , ..., νm ), i.e.  is a subsystem of . Corollary 23. The subsystems of the 1-combinational system  : Bn × Bm → Bn coincide with those 1-combinational systems  : Bn1 × Bm → Bn1 , where 0 < n1 ≤ n, that fulfill ∀i ∈ {1, ..., n1 }, χ,i = χ,i . Proof. If  is a subsystem of , then Theorem 257 shows us that  does not depend on μ1 , ..., μn1 , i.e. it is 1-combinational. In addition, we can write for arbitrary (μ1 , ..., μn1 , μn1 +1 , ..., μn ) ∈ Bn , (ν1 , ..., νm ) ∈ Bm that ∀i ∈ {1, ..., n1 }, χ,i (ν1 , ..., νm ) = i (μ1 , ..., μn1 , ν1 , ..., νm ) = i (μ1 , ..., μn1 , μn1 +1 , ..., μn , ν1 , ..., νm ) = χ,i (ν1 , ..., νm ).

36 Combinational systems The system  : Bn × Bm → Bn is called p-combinational, p ≥ 1, if it has a state space decomposition  1 ,  2 , ...,  p of 1-combinational systems. This decomposition is proved to be unique, and  1 ,  2 , ...,  p are called the levels of . These concepts are introduced, together with some examples, in the first three sections. Like in the case of the 1-combinational systems, if  is p-combinational then an inputoutput function χ : Bm → Bn may be defined. Function χ characterizes the stability of these systems, and this is shown in Section 36.4. The hazards of the p-combinational systems, p ≥ 2, represent the situations when the convergence φ α (μ, u, k) → lim φ α (μ, u, k) under constant input (indicating stability) is not k→∞

coordinate-wise monotonic, and they are mentioned in Section 36.5. In Section 36.6 we prove that the Cartesian product of combinational systems is combinational. We characterize in Section 36.7 the predecessors and successors of the points μ ∈ Bn , when  is p-combinational. We do not know exactly which is the relation between the isomorphisms and the pcombinational systems and we stated this problem, together with an example, in Sections 36.8, 36.9. An invariance property of O + (μ, u) when  is p-combinational and u is constant is given in Section 36.10. The basins of attraction and the attractors of the combinational systems are addressed in Section 36.11. The subsystems of the combinational systems are the topic of Section 36.12. In Section 36.13, we show the manner that the fundamental operating mode works in the case of the combinational systems, due to their stability.

36.1 Definition   Notation 50. The function  : Bn × Bm → Bn is given. We denote with L 1 , L2 , ..., Lp , ... p ≥ 1 the following sequence of sets:

m L 1 = {i|i ∈ {1, ..., n}, ∃f : B → B, ∀μ ∈ Bn , ∀ν ∈ Bm , i (μ1 , ..., μn , ν1 , ..., νm ) = f (ν1 , ..., νm )}, Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00042-8 Copyright © 2023 Elsevier Inc. All rights reserved.

335

336

Boolean Systems

and we suppose that L 1 = {j1 , ..., jn1 }, ∀μ ∈ Bn , ∀ν

 n1 +m → B, L 2 = {i|i ∈ {1, ..., n}  L1 , ∃f : B m ∈ B , i (μ1 , ..., μn , ν1 , ..., νm ) = f (μj1 , ..., μjn1 , ν1 , ..., νm )},

...  and we suppose that L 1 ∨ ... ∨ Lp = {j1 , ..., jnp },   np +m → B, L p+1 = {i|i ∈ {1, ..., n}  L1 ∨ ... ∨ Lp , ∃f : B n m ∀μ ∈ B , ∀ν ∈ B , i (μ1 , ..., μn , ν1 , ..., νm ) = f (μj1 , ..., μjnp , ν1 , ..., νm )},

... Theorem 258. (a) ∃p ≥ 1, L p = ∅,

≥ p, L = ∅, = ∅ =⇒ ∀p (b) ∀p ≥ 1, L p p

  (c) ∀p ≥ 1, L1 ∨ ... ∨ Lp = {1, ..., n} =⇒ ∀p ≥ p + 1, L p = ∅, (d) we can define a function  Fn,m   −→

0, if L 1 = ∅, ∈ {0, 1, ..., n}, max{p|p ≥ 1, L p = ∅}, otherwise

(e) if L 1 = ∅, we have n m j ∀i ∈ L 1 , ∀j ∈ {1, ..., n}, ∀μ ∈ B , ∀ν ∈ B , i (μ ⊕ ε , ν) = i (μ, ν), 1 and if L p = ∅, p ≥ 2, then  n m j ∀i ∈ L p , ∃j ∈ Lp−1 , ∃μ ∈ B , ∃ν ∈ B , i (μ ⊕ ε , ν) = i (μ, ν).  Proof. (a) This is a consequence of the inclusion ∀p ≥ 1, L 1 ∨ ... ∨ Lp ⊂ {1, ..., n}, where the  sets Li are disjoint two by two. 

(b) If L p = ∅, then Lp+1 = ∅ and the proof is made by induction on p ≥ p.  (e) We ask that L1 = ∅. The first statement is a consequence of the fact that for any m i ∈ L 1 , a function f : B → B exists such that

∀j ∈ {1, ..., n}, ∀μ ∈ Bn , ∀ν ∈ Bm , i (μ ⊕ ε j , ν) = f (ν) = i (μ, ν).  For the second statement, we ask that L 2 = ∅. We denote L1 = {j1 , ..., jn1 } and we sup m pose, against all reason, that i ∈ L2 exists such that ∀ν ∈ B ,

∀j ∈ {j1 , ..., jn1 }, ∀μ ∈ Bn , i (μ ⊕ ε j , ν) = i (μ, ν). 1

(36.1.1)

∂i (μ,ν) = i (μ ⊕ εj , ν) ⊕ i (μ, ν). Then the statements i (μ ⊕ εj , ν) = i (μ, ν), i (μ ⊕ ∂μj ∂i (μ,ν) ∂i (μ,ν) = 0, ∂μ = 1. εj , ν) = i (μ, ν) become ∂μ j j

We can denote

Chapter 36 • Combinational systems

337

n1 +m → B exists with ∀μ ∈ Bn , ∀ν ∈ Bm , From the way that L 2 is defined, the function g : B

g(μj1 , ..., μjn1 , ν1 , ..., νm ) = i (μ1 , ..., μj1 , ..., μjn1 , ..., μn , ν1 , ..., νm ). This shows that ∀ν ∈ Bm , ∀j ∈ {1, ..., n}  {j1 , ..., jn1 }, ∀μ ∈ Bn , i (μ ⊕ ε j , ν) = i (μ, ν).

(36.1.2)

We fix μ ∈ Bn arbitrary, and we define the function f : Bm → B by ∀ν ∈ Bm , f (ν) = i (μ , ν). We fix an arbitrary ν ∈ Bm now and we want to prove that for any μ

∈ Bn we have f (ν) = i (μ

, ν).

(36.1.3)

Indeed, if μ

= μ , then (36.1.3) is true, thus we can suppose that μ

= μ . This means the existence of i1 , ..., ik ∈ {1, ..., n} with the property that μ

= μ ⊕ ε i1 ⊕ ... ⊕ ε ik .

(36.1.4)

We infer f (ν) = i (μ , ν) (36.1.1),(36.1.2)

=

(36.1.1),(36.1.2)

=

i (μ ⊕ ε i1 , ν)

i (μ ⊕ ε i1 ⊕ ... ⊕ ε ik , ν)

(36.1.1),(36.1.2)

(36.1.4)

=

=

...

i (μ

, ν).

Statement (36.1.3) is proved, wherefrom we get that i ∈ L 1 , contradiction. The proof for p ≥ 3 arbitrary is similar. Theorem 259.  is 1-combinational if and only if L 1 = {1, ..., n}. Proof. If. From the fact that L 1 = {1, ..., n} we infer the existence of the functions χ,1 , ..., χ,n : Bm → B with the property that ∀μ ∈ Bn , ∀ν ∈ Bm , (μ, ν) = χ (ν) i.e.  is 1combinational. Only if. The existence of the input-output function χ : Bm → Bn shows that L 1 = {1, ..., n}.   Definition 139. (a) In case that p ≥ 1 exists such that L p = ∅ and L1 ∨ ... ∨ Lp = {1, .., n},  is said to be p-combinational, shortly combinational.    (b) If p ≥ 1 exists with L p = ∅, Lp+1 = ∅ and L1 ∨ ... ∨ Lp  {1, .., n},  is called partially combinational. (c) If L 1 = ∅,  is called non-combinational.

Remark 235. Theorem 259 shows that the definition of the 1-combinational functions  may be done as functions for which L 1 = {1, ..., n} and from this point of view Definition 139 represents a generalization from p = 1 to p ≥ 1. This generalization takes into ac    count Theorem 258, i.e. if L 1 ∨ ... ∨ Lp = {1, ..., n}, Lp = ∅, then all L1 , ..., Lp are nonempty, 

and on the other hand ∀p ≥ p + 1, Lp = ∅.

338

Boolean Systems

Example 177. The functions , , : B3 ×B2 → B3 defined by ∀(μ1 , μ2 , μ3 ) ∈ B3 , ∀(ν1 , ν2 ) ∈ B2 , ((μ1 , μ2 , μ3 ), (ν1 , ν2 )) = (ν1 ν2 , μ1 ν2 , μ2 ν2 ), ((μ1 , μ2 , μ3 ), (ν1 , ν2 )) = (ν1 ν2 , μ1 μ3 , μ2 ν2 ),

((μ1 , μ2 , μ3 ), (ν1 , ν2 )) = (ν1 μ3 , μ1 ν2 , μ2 ν2 )     fulfill L 1 = {1}, L2 = {2}, L3 = {3}, thus  is 3-combinational; L1 = {1}, L2 = ∅, thus  is

partially combinational; L1 = ∅, thus is non-combinational.

Remark 236. For an easy handling of the combinational functions  : Bn × Bm → Bm , we shall suppose without losing the generality that  L 1 = ∅ =⇒ ∃n1 , L1 = {1, ..., n1 },

 L 2 = ∅ =⇒ ∃n2 > n1 , L2 = {n1 + 1, ..., n2 },

... in particular if  is p-combinational, p ≥ 2 then the numbers 0 = n0 < n1 < n2 < ... < np = n

(36.1.5)

exist such that L 1 = {n0 + 1, .., n1 }, L 2 = {n1 + 1, ..., n2 }, ... = {n + 1, ..., np }. L p−1 p

(36.1.6)

36.2 Levels Remark 237. Let  : Bn × Bm → Bn . For p ≥ 1 and 0 = n0 < n1 < n2 < ... < np = n, the functions  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 ,  2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ...,  p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 represent a state space decomposition of  (see Definition 135, page 316) in 1-combinational functions, if the following statements are fulfilled:

Chapter 36 • Combinational systems

339

∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, ⎧ 1 i−n (μ1 , ν), if i ∈ {n0 + 1, .., n1 }, ⎪ 0 ⎪ ⎪ ⎪ 2 2 ⎪ ⎨ i−n1 (μ , (μ1 , ν)), if i ∈ {n1 + 1, ..., n2 }, i ((μ1 , ..., μp ), ν) = ... ⎪ p ⎪ p , (μ1 , ..., μp−1 , ν)), ⎪ (μ  ⎪ i−np−1 ⎪ ⎩ if i ∈ {np−1 + 1, ..., np },

(36.2.1)

and χ 1 : Bn0 +m → Bn1 −n0 , χ 2 : Bn1 +m → Bn2 −n1 , ..., χ p : Bnp−1 +m → Bnp −np−1 exist such that ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm ,  1 (μ1 , ν) = χ 1 (ν),

(36.2.2)

 2 (μ2 , (μ1 , ν)) = χ 2 (μ1 , ν),

(36.2.3)

...  p (μp , (μ1 , ..., μp−1 , ν)) = χ p (μ1 , ..., μp−1 , ν).

(36.2.4)

Theorem 260. Function  is p-combinational if and only if it has a state space decomposition  1 ,  2 , ...,  p in 1-combinational functions. Proof. For p = 1, the state space decomposition of  from Remark 237 consists in the 1combinational function  1 : Bn × Bm → Bn that is equal with  and the result is trivial, thus we suppose in the rest of the proof that p ≥ 2. If. The hypothesis asks that the state space decomposition  1 ,  2 , ...,  p from Remark 237 exists. We infer 1 n1 −n0 L , ∀μ2 ∈ Bn2 −n1 , ..., 1 = {i|i ∈ {1, ..., n}, ∀μ ∈ B

∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , i ((μ1 , ..., μp ), ν) = χ 1 ,i−n0 (ν)} (36.2.1),(36.2.2)

=

{n0 + 1, ..., n1 },

1 n1 −n0 L , ∀μ2 ∈ Bn2 −n1 , ..., 2 = {i|i ∈ {n1 + 1, ..., n}, ∀μ ∈ B

∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , i ((μ1 , ..., μp ), ν) = χ 2 ,i−n1 (μ1 , ν)} (36.2.1),(36.2.3)

=

{n1 + 1, ..., n2 }, ...

1 n1 −n0 L , ∀μ2 ∈ Bn2 −n1 , ..., p = {i|i ∈ {np−1 + 1, ..., n}, ∀μ ∈ B

∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , i ((μ1 , ..., μp ), ν) = χ p ,i−np−1 (μ1 , ..., μp−1 , ν)}

340

Boolean Systems

(36.2.1),(36.2.4)

=

{np−1 + 1, ..., n},

thus   L 1 ∨ L2 ∨ ... ∨ Lp = {n0 + 1, ..., n1 } ∨ {n1 + 1, ..., n2 } ∨ ... ∨ {np−1 + 1, ..., np } = {1, ..., n},

i.e.  is p-combinational. Only if. Function  is p-combinational, thus we have the existence of the numbers   (36.1.5) such that L 1 , L2 , ..., Lp fulfill (36.1.6). This means the existence of the funcn +m n −n 0 1 0 →B , χ 2 : Bn1 +m → Bn2 −n1 , ..., χ p : Bnp−1 +m → Bnp −np−1 that satisfy: tions χ 1 : B 1 n −n 2 n −n 1 0 2 1 , ∀μ ∈ B , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀μ ∈ B ∀i ∈ {1, ..., n1 − n0 }, χ 1 ,i (ν) = i+n0 ((μ1 , ..., μp ), ν), ∀i ∈ {1, ..., n2 − n1 }, χ 1 ,i (μ1 , ν) = i+n1 ((μ1 , ..., μp ), ν), ... ∀i ∈ {1, ..., np − np−1 }, χ p ,i (μ1 , ..., μp−1 , ν) = i+np−1 ((μ1 , ..., μp ), ν). The functions  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 ,  2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ...,  p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 defined by ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , (36.2.2), (36.2.3), ..., (36.2.4), satisfy (36.2.1).  1 ,  2 , ...,  p is a state space decomposition of  in 1-combinational functions. Theorem 261. The state space decomposition of the p-combinational function  in 1combinational functions  1 ,  2 , ...,  p is unique. Proof. We suppose that  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 ,  2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 ,









...,  p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 and 1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 , 2 : Bn2 −n1 ×











n −n n +m n −n → B p p −1 are two decompositions of  Bn1 +m → Bn2 −n1 , ..., p : B p p −1 × B p −1 in 1-combinational functions, where 0 = n0 < n1 < n2 < ... < np = n, 0 = n 0 < n 1 < n 2 < ... < n p = n hold. We have n0 = 0 = n 0 ,



n1 = n0 + card(L 1 ) = n0 + card(L1 ) = n1 ,



n2 = n1 + card(L 2 ) = n1 + card(L2 ) = n2 ,

Chapter 36 • Combinational systems

341

...



n = np = np−1 + card(L p ) = np−1 + card(Lp ) = np



1 n1 −n0 , μ2 ∈ and, as L p

= ∅, ∀p > p, we infer that p = p. Furthermore, for any μ ∈ B Bn2 −n1 , ..., μp ∈ Bnp −np−1 , ν ∈ Bm , we obtain 1 (μ1 , ν) ∀i ∈ {n0 + 1, ..., n1 }, i−n 0 1 1 p = i ((μ , ..., μ ), ν) = i−n0 (μ1 , ν), 2 ∀i ∈ {n1 + 1, ..., n2 }, i−n (μ2 , (μ1 , ν)) 1 2 1 p = i ((μ , ..., μ ), ν) = i−n1 (μ2 , (μ1 , ν)),

... p

∀i ∈ {np−1 + 1, ..., np }, i−np−1 (μp , (μ1 , ..., μp−1 , ν)) p = i ((μ1 , ..., μp ), ν) = i−np−1 (μp , (μ1 , ..., μp−1 , ν)). Definition 140. If  is p-combinational, then  1 ,  2 , ...,  p giving its state space decomposition in 1-combinational functions are called the first level of , the second level of , ..., the p-th level of . Remark 238. In the following commutative diagrams that characterize the levels of a combinational function Bn × Bm



- Bn

Bn × Bm

? ?  1 - n1 −n0 B Bn1 −n0 × Bn0 +m χ 1

? Bn1 +m ... 

- Bn

? ? p Bnp −np−1 × Bnp−1 +m - Bnp −np−1 χ p ?

np−1 +m

B

- Bn

? ?  2 - n2 −n1 B Bn2 −n1 × Bn1 +m χ 2

? Bn0 +m

Bn × Bm



342

Boolean Systems

the vertical arrows are projections, or projections combined with the inclusion of some state coordinates of  between the input coordinates of  k , k ∈ {1, ..., p}. The meaning of the commutative squares is that of indicating that  1 ,  2 , ...,  p are a state space decomposition of , see (36.2.1), while the meaning of the lower commutative triangles is to point out that  1 ,  2 , ...,  p are 1-combinational (they do not depend on their state coordinates), see (36.2.2), (36.2.3), ..., (36.2.4).

36.3 Example Example 178. We consider the function  : B3 × B → B3 , defined as ∀(μ1 , μ2 , μ3 ) ∈ B3 , ∀ν ∈ B, ((μ1 , μ2 , μ3 ), ν) = (ν, μ1 ∪ ν, μ2 ν),   which has L 1 = {1}, L2 = {2}, L3 = {3}. For

0 = n0 < n1 = 1 < n2 = 2 < n3 = 3, we can define the 1-combinational functions  1 : B × B → B,  2 : B × B2 → B,  3 : B × B3 → B and their input-output functions χ 1 : B → B, χ 2 : B2 → B, χ 3 : B3 → B like this:  1 (μ1 , ν) = χ 1 (ν) = ν,  2 (μ2 , (μ1 , ν)) = χ 2 (μ1 , ν) = μ1 ∪ ν,  3 (μ3 , (μ1 , μ2 , ν)) = χ 3 (μ1 , μ2 , ν) = μ2 ν, where μ1 , μ2 , μ3 , ν ∈ B. We notice that 1 ((μ1 , μ2 , μ3 ), ν) =  1 (μ1 , ν),

2 ((μ1 , μ2 , μ3 ), ν) =  2 (μ2 , (μ1 , ν)), 3 ((μ1 , μ2 , μ3 ), ν) =  3 (μ3 , (μ1 , μ2 , ν)), therefore  1 ,  2 ,  3 are the levels of .

Chapter 36 • Combinational systems

343

The state portrait family of  consists in two state portraits, of which this is the one (0, 1, 1)



-

(1, 1, 0)  - 6

(0, 1, 0)

-

(0, 0, 1)

-



-

? (1, 0, 1)

 - (1, 0, 0) 

-

-

? (0, 0, 0)



? (1, 1, 1) ν=0 of 0 : B3 → B3 , ∀(μ1 , μ2 , μ3 ) ∈ B3 , ((μ1 , μ2 , μ3 ), 0) = (1, μ1 , 0).

36.4 The input-output function. Stability Remark 239. The next definition makes use of the fact that the p-combinational functions  have a unique state space decomposition in levels  1 ,  2 , ...,  p , and the fact that  1 ,  2 , ...,  p are 1-combinational, with the input-output functions χ 1 , χ 2 , ..., χ p , see the notations from Remark 237, page 338. Definition 141. If  : Bn × Bm → Bn is p-combinational, p ≥ 1, then its input-output function χ : Bm → Bn is defined as ∀ν ∈ Bm , ∀i ∈ {1, ..., n},

χ,i (ν) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

χ 1 ,i−n0 (ν), if i ∈ {n0 + 1, .., n1 }, χ 2 ,i−n1 (χ 1 (ν), ν), if i ∈ {n1 + 1, ..., n2 }, ... χ p ,i−np−1 (χ 1 (ν), ..., χ p−1 (χ 1 (ν), ..., ν), ν), if i ∈ {np−1 + 1, ..., np }.

Theorem 262. If  : Bn × Bm → Bn is combinational and ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν, then ∀α ∈ n , ∀μ ∈ Bn , lim φ α (μ, u, k) = χ (ν).

k→∞

344

Boolean Systems

Proof. The property was already addressed for  1-combinational in Theorem 252, page 329, so that we shall suppose that  is p-combinational, with p ≥ 2. We have a unique state space decomposition of  in 1-combinational functions: the numbers 0 = n0 < n1 < n2 < ... < np = n and the 1-combinational functions  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 ,  2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ...,  p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 exist such that ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀ν ∈ Bm , ∀i ∈ {1, ..., n}, ⎧ 1 i−n (μ1 , ν), if i ∈ {n0 + 1, .., n1 }, ⎪ 0 ⎪ ⎪ ⎪  2 (μ2 , (μ1 , ν)), if i ∈ {n + 1, ..., n }, ⎪ ⎨ i−n1 1 2 1 p (36.4.1) i ((μ , ..., μ ), ν) = ... ⎪ p ⎪ p , (μ1 , ..., μp−1 , ν)), ⎪ (μ  ⎪ i−np−1 ⎪ ⎩ if i ∈ {np−1 + 1, ..., np }, see Theorem 260, page 339, and Theorem 261, page 340. We know from Theorem 248, page 317 that ∀α 1 ∈ n1 −n0 , ∀α 2 ∈ n2 −n1 , ..., ∀α p ∈ np −np−1 , ∀μ1 ∈ Bn1 −n0 , ∀μ2 ∈ Bn2 −n1 , ..., ∀μp ∈ Bnp −np−1 , ∀u ∈ S (m) , with the notations ∀k ∈ N, v 1 (k) = u(k), 1

v 2 (k) = ((ψ 1 )α (μ1 , v 1 , k), u(k)), ... 1

v p (k) = ((ψ 1 )α (μ1 , v 1 , k), ..., (ψ p−1 )α

p−1

(μp−1 , v p−1 , k), u(k)),

where v 1 ∈ S (n0 +m) , v 2 ∈ S (n1 +m) , ..., v p ∈ S (np−1 +m) , we have ∀k ∈ N, 1

2

p

φ (α ,α ,...,α ) ((μ1 , μ2 , ..., μp ), u, k) 2 p 1 1 = ((ψ ) (μ , v 1 , k), (ψ 2 )α (μ2 , v 2 , k), ..., (ψ p )α (μp , v p , k)). α1

(36.4.2)

The hypothesis states that u is constant and equal with ν, and let α ∈ n , α = μ ∈ Bn , μ = (μ1 , μ2 , ..., μp ) arbitrary, fixed. We use Theorem 252, page 329 and we have in succession: (α 1 , α 2 , ..., α p ),

∀k ≥ 0, v 1 (k) = ν,  ∃k1 ∈ N, ∀k ≥ k1 ,  ∃k2

∈ N, ∀k

≥ k1

+ k2 ,

1

(ψ 1 )α (μ1 , v 1 , k) = χ 1 (ν), v 2 (k) = (χ 1 (ν), ν),

(ψ 2 )α (μ2 , v 2 , k) = χ 2 (v 2 (k1 )) = χ 2 (χ 1 (ν), ν), v 3 (k) = (χ 1 (ν), χ 2 (χ 1 (ν), ν), ν), 2

Chapter 36 • Combinational systems

345

...



∈ N, ∀k ≥ k1 + ... + kp−1 , v p (k) = (χ 1 (ν), ..., χ p−1 (χ 1 (ν), ..., ν), ν), ∃kp−1

)) ∃kp ∈ N, ∀k ≥ k1 + ... + kp , (ψ p )α (μp , v p , k) = χ p (v p (k1 + ... + kp−1 p

= χ p (χ 1 (ν), ..., χ p−1 (χ 1 (ν), ..., ν), ν). We conclude that ∀k ≥ k1 + ... + kp , we can write φ α (μ, u, k) = φ (α (36.4.2)

=

1 ,α 2 ,...,α p )

1

((μ1 , μ2 , ..., μp ), u, k)

2

p

((ψ 1 )α (μ1 , v 1 , k), (ψ 2 )α (μ2 , v 2 , k), ..., (ψ p )α (μp , v p , k))

= (χ 1 (ν), χ 2 (χ 1 (ν), ν), ..., χ p (χ 1 (ν), ..., χ p−1 (χ 1 (ν), ..., ν), ν)) Definition 141, page 343

=

χ (ν).

Remark 240. The previous property of global asymptotic stability expresses, like in the case of the 1-combinational systems, the fact that all the dependence on the initial conditions properties (13.1.1)page 119 –(13.1.6)page 120 are false under constant input. The global asymptotic stability and the dependence on the initial conditions refer to the system ν . Corollary 24. We suppose that  : Bn × Bm → Bn is combinational and we define the 1combinational system  : Bn × Bm → Bn by ∀μ ∈ Bn , ∀ν ∈ Bm , (μ, ν) = χ (ν). We ask that ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν. Then ∀α ∈ n , ∀μ ∈ Bn , lim φ α (μ, u, k) = lim ψ α (μ, u, k).

k→∞

k→∞

Proof. This follows from Theorem 252, page 329 and Theorem 262. Example 179. We continue the explanations related with Example 178, page 342 now. Obviously, χ : B → B3 from there is given by ∀ν ∈ B, χ (ν) = (ν, ν ∪ ν, (ν ∪ ν)ν) = (ν, 1, ν). When ∀k ∈ N, u(k) = ν, the concept of attractor makes sense, and it refers to the system ν , i.e. 0 in this case. The global attractor noticed in the state portrait is {(1, 1, 0)} (in other words W 3 (1, 1, 0) = B3 ), as χ (0) = (1, 1, 0), due to the global asymptotic stability property that was proved at Theorem 262.

346

Boolean Systems

For ν = 1, the global attractor is {(0, 1, 1)}, as χ (1) = (0, 1, 1). Corollary 24 shows that a certain identification is possible between  and  : B3 × B → B3 , ∀μ ∈ B3 , ∀ν ∈ B, (μ, ν) = (ν, 1, ν). For example, via this identification, the attractors (which are global), of ν , ν , ν ∈ B coincide.

36.5 Hazards Definition 142. The combinational system  : Bn × Bm → Bn is given. If ∀α ∈ n , ∀μ ∈ Bn , ∀ν ∈ Bm such that ∀k ∈ N, u(k) = ν, we have lim φ α (μ, u, k) = χ (ν) coordinate-wise monotonically, the system  is called k→∞

hazard free (we say that there is no hazard), otherwise we say that a hazard takes place. Remark 241. The coordinate-wise monotonicity of the state function of a system which is 1-combinational, stated in Theorem 252, page 329 is considered to be natural, desirable, for the combinational systems in general. Hazards represent the situation when this property does not hold, and they refer to p-combinational systems, p ≥ 2. Example 180. In Example 178, page 342, the existence for ν = 0 of the path (0, 1, 1) → (0, 0, 1) → (1, 0, 0) → (1, 1, 0) shows that a hazard exists, since the convergence ψ2α ((0, 1, 1), u, k) → χ,2 (0) = 1 is not monotonic. This is called a static 1 hazard. Several other paths there show the existence of hazards too.

36.6 Cartesian product Lemma 6. The functions h : Bn × Bm → B, g : Bm × Bq → B are given. The following statements are equivalent: (a) ∀μ ∈ Bn , ∀ν ∈ Bm , ∀ξ ∈ Bq , h(μ, ν) = g(ν, ξ ), (b) the function f : Bm → B exists, such that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀ξ ∈ Bq , h(μ, ν) = g(ν, ξ ) = f (ν). Proof. (a)=⇒(b) We fix μ ∈ Bn and define f : Bm → B by ∀ν ∈ Bm , f (ν) = h(μ , ν).

Chapter 36 • Combinational systems

347

For any μ ∈ Bn , ν ∈ Bm , ξ ∈ Bq , we have (a)

(a)

h(μ, ν) = g(ν, ξ ) = h(μ , ν) = f (ν). Theorem 263. We consider the functions  : Bn × Bm → Bn , : Bp × Bq → Bp . (a) If , are combinational, then the Cartesian product  × is combinational, (b) we suppose that  has n levels, and has p levels. Then  × has max{n , p } levels and, (b.1) if n = p , then the 1-st level of  × is the product of the 1-st levels of , , ..., the

n -th level of  × is the product of the n -th levels of , , (b.2) if n < p , then the 1-st level of  × is the product of the 1-st levels of , , ..., the

n -th level of  × is the product of the n -th levels of , , the n + 1-th levels of  × and

coincide, ..., the p -th levels of  × and coincide, (b.3) if n > p , the situation is similar with (b.2), (c) the input-output function χ× : Bm+q → Bn+p of  × fulfills χ× = χ × χ . Proof. (a) We use the fact that  × : Bn+p × Bm+q → Bn+p fulfills ∀μ ∈ Bn , ∀ν ∈ Bm , ∀δ ∈ Bp , ∀ξ ∈ Bq , ∀i ∈ {1, ..., n + p},  i (μ, ν), if i ∈ {1, ..., n}, ( × )i ((μ, δ), (ν, ξ )) = (36.6.1)

i−n (δ, ξ ), if i ∈ {n + 1, ..., n + p}. We get L×

= {i|i ∈ {1, ..., n + p}, ∃f : Bm+q → B, ∀μ ∈ Bn , ∀ν ∈ Bm , 1 p ∀δ ∈ B , ∀ξ ∈ Bq , ( × )i ((μ1 , ..., μn , δ1 , ..., δp ), (ν1 , ..., νm , ξ1 , ..., ξq )) = f (ν1 , ..., νm , ξ1 , ..., ξq )} (36.6.1)

=

{i|i ∈ {1, ..., n}, ∃f : Bm+q → B, ∀μ ∈ Bn , ∀ν ∈ Bm , ∀ξ ∈ Bq , i (μ1 , ..., μn , ν1 , ..., νm ) = f (ν1 , ..., νm , ξ1 , ..., ξq )}

{i|i ∈ {n + 1, ..., n + p}, ∃f : Bm+q → B, ∀ν ∈ Bm , ∨ ∀δ ∈ Bp , ∀ξ ∈ Bq , i−n (δ1 , ..., δp , ξ1 , ..., ξq ) = f (ν1 , ..., νm , ξ1 , ..., ξq )} Lemma 6

=

∨

{i|i ∈ {1, ..., n}, ∃f : Bm → B, ∀μ ∈ Bn , ∀ν ∈ Bm , i (μ1 , ..., μn , ν1 , ..., νm ) = f (ν1 , ..., νm )}

{i|i ∈ {n + 1, ..., n + p}, ∃f : Bq → B, ∀δ ∈ Bp , ∀ξ ∈ Bq ,

i−n (δ1 , ..., δp , ξ1 , ..., ξq ) = f (ξ1 , ..., ξq )}.

We use the following notation for k = 1, 2, ...  ∅, if L k = ∅,

n + Lk = {n + i|i ∈ L k }, if L k = ∅

348

Boolean Systems

and we conclude that

= L L×

1 ∨ (n + L1 ). 1

Furthermore, if we suppose without losing the generality that L 1 = {1, ..., n1 }, L1 = {1, ..., p1 }, we infer:

L×

= {i|i ∈ {1, ..., n + p}  L×

, ∃f : Bn1 +p1 +m+q → B, ∀μ ∈ Bn , ∀ν ∈ Bm , 2 1 p q ∀δ ∈ B , ∀ξ ∈ B , ( × )i ((μ1 , ..., μn , δ1 , ..., δp ), (ν1 , ..., νm , ξ1 , ..., ξq )) = f (μ1 , ..., μn1 , δ1 , ..., δp1 , ν1 , ..., νm , ξ1 , ..., ξq )} (36.6.1)

=

n1 +p1 +m+q → B, ∀μ ∈ Bn , ∀ν ∈ Bm , {i|i ∈ {1, ..., n}  L 1 , ∃f : B p q ∀δ ∈ B , ∀ξ ∈ B , i (μ1 , ..., μn , ν1 , ..., νm ) = f (μ1 , ..., μn1 , δ1 , ..., δp1 , ν1 , ..., νm , ξ1 , ..., ξq )}

{i|i ∈ {n + 1, ..., n + p}  (n + L 1 ), ∃f : Bn1 +p1 +m+q → B, ∀μ ∈ Bn , ∀ν ∈ Bm , ∨ ∀δ ∈ Bp , ∀ξ ∈ Bq , i−n (δ1 , ..., δp , ξ1 , ..., ξq ) = f (μ1 , ..., μn1 , δ1 , ..., δp1 , ν1 , ..., νm , ξ1 , ..., ξq )} Lemma 6

=

∨

{i|i ∈ {n1 + 1, ..., n}, ∃f : Bn1 +m → B, ∀μ ∈ Bn , ∀ν ∈ Bm , i (μ1 , ..., μn , ν1 , ..., νm ) = f (μ1 , ..., μn1 , ν1 , ..., νm )}

{i|i ∈ {n + p1 + 1, ..., n + p}, ∃f : Bp1 +q → B, ∀δ ∈ Bp , ∀ξ ∈ Bq ,

i−n (δ1 , ..., δp , ξ1 , ..., ξq ) = f (δ1 , ..., δp1 , ξ1 , ..., ξq )}

= L 2 ∨ (n + L2 )

 ... and the situation continues, until one or both of L k , Lk are empty. If, for example, Lk = ×





= Lk ∨ (n + Lk ) = n + Lk and so on. ∅, then Lk The fact that , are combinational means that the numbers n , p exist with the property that   L 1 ∨ L2 ∨ ... ∨ Ln = {1, ..., n},

L 1 ∨ L 2 ∨ ... ∨ L p = {1, ..., p}. These imply: ∨ L×

∨ ... ∨ L×

L×

1 2 max{n ,p }  



= L 1 ∨ L2 ∨ ... ∨ Lmax{n ,p } ∨ (n + L1 ) ∨ (n + L2 ) ∨ ... ∨ (n + Lmax{n ,p } )

= {1, ..., n} ∨ {n + 1, ..., n + p} = {1, ..., n + p}, thus  × is combinational. (b) The first level of  × is given by ∀i ∈ L×

, 1  ( × )i ((μ, δ), (ν, ξ )) =

i (μ, ν), if i ∈ L 1,

i−n (δ, ξ ), if i ∈ (n + L 1 )

Chapter 36 • Combinational systems  =

349

i1 (μ1 , ..., μn1 , ν1 , ..., νm ), if i ∈ {1, ..., n1 }, 1 ϒi−n (δ1 , ..., δp1 , ξ1 , ..., ξq ), if i ∈ {n + 1, ..., n + p1 }

where the 1-combinational functions  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 and ϒ 1 : Bp1 −p0 × Bp0 +q → Bp1 −p0 are the first levels of  and etc. (c) If 1 = n0 < n1 < ... < nn = n, 1 = p0 < p1 < ... < pp = p



and  1 ,  2 , ...,  n , ϒ 1 , ϒ 2 , ..., ϒ p are the levels of , , we obtain from (36.6.1) that ∀μ ∈ Bn , ∀ν ∈ Bm , ∀δ ∈ Bp , ∀ξ ∈ Bq , ∀i ∈ {1, ..., n + p}, ⎧ 1 ⎪ i−n ((μ1 , ..., μn1 ), (ν1 , ..., νm )), if i ∈ {n0 + 1, ..., n1 }, ⎪ 0 ⎪ ⎪ ⎪ ... ⎪ ⎪

⎪ n ⎪ ⎪ i−n ((μnn −1 +1 , ..., μnn ), (μ1 , ..., μnn −1 , ν1 , ..., νm )), ⎪ n −1 ⎪ ⎪ ⎪ ⎪ if i ∈ {nn −1 + 1, ..., nn }, ⎨ 1 ϒi−n−p0 ((δ1 , ..., δp1 ), (ξ1 , ..., ξq )), ( × )i ((μ, δ), (ν, ξ )) = ⎪ ⎪ ⎪ if i ∈ {n + p0 + 1, ..., n + p1 }, ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ p

⎪ ⎪ ϒi−n−p ((δpp −1 +1 , ..., δpp ), (δ1 , ..., δpp −1 , ξ1 , ..., ξq )), ⎪ ⎪ p −1 ⎪ ⎩ if i ∈ {n + pp −1 + 1, ..., n + pp }, wherefrom ∀ν ∈ Bm , ∀ξ ∈ Bq , ∀i ∈ {1, ..., n + p},

χ× ,i (ν, ξ ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

χ 1 ,i−n0 (ν), if i ∈ {n0 + 1, ..., n1 }, ... χ n ,i−n (χ 1 (ν), ..., χ n −1 (χ 1 (ν), ..., ν), ν), n −1

if i ∈ {nn −1 + 1, ..., nn }, χϒ 1 ,i−n−p0 (ξ ), if i ∈ {n + p0 + 1, ..., n + p1 }, ... χϒ p ,i−n−p (χϒ 1 (ξ ), ..., χϒ p −1 (χϒ 1 (ξ ), ..., ξ ), ξ ), p −1

if i ∈ {n + pp −1 + 1, ..., n + pp } = (χ × χ )i (ν, ξ ).

36.7 Predecessors and successors Theorem 264. The p-combinational function  : Bn × Bm → Bn is given, and we suppose that ν ∈ Bm exists such that ∀k ∈ N, u(k) = ν.

350

Boolean Systems

Let  1 : Bn1 −n0 × Bn0 +m → Bn1 −n0 ,  2 : Bn2 −n1 × Bn1 +m → Bn2 −n1 , ...,  p : Bnp −np−1 × Bnp−1 +m → Bnp −np−1 be the levels of , 0 = n0 < n1 < n2 < ... < np = n. We denote for an arbitrary μ ∈ Bn with μ1 ∈ Bn1 −n0 , μ2 ∈ Bn2 −n1 , ..., μp ∈ Bnp −np−1 the coordinates μ1 = (μn0 +1 , ..., μn1 ), μ2 = (μn1 +1 , ..., μn2 ), ..., μp = (μnp−1 +1 , ..., μnp ). (a) The following statements are true for any μ ∈ Bn : 1 2 1 p 1 p−1 p μ+ , ν)], ν = [μ , χ 1 (ν)] × [μ , χ 2 (μ , ν)] × ... × [μ , χ (μ , ..., μ 1 2 1 1 p−1 , ν), μp ]. p μ− ν = [χ 1 (ν), μ ] × [χ 2 (μ , ν), μ ] × ... × [χ (μ , ..., μ

(b) In the special case when at (a) we have μ = χ (ν), we infer O + (χ (ν), u) = {χ (ν)}, and ∀α ∈ n , ∀k ∈ N, φ α (χ (ν), u, k) = χ (ν). Proof. (a) We can write, taking into account Theorem 3, page 4 μ+ ν = [μ, (μ, ν)] = [(μ1 , μ2 , ..., μp ), ( 1 (μ1 , ν),  2 (μ2 , (μ1 , ν)), ...,  p (μp , (μ1 , ..., μp−1 , ν))] = [μ1 ,  1 (μ1 , ν)] × [μ2 ,  2 (μ2 , (μ1 , ν))] × ... × [μp ,  p (μp , (μ1 , ..., μp−1 , ν))] = [μ1 , χ 1 (ν)] × [μ2 , χ 2 (μ1 , ν)] × ... × [μp , χ p (μ1 , ..., μp−1 , ν)], and similarly μ− ν = [(μ, ν), μ] = [( 1 (μ1 , ν),  2 (μ2 , (μ1 , ν)), ...,  p (μp , (μ1 , ..., μp−1 , ν)), (μ1 , μ2 , ..., μp )] = [( 1 (μ1 , ν),  2 (μ2 , (μ1 , ν)), ...,  p (μp , (μ1 , ..., μp−1 , ν)), (μ1 , μ2 , ..., μp )] = [ 1 (μ1 , ν), μ1 ] × [ 2 (μ2 , (μ1 , ν)), μ2 ] × ... × [ p (μp , (μ1 , ..., μp−1 , ν)), μp ] = [χ 1 (ν), μ1 ] × [χ 2 (μ1 , ν), μ2 ] × ... × [χ p (μ1 , ..., μp−1 , ν), μp ]. (b) We use Definition 141, page 343 first and get that ∀i ∈ {1, ..., n}, ⎧ ⎪ χ 1 ,i−n0 (ν), if i ∈ {n0 + 1, .., n1 }, ⎪ ⎪ ⎪ ⎪ χ 2 ⎨  ,i−n1 (χ 1 (ν), ν), if i ∈ {n1 + 1, ..., n2 }, hyp μi = χ,i (ν) = ... ⎪ ⎪ ⎪ χ p ,i−np−1 (χ 1 (ν), ..., χ p−1 (χ 1 (ν), ..., ν), ν), ⎪ ⎪ ⎩ if i ∈ {np−1 + 1, ..., np }, in other words μ1 = χ 1 (ν),

Chapter 36 • Combinational systems

351

μ2 = χ 2 (μ1 , ν), ... μp = χ p (μ1 , ..., μp−1 , ν). This and item (a) imply 1 1 2 2 p p 1 2 p μ+ ν = [μ , μ ] × [μ , μ ] × ... × [μ , μ ] = {μ } × {μ } × ... × {μ } = {μ}.

We use Theorem 88, page 103 now. We denote with φν the flow of ν and deduce: + O + (χ (ν), u) = O + (μ, u) = O (μ) = {μ} = χ (ν), ν

and also ∀α ∈ n , ∀k ∈ N, φ α (χ (ν), u, k) = (φν )α (μ, k) = μ = χ (ν). Problem 21. In the circumstances of Theorem 264 (a), to be studied O + (μ, u) and O − (μ, u).

36.8 Isomorphisms Problem 22. To be studied, taking into account Theorem 255, page 332, the way that the isomorphisms of combinational functions act on levels. An example for this will be given in the next section.

36.9 Symmetry relative to translations Example 181. We consider the 2-combinational functions , : B2 × B → B2 , ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ((μ1 , μ2 ), ν) = (ν, μ1 ν),

((μ1 , μ2 ), ν) = (ν, μ1 ν), which have the first levels  1 , 1 : B × B → B,∀μ1 ∈ B,∀ν ∈ B,  1 (μ1 , ν) = ν, 1 (μ1 , ν) = ν, and the second levels  2 , 2 : B × B2 → B,∀μ2 ∈ B,∀(μ1 , ν) ∈ B2 ,  2 (μ2 , (μ1 , ν)) = μ1 ν, 2 (μ2 , (μ1 , ν)) = μ1 ν.

352

Boolean Systems

Looking at the state portrait families of ,

(0, 0) 6

(0, 1) 6

(1, 0) 

(1, 1)

(0, 0) 

? (1, 0)

ν=0

(0, 1)

? - (1, 1) ν=1

(1, 0) 6

(1, 1) 6

(0, 0) 

(0, 1)

(1, 0) 

? (0, 0)

ν=1

(1, 1)

? - (0, 1) ν=0

that were drawn in the order ν = 0, ν = 1 for  and ν = 1, ν = 0 for , we note that an isomorphism (θ (1,0) , 1B2 , h

) :  → exists, for h

: B → B given by ∀ν ∈ B, h

(ν) = ν. Indeed, we can check that ∀(μ1 , μ2 ) ∈ B2 , ∀ν ∈ B, ∀(λ1 , λ2 ) ∈ B2 , θ (1,0) ((λ1 ,λ2 ) ((μ1 , μ2 ), ν)) = (λ1 ν ∪ λ1 μ1 ∪ μ1 ν, λ2 μ2 ∪ λ2 μ1 ν) = (λ1 μ1 ∪ λ1 ν, λ2 μ2 ∪ λ2 μ1 ν) = (λ1 ,λ2 ) ((μ1 , μ2 ), ν). To be noticed also the commutativity ∀δ ∈ B of the diagrams B×B 1B × 1B

( 1 )δ B

? B×B

1B ? (1 )1B (δ)B

B × B2

( 2 )δ B

1B × g

1B ? (2 )1B (δ)- ? B B × B2

and we have denoted with g

: B2 → B2 the function ∀(μ1 , ν) ∈ B2 , g

(μ1 , ν) = (μ1 , ν). This means that (1B , 1B , 1B ) :  1 → 1 and (1B , 1B , g

) :  2 → 2 are isomorphisms.

Chapter 36 • Combinational systems

353

36.10 Invariance Theorem 265. Let  : Bn × Bm → Bn be a combinational system. For any ν ∈ Bm such that ∀k ∈ N, u(k) = ν, the following 3-invariance property is true: ∀μ ∈ Bn , ∀ξ ∈ O + (μ, u), ∀α ∈ n , O α (ξ, u) ⊂ O + (μ, u). + + (μ, u) = O (μ) and we apply TheoProof. We fix an arbitrary μ ∈ Bn . We see that O ν rem 128, page 160.

36.11 Basins of attraction, attractors Theorem 266. The combinational system  : Bn × Bm → Bn , γ ∈ n and ν ∈ Bm are given. We ask that ∀k ∈ N, u(k) = ν. (a) We have ν (χ (ν)) = χ (ν) and W 1 (χ (ν)) = Wγ2 (χ (ν)) = W 3 (χ (ν)) = W 4 (χ (ν)) = Bn , W 5 (χ (ν)) = {χ (ν)} are true. (b) ∀μ ∈ Bn , ∀α ∈ n , W 1 [φ α (μ, u, ·)] = Wγ2 [φ α (μ, u, ·)] = W 3 [φ α (μ, u, ·)] = W 4 [φ α (μ, u, ·)] = Bn , W 5 [φ α (μ, u, ·)] = {χ (ν)}. (c) {χ (ν)} is attractor. Proof. (a) Theorem 262, page 343, together with Theorem 89, page 104, shows that ν (χ (ν)) = χ (ν), and since  is combinational, χ (ν) is the only fixed point of ν . We get α (μ) ⊂ {χ (ν)}} W 3 (χ (ν)) = {μ|μ ∈ Bn , ∀α ∈ n , ω ν

= {μ|μ ∈ Bn , ∀α ∈ n , lim φ α (μ, u, k) = χ (ν)} k→∞

Theorem 262

=

Bn

and, by taking into account Theorem 187, page 222, we infer W 4 (χ (ν)) = Bn = W 3 (χ (ν)) ⊂ Wγ2 (χ (ν)) ⊂ W 1 (χ (ν)) ⊂ Bn .

354

Boolean Systems

In addition W 5 (χ (ν)) = {χ (ν)} results from the fact that χ (ν) is a fixed point of ν and Theorem 194, page 231. (b) For arbitrary μ ∈ Bn , α ∈ n we have W 3 [φ α (μ, u, ·)] = {μ |μ ∈ Bn , ∀β ∈ n , φ β (μ , u, ·) ≈ φ α (μ, u, ·)} (a)

= {μ |μ ∈ Bn , ∀β ∈ n , lim φ β (μ , u, k) = χ (ν)} = Bn . k→∞

We make use of Theorem 196, page 238, and we get W 4 [φ α (μ, u, ·)] = Bn = W 3 [φ α (μ, u, ·)] ⊂ Wγ2 [φ α (μ, u, ·)] ⊂ W 1 [φ α (μ, u, ·)] ⊂ Bn . On the other hand, Theorem 200, page 242 implies W 5 [φ α (μ, u, ·)] = W 5 [(φν )α (μ, ·)] = W 5 (χ (ν)) = {χ (ν)}, where φν is the flow generated by ν . (c) As ν (χ (ν)) = χ (ν), we infer that {χ (ν)} is attractor.

36.12 Subsystem Theorem 267. We suppose that : Bp × Bm −→ Bp is a q-combinational system and  : Bn × Bm −→ Bn is a subsystem, n ≤ p. Then  is r-combinational, r ≤ q. In addition, if  1 ,  2 , ...,  q ; 1 , 2 , ..., r are the levels of ; , then 1 ⊂  1 , 2 ⊂  2 , ..., r ⊂  r , i.e. the levels of  are subsystems of the levels of . Proof. The hypothesis implies the existence of the numbers 0 = p0 < p1 < ... < pq = p for which L 1 = {p0 + 1, ..., p1 }, L 2 = {p1 + 1, ..., p2 }, ..., L q = {pq−1 + 1, ..., pq } and the levels of have been denoted with  1 : Bp1 −p0 × Bm → Bp1 −p0 ,  2 : Bp2 −p1 × Bp1 +m → Bp2 −p1 , ...,  q : Bpq −pq−1 × Bpq−1 +m → Bpq −pq−1 . As  is a subsystem of , j1 , ..., jn ∈ {1, ..., p} exist, j1 < ... < jn such that ∀μ ∈ Bp , ∀ν ∈ m B , ∀i ∈ {1, ..., n},

ji ((μ1 , ..., μj1 , ..., μjn , ..., μp ), (ν1 , ..., νm )) = i ((μj1 , ..., μjn ), (ν1 , ..., νm )).

(36.12.1)

We need this initial interpretation of the subsystems from Theorem 68 and Definition 64, page 74. We denote with r the greatest s ∈ {1, ..., q} with the property that {j1 , ..., jn } ∧ {ps−1 + 1, ..., ps } = ∅.

Chapter 36 • Combinational systems

355

Case r = 12 If {j1 , ..., jn } = {p0 + 1, ..., p1 }, then the statement of the theorem is true for  = 1 =  1 , thus we can suppose that {j1 , ..., jn }  {p0 + 1, ..., p1 }. We fix μk ∈ B arbitrary for all k ∈ {p0 + 1, ..., p1 }  {j1 , ..., jn }. We obtain ∀(μj1 , ..., μjn ) ∈ Bn , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {1, ..., n}, i ((μj1 , ..., μjn ), (ν1 , ..., νm ))

(36.12.1)

=

j1i ((μ1 , ..., μj1 , ..., μjn , ..., μp1 ), (ν1 , ..., νm ))

= χ 1 ,ji (ν1 , ..., νm ). Then  is 1-combinational, its state space decomposition in 1-combinational systems consists in 1 : Bn × Bm −→ Bn where by definition 1 = , thus 1 ⊂  1 holds. Case r = 2 We claim that {j1 , ..., jn } ∧ {p0 + 1, ..., p1 } = ∅, {j1 , ..., jn } ∧ {p1 + 1, ..., p2 } = ∅ are both true. Let us suppose against all reason that this is false, i.e. that {j1 , ..., jn } ⊂ {p1 + 1, ..., p2 }. Theorem 258 (e), page 336 shows that ∀i ∈ L 2 , ∃j ∈ L 1 , ∃μ ∈ Bp , ∃ν ∈ Bm , i (μ ⊕ ε j , ν) = i (μ, ν)

(36.12.2)

and we fix such an i ∈ {j1 , ..., jn } ⊂ {p1 + 1, ..., p2 } = L 2 , namely i = jk . We get the existence of j ∈ {p0 + 1, ..., p1 } = L 1 , μ ∈ Bp and ν ∈ Bm , with k ((μj1 , ..., μjn ), (ν1 , ..., νm )) = jk ((μ1 , ..., μj , ..., μj1 , ..., μjn , ..., μp ), (ν1 , ..., νm ))

= jk ((μ1 , ..., μj , ..., μj1 , ..., μjn , ..., μp ), (ν1 , ..., νm )) = k ((μj1 , ..., μjn ), (ν1 , ..., νm )), contradiction. We denote n1 = card({j1 , ..., jn } ∧ {p0 + 1, ..., p1 }), and we have the numbers 0 = n0 < n1 < n2 = n. Two functions 1 : Bn1 −n0 × Bm −→ Bn1 −n0 , 2 : Bn2 −n2 × Bn1 +m −→ Bn2 −n1 must be defined. If {j1 , ..., jn } = {p0 + 1, ..., p2 }, then the theorem is true with n1 = p1 , n2 = p2 and 1 =  1 , 2 =  2 , thus we can suppose that {j1 , ..., jn }  {p0 + 1, ..., p2 }. We fix μk ∈ B arbitrary for all k ∈ {p0 + 1, ..., p2 }  {j1 , ..., jn }. We see that ∀(μj1 , ..., μjn ) ∈ Bn , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {n0 + 1, ..., n1 }, i ((μj1 , ..., μjn ), (ν1 , ..., νm )) (36.12.1)

=

j1i ((μ1 , ..., μj1 , ..., μjn1 , ..., μp1 ), (ν1 , ..., νm )) = χ 1 ,ji (ν1 , ..., νm )

is true. We define ∀(μj1 , ..., μjn1 ) ∈ Bn1 −n0 , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {n0 + 1, ..., n1 }, 1i ((μj1 , ..., μjn1 ), (ν1 , ..., νm )) = χ 1 ,ji (ν1 , ..., νm ), 2

See also Theorem 257, page 334 and Corollary 23.

356

Boolean Systems

and also χ1 : Bm → Bn1 −n0 by ∀(μj1 , ..., μjn1 ) ∈ Bn1 −n0 , ∀(ν1 , ..., νm ) ∈ Bm , χ1 (ν1 , ..., νm ) = 1 ((μj1 , ..., μjn1 ), (ν1 , ..., νm )). Similarly, ∀(μj1 , ..., μjn ) ∈ Bn , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {n1 + 1, ..., n2 }, i ((μj1 , ..., μjn ), (ν1 , ..., νm )) (36.12.1)

=

j2i ((μp1 +1 , ..., μjn1 +1 , ..., μjn2 , ..., μp2 ), (μ1 , ..., μj1 , ..., μjn1 , ..., μp1 , ν1 , ..., νm )) = χ 2 ,ji (μ1 , ..., μj1 , ..., μjn1 , ..., μp1 , ν1 , ..., νm )

is true, thus we define ∀(μj1 , ..., μjn1 ) ∈ Bn1 −n0 , ∀(μjn1 +1 , ..., μjn2 ) ∈ Bn2 −n1 , ∀(ν1 , ..., νm ) ∈ Bm , ∀i ∈ {n1 + 1, ..., n2 }, 2i−n1 ((μjn1 +1 , ..., μjn2 ), (μj1 , ..., μjn1 , ν1 , ..., νm )) = χ 2 ,ji (μ1 , ..., μj1 , ..., μjn1 , ..., μp1 , ν1 , ..., νm ) and also χ2 : Bn1 +m → Bn2 −n1 by ∀(μj1 , ..., μjn1 ) ∈ Bn1 −n0 , ∀(μjn1 +1 , ..., μjn2 ) ∈ Bn2 −n1 , ∀(ν1 , ..., νm ) ∈ Bm , χ2 (μj1 , ..., μjn1 , ν1 , ..., νm ) = 2 ((μjn1 +1 , ..., μjn2 ), (μj1 , ..., μjn1 , ν1 , ..., νm )). Both 1 , 2 are 1-combinational. The system  is in this case 2-combinational, 1 , 2 are its levels and the inclusions 1 ⊂  1 , 2 ⊂  2 hold. Case r ≥ 3 In a similar manner we can prove that {j1 , .., jn } ∧ {p0 + 1, ..., p1 } = ∅, {j1 , .., jn } ∧ {p1 + 1, ..., p2 } = ∅, ..., {j1 , .., jn } ∧ {pr−1 + 1, ..., pr } = ∅, and show the existence of 1 ⊂  1 , 2 ⊂  2 , ..., r ⊂  r , 1-combinational.

36.13 The fundamental operating mode Theorem 268. We consider the combinational function  : Bn × Bm → Bn . (a) For arbitrary p ≥ 1, α 0 , α 1 , ..., α p ∈ n , μ ∈ Bn , u0 , u1 , ..., up ∈ S (m) , ν 0 , ν 1 , ..., ν p ∈ Bm , if ∀k ∈ N, u0 (k) = ν 0 , ∀k ∈ N, u1 (k) = ν 1 , ... ∀k ∈ N, up (k) = ν p , then k0 , k1 , ..., kp ≥ 1 exist that satisfy φ α (μ, u0 , k0 ) = χ (ν 0 ), 0

(36.13.1)

Chapter 36 • Combinational systems

φ α (χ (ν 0 ), u1 , k1 ) = χ (ν 1 ), 1

357

(36.13.2)

... φ α (χ (ν p−1 ), up , kp ) = χ (ν p ); p

in addition, for arbitrary k0 ≥ k0 , k1 ≥ k1 , ..., kp−1 ≥ kp−1 , if we define γ ∈ n , v ∈ S (m) by

γk =

v(k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

α 0,k , if k ∈ {0, ..., k0 − 1}, if k ∈ {k0 , ..., k0 + k1 − 1}, ... p−1,k−k −...−k 0 p−2 , if k ∈ {k0 + ... + kp−2 , α ..., k0 + ... + kp−1 − 1}, α p,k−k0 −...−kp−1 , if k ≥ k0 + ... + kp−1 , α 1,k−k0 ,

ν 0 , if k ∈ {0, ..., k0 − 1}, ν 1 , if k ∈ {k0 , ..., k0 + k1 − 1}, ... ν p−1 , if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 − 1}, ν p , if k ≥ k0 + ... + kp−1 ,

(36.13.3)

(36.13.4)

we have

φ γ (μ, v, k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

φ α (μ, u0 , k), if k ∈ {0, ..., k0 }, φ (χ (ν 0 ), u1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p−1 φ α (χ (ν p−2 ), up−1 , k − k0 − ... − kp−2 ), if k ∈ {k0 + ... + kp−2 , ..., k0 + ... + kp−1 }, p φ α (χ (ν p−1 ), up , k − k0 − ... − kp−1 ), if k ≥ k0 + ... + kp−1 . α1

(b) We take α 0 , α 1 , ..., α p , ... ∈ n , μ ∈ Bn , u0 , u1 , ..., up , ... ∈ S (m) , ν 0 , ν 1 , ..., ν p , ... ∈ Bm arbitrary, with ∀k ∈ N, u0 (k) = ν 0 , ∀k ∈ N, u1 (k) = ν 1 , ... ∀k ∈ N, up (k) = ν p , ...

358

Boolean Systems

Then k0 , k1 , ..., kp , ... ≥ 1 exist with φ α (μ, u0 , k0 ) = χ (ν 0 ),

(36.13.5)

φ α (χ (ν 0 ), u1 , k1 ) = χ (ν 1 ),

(36.13.6)

0

1

... φ α (χ (ν p−1 ), up , kp ) = χ (ν p ), p

(36.13.7)

... Furthermore, for arbitrary k0 ≥ k0 , k1 ≥ k1 , ..., kp ≥ kp , ..., we define the computation function γ ∈ n and the input v ∈ S (m) as: ⎧ ⎪ α 0,k , if k ∈ {0, ..., k0 − 1}, ⎪ ⎪ ⎪ 1,k−k 0 , if k ∈ {k , ..., k + k − 1}, ⎪ α ⎨ 0 0 1 γk = (36.13.8) ... ⎪ p,k−k −...−k ⎪ 0 p−1 ⎪ , if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, α ⎪ ⎪ ⎩ ...

v(k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ν 0 , if k ∈ {0, ..., k0 − 1}, if k ∈ {k0 , ..., k0 + k1 − 1}, ... ν p , if k ∈ {k0 + ... + kp−1 , ..., k0 + ... + kp − 1}, ... ν1,

(36.13.9)

We have

φ γ (μ, v, k) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

φ α (μ, u0 , k), if k ∈ {0, ..., k0 }, 1 α φ (χ (ν 0 ), u1 , k − k0 ), if k ∈ {k0 , ..., k0 + k1 }, ... p α p−1 p ), u , k − k0 − ... − kp−1 ), if k ∈ {k0 + ... + kp−1 , φ (χ (ν ..., k0 + ... + kp }, ...

Proof. (b) From Theorem 262, page 343 we get the existence of k0 , k1 , ..., kp , ... ≥ 1 such that ∀k ≥ k0 , φ α (μ, u0 , k) = χ (ν 0 ),

(36.13.10)

∀k ≥ k1 , φ α (χ (ν 0 ), u1 , k) = χ (ν 1 ),

(36.13.11)

0

1

...

Chapter 36 • Combinational systems

∀k ≥ kp , φ α (χ (ν p−1 ), up , k) = χ (ν p ), p

359

(36.13.12)

... We apply Theorem 251, page 323. Remark 242. The previous theorem characterizes the fundamental operating mode (γ , μ, v). The combinational system  accepts at (b) a fundamental mode along μ, χ (ν 0 ), ..., χ (ν p ), ... ∈ Bn for any μ and any ν 0 , ν 1 , ..., ν p , ... But we can abusively say, see Remark 228, page 325, that  accepts at (a) also a fundamental mode along μ, χ (ν 0 ), ..., χ (ν p ) ∈ Bn for any μ and any ν 0 , ν 1 , ..., ν p .3

3

Characterizing the circumstances of this abuse might be interesting.

37 Wires, gates, and flip flops The purpose of this chapter is to give examples of modeling very simple circuits, and the first two sections are introductory. The delay element is defined in Section 37.3. Several examples of circuits with delay elements are analyzed. The gates, presented in Section 37.4, are the devices that compute the Boolean functions. Of them, the NAND gate is probably the most important, since it computes the function of Sheffer. Sections 37.5, 37.6, and 37.7 are dedicated to the SR latch, the gated SR flip flop and the D type flip flop.

37.1 Circuits Two things are understood by the word circuit: a physical device and its model. Let us recall that other usual names for the model are system and network. Unlike the word system, that looks more abstract, the word circuit seems concrete and abstract at the same time, and allows the identification between the physical device and the model, when this identification is convenient.

37.2 The wire The wire is a component of a more complex physical device and acts as ideal connector, indicating the existence of the same voltages, i.e. the same signals in several parts of a circuit. The symbol of the wire was drawn in Fig. 1, where two signals u, x ∈ S were also specified. As u and x are connected by a wire, Fig. 1 is associated to the equation ∀k ∈ N, x(k) = u(k). The possibility exists that the symbol of the wire has no labels of signals. The symbol of the wire, by its symmetry, does not suggest the sense of the flow of information through it, since this is not necessary.

FIGURE 1 The wire. Boolean Systems. https://doi.org/10.1016/B978-0-32-395422-8.00043-X Copyright © 2023 Elsevier Inc. All rights reserved.

361

362

Boolean Systems

FIGURE 2 The delay element.

37.3 The delay element 37.3.1 The delay element Definition 143. The delay element (or delay circuit, or delay buffer) is the system defined by any of (a) the generator function  : B × B → B, (x(k), u(k)) = u(k), (b) the equation of evolution ∀k ∈ N, x(k + 1) = u(k)α k ∪ x(k)α k ,

(37.3.1)

where x(0) = x 0 ∈ B and α ∈  are parameters, (c) the state portrait family 0

1

- 1

0

u=0

u=1

The symbol of the delay element is drawn in Fig. 2. Remark 243. There are two signals u, x ∈ S in Fig. 2, and there is also a suggestion given by the asymmetry of the symbol, that the input is u, the output is x, and the sense of the flow of information through the delay element is from the input to the output. Remark 244. We have used the words input (in a delay element) and output (from a delay element), as opposite. We can surely call x state as well. Remark 245. In Definition 143, instead of ∀μ ∈ B, ∀ν ∈ B, (μ, ν) = ν, we have written (x(k), u(k)) = u(k). This notation aims to put (a) and (b) in Definition 143 closer to each other. Remark 246. The delay element is a 1-combinational system, and its input-output function χ : B → B is the identity, χ = 1B . Remark 247. Note in Fig. 2 the existence of the wires, they are always necessary. This is somehow analogue to the connectors from logic. Theorem 269. In Eq. (37.3.1), (a) if ∀k ∈ N, α k = 1, then the solution x fulfills ∀k ≥ 1, x(k) = u(k − 1),

(37.3.2)

Chapter 37 • Wires, gates, and flip flops

363

(b) in case that ∃k ∈ N, α k = 0, u exists such that x does not fulfill (37.3.2), (c) if ∃k  ∈ N, ∀k ≥ k  , u(k) = u(k  ), 

then σ k (x)(k) → u(k  ) monotonically. Proof. (c) This results from Theorem 252, page 329, but we can make also the following reasoning. For any k ≥ k  , we have from (37.3.1): x(k + 1) = u(k  )α k ∪ x(k)α k and we denote k1 = min{1 + k|k ≥ k  , α k = 1}. We infer that



x(k  ), if k ∈ {k  , ..., k1 − 1}, u(k  ), if k ≥ k1

x(k) = therefore ∀k ∈ N, k



σ (x)(k) = x(k + k ) =



x(k  ), if k ∈ {0, ..., k1 − k  − 1}, u(k  ), if k ≥ k1 − k  .

Remark 248. The interpretation of the previous theorem is: the input values u(k) are reproduced at the output x with a delay equal with 1, provided that α = 1, 1, 1, ... When some values α k are 0, the appropriate values u(k) are filtered out, they are lost. The phenomenon is called inertia. Since a delay exists also, the two phenomena, delay and inertia overlap as: at (a) the delay is said to be pure, and at (b) the delay is said to be inertial. Remark 249. We interpret the state portrait family of the delay element now, by taking a look at the previous theorem: it shows that keeping u constant 0 long enough produces an eventual switch of x from 1 to 0, if it was 1 and respectively that keeping u constant 1 long enough produces an eventual switch of x from 0 to 1, if it was 0. The state portrait family is relevant for k ≥ k  . Therefore two global attractors exist, see Theorem 266, page 353: 0 if ∀k ≥ k  , u(k) = 0 and 1 if ∀k ≥ k  , u(k) = 1. Remark 250. The word ‘delay’ has three meanings in this text,1 but these distinct meanings will create no misunderstandings: (a) it is the name of a system, which is defined at Definition 143, (b) it is a natural number, equal with 1 at Remark 248, (c) when classifying in Remark 248 the states of the system from Definition 143 in pure delays and inertial delays, delay means state. 1

And other texts also.

364

Boolean Systems

FIGURE 3 Delay element with feedback.

37.3.2 Circuit with a delay element and feedback Definition 144. We define the system by any of (a) the generator function  : B → B, (x(k)) = x(k), (b) the equation ∀k ∈ N, x(k + 1) = x(k)

(37.3.3)

with x(0) = x 0 ∈ B parameter, (c) the state portrait 0

1

The symbol of the circuit is drawn in Fig. 3. Remark 251. Eq. (37.3.3) results from (37.3.1) for u = x, the feedback loop: ∀α ∈ , ∀k ∈ N, x(k + 1) = x(k)α k ∪ x(k)α k = x(k)(α k ∪ α k ) = x(k). Remark 252. The solution of (37.3.3) is ∀k ∈ N, x(k) = x 0 , and the absence in the state portrait of the underlined coordinates indicates the constancy of x.

37.3.3 Circuit with two delay elements Definition 145. The system is defined by any of (a) the generator function  : B2 × B → B2 , ((x(k), y(k)), u(k)) = (u(k), x(k)), (b) the equation of evolution ∀k ∈ N, 

x(k + 1) = u(k)α k ∪ x(k)α k , y(k + 1) = x(k)β k ∪ y(k)β k

with x(0) = x 0 ∈ B, y(0) = y 0 ∈ B, α ∈ , β ∈  parameters,

(37.3.4)

Chapter 37 • Wires, gates, and flip flops

365

FIGURE 4 Two delay elements.

(c) the state portrait family (0, 0)  6

(1, 0)

(0, 1) - 6 - (1, 1)

u=0

(0, 0) 

? (1, 0)

(0, 1)

? - (1, 1) u=1

which was drawn in the plane (x, y). The symbol of the circuit is given in Fig. 4. Remark 253. The system is 2-combinational, its levels are 1 : B × B → B, 2 : B × B2 → B, ∀μ1 ∈ B, ∀μ2 ∈ B, ∀ν ∈ B, 1 (μ1 , ν) = ν, 2 (μ2 , (μ1 , ν)) = μ1 , for which χ 1 : B → B, χ 2 : B2 → B are ∀μ1 ∈ B, ∀ν ∈ B, χ 1 (ν) = ν, χ 2 (μ1 , ν) = μ1 and the input-output function of  is χ : B → B2 , ∀ν ∈ B, χ (ν) = (χ 1 (ν), χ 2 (χ 1 (ν), ν)) = (ν, ν). Remark 254. The formula ⎧ y 0 , if k = 0, ⎪ ⎪ ⎪ ⎨ x 0 β 0 ∪ y 0 β 0 , if k = 1, y(k) = k−2 k−1 ⎪ u(k − 2)α β ∪ x(k − 2)α k−2 β k−1 ∪ y(k − 1)β k−1 , ⎪ ⎪ ⎩ if k ≥ 2

(37.3.5)

shows that when ∀k ≥ 2, α k−2 β k−1 = 1, the input values u(k) are reproduced by the output y with a pure delay of 2 time units. And other α, β make some input values be lost, inertia. Remark 255. The state portrait family of  indicates the existence of two global attractors, (0, 0) = χ (0) if ∀k ∈ N, u(k) = 0 and (1, 1) = χ (1) if ∀k ∈ N, u(k) = 1.

37.3.4 Circuit with two delay elements and feedback Definition 146. The system is defined by any of (a) the generator function  : B2 → B2 , (x(k), y(k)) = (y(k), x(k)), (b) the equation ∀k ∈ N,  x(k + 1) = y(k)α k ∪ x(k)α k , y(k + 1) = x(k)β k ∪ y(k)β k ,

(37.3.6)

366

Boolean Systems

FIGURE 5 Two delay elements with feedback.

FIGURE 6 Two delay elements with feedback.

with x(0) = x 0 ∈ B, y(0) = y 0 ∈ B, α ∈ , β ∈  parameters, (c) the state portrait - (1, 0) (0, 1) 

? (0, 0)

- ? (1, 1)

which was drawn in the plane (x, y). There are two symbols of the circuit, drawn in Figs. 5, 6. Remark 256. Eq. (37.3.6) results from (37.3.4) for u = y (the feedback). Remark 257. The state portrait shows that two possibilities exist: either stability, when one of the fixed points (0, 0), (1, 1) is reached, and (29.1.3)page 275 is true for α = (1, 0), (1, 1), (1, 1), (1, 1), ... or instability, when (x(k), y(k)) switches between (1, 0) and (0, 1). Theorem 270. In Eq. (37.3.6), (a) the following statements are equivalent: (a.1) ∃k  ∈ N, x(k  ) = y(k  ), (a.2) one of lim (x(k), y(k)) = (0, 0), lim (x(k), y(k)) = (1, 1) holds, k→∞

k→∞

(b) the following statements are also equivalent: (b.1) ∀k ∈ N, x(k) = y(k), (b.2) x 0 = y 0 and α = β, (b.3) lim (x(k), y(k)) does not exist. k→∞

Chapter 37 • Wires, gates, and flip flops

367

Proof. (a.1)=⇒(a.2) We suppose that k  ∈ N exists such that x(k  ) = y(k  ). We have    x(k  + 1) = x(k  )α k ∪ x(k  )α k = x(k  ),   y(k  + 1) = y(k  )β k ∪ y(k  )β k = y(k  ) and we can prove by induction on k ≥ k  that (x(k), y(k)) = (x(k  ), y(k  )). (a.2)=⇒(a.1) is obvious. (b.1)=⇒(b.2) The hypothesis states that ∀k ∈ N, x(k) = y(k). Then x 0 = y 0 and ∀k ∈ N, y(k)α k ∪ x(k)α k = x(k + 1) = y(k + 1) = x(k)β k ∪ y(k)β k = (x(k) ∪ β k )(y(k) ∪ β k ) = x(k) y(k) ∪ y(k) β k ∪ x(k)β k , in other words, since y(k) = x(k), x(k) y(k) = 0, we infer x(k)α k ∪ x(k)α k = x(k)β k ∪ x(k)β k , thus x(k) ⊕ α k = x(k) ⊕ β k is true. We have obtained that α k = β k and, as far as k was arbitrary, that α = β. (b.2)=⇒(b.1) We can write x(1) ⊕ y(1) = (y 0 α 0 ∪ x 0 α 0 ) ⊕ (x 0 β 0 ∪ y 0 β 0 ) = (x 0 α 0 ∪ x 0 α 0 ) ⊕ (y 0 β 0 ∪ y 0 β 0 ) = (x 0 ⊕ α 0 ) ⊕ (y 0 ⊕ β 0 ) = x 0 ⊕ y 0 = 1 i.e. x(1) = y(1). The result is proved by induction on k ≥ 0. (b.1)=⇒(b.3) We suppose against all reason that lim (x(k), y(k)) exists: k→∞

∃k  ∈ N, ∀k ≥ k  , (x(k), y(k)) = (x(k  ), y(k  )). The possibilities (x(k  ), y(k  )) = (0, 0), (x(k  ), y(k  )) = (1, 1) are excluded, thus we analyze Case (x(k  ), y(k  )) = (0, 1) We infer ∀k ≥ k  , 0 = x(k + 1) = α k , contradiction with the progressiveness of α. Case (x(k  ), y(k  )) = (1, 0) In this case we get ∀k ≥ k  , 1 = x(k + 1) = α k , representing a contradiction with the progressiveness of α again. (b.3)=⇒(b.1) If we suppose against all reason that ∃k  ∈ N, x(k  ) = y(k  ), then we infer, from (a), that lim (x(k), y(k)) exists, contradiction. k→∞

Remark 258. In Theorem 270, (a) refers to stable states, and (b) refers to unstable states. Note at (b.2) the condition α = β meaning that the two delay elements must be identical in order to have an unstable system.

368

Boolean Systems

37.4 Gates 37.4.1 Gates Definition 147. The 1-combinational functions  : B × Bm → B are also called (logical) gates. Remark 259. The gates are characterized by the equations ∀k ∈ N, x(k + 1) = χ (u(k))α k ∪ x(k)α k ,

(37.4.1)

where α ∈ , x(0) = x 0 ∈ B and u ∈ S (m) . For example the delay element is a gate with χ = 1B , see (37.3.1)page 362 . In order to explain the model, we decompose (37.4.1) in two equations: v(k) = χ (u(k)), considered to be the ideal computation of χ : Bm → B, since there is no delay and no loss of data (no inertia), and x(k + 1) = v(k)α k ∪ x(k)α k , the delay element, that concentrates the delay and the possible inertia of the gate. In other words, starting from the necessity of computing χ , we have supposed that this computation is made ideally, and afterwards we have inserted the delay element. Remark 260. What we have indicated previously is just a choice. Another possibility of modeling, not used in our work, but suggested in literature, is that of introducing delay elements before u, u1 (k + 1) = w1 (k)β1k ∪ u1 (k)β1k , ... k k, um (k + 1) = wm (k)βm ∪ um (k)βm

where u1 (0) = u01 ∈ B, ..., um (0) = u0m ∈ B, β1 ∈ , ..., βm ∈  are parameters and the new input in the circuit that computes χ is w ∈ S (m) , while the output is x(k) = χ (u(k)). And there is still an unused possibility, of introducing delay elements before and after the ideal computation of χ .

Chapter 37 • Wires, gates, and flip flops

369

FIGURE 7 The NOT gate.

37.4.2 The NOT gate Definition 148. The NOT gate (or the inverter) is the system defined by any of (a) the generator function  : B × B → B, (x(k), u(k)) = u(k), (b) the equation of evolution ∀k ∈ N, x(k + 1) = u(k)α k ∪ x(k)α k ,

(37.4.2)

with x(0) = x 0 ∈ B, α ∈  parameters, (c) the state portrait family - 1

0

0

u=0

1 u=1

The symbol of the NOT gate is given in Fig. 7. Remark 261. To be compared (37.4.2) with (37.4.1) and also with (37.3.1)page 362 . Remark 262. To be noticed in Fig. 7 the two wires labeled u, x, and also the similarities with the symbol of the delay element from Fig. 2, page 362. Remark 263. Theorem 269, page 362, adapted to (37.4.2), is still true: the values u(k) are reproduced at the output x with a delay equal with 1, provided that α = 1, 1, 1, ... etc., respectively: if k  ∈ N exists with ∀k ≥ k  , u(k) = u(k  ), 

then σ k (x)(k) → u(k  ) monotonically.

37.4.3 Circuit with a Not gate and feedback Definition 149. The system is defined by any of (a) the generator function is  : B → B, (x(k)) = x(k), (b) the equation ∀k ∈ N, x(k + 1) = x(k)α k ∪ x(k)α k , where x(0) = x 0 ∈ B and α ∈ , (c) the state portrait 0

-

1

(37.4.3)

370

Boolean Systems

FIGURE 8 Not gate and feedback.

The symbol of the circuit is drawn in Fig. 8. Remark 264. Eq. (37.4.3) represents (37.4.2) rewritten for u = x, the feedback loop. Remark 265. The state portrait indicates the instability of the system. Theorem 271. The solution x of Eq. (37.4.3) fulfills: ∀k ∈ N, ∃k  > k, x(k) = x(k  ). Proof. Let k ∈ N arbitrary and we denote k  = min{1 + p|p ≥ k, α p = 1}. We rewrite (37.4.3) as x(k + 1) = x(k) ⊕ α k and we obtain x(k  ) (37.4.3)

=

(37.4.3)

=



x(k  − 1) ⊕ α k −1

x(k) ⊕ α k ⊕ ... ⊕ α

k  −2

(37.4.3)





(37.4.3)

=

x(k  − 2) ⊕ α k −2 ⊕ α k −1

k  −1

= x(k) ⊕ 0 ⊕ ... ⊕ 0 ⊕ 1 = x(k) ⊕ 1.

⊕α

=

...

Remark 266. A concise way of expressing the previous theorem is saying that the solution x of (37.4.3) is orbitally equivalent with x 0 , x 0 , x 0 , x 0 , x 0 , ...

37.4.4 Circuit with two NOT gates Definition 150. The system is defined by any of (a) the generator function  : B2 × B → B2 , ((x(k), y(k)), u(k)) = (u(k), x(k)), (b) the equation ∀k ∈ N,  x(k + 1) = u(k)α k ∪ x(k)α k , y(k + 1) = x(k)β k ∪ y(k)β k with the parameters x(0) = x 0 ∈ B, y(0) = y 0 ∈ B, α ∈  and β ∈ , (c) the following state portrait family, drawn in the plane (x, y) - (0, 1)

(0, 0)

? (1, 0)  u=0

- ? (1, 1)

- (0, 1) 6

(0, 0) 6 (1, 0) 

u=1

(1, 1)

(37.4.4)

Chapter 37 • Wires, gates, and flip flops

371

FIGURE 9 Two NOT gates.

The symbol of the circuit is given in Fig. 9. Remark 267. The system is 2-combinational, with χ : B → B2 , ∀ν ∈ B, χ (ν) = (ν, ν). Remark 268. The analysis of the state portrait indicates that two global attractors exist, (1, 0) = χ (0) for ∀k ∈ N, u(k) = 0 and (0, 1) = χ (1) for ∀k ∈ N, u(k) = 1. Remark 269. We get ⎧ y 0 , if k = 0, ⎪ ⎪ ⎪ ⎨ 0 0 x β ∪ y 0 β 0 , if k = 1, y(k) = ⎪ u(k − 2)(x(k − 2) ∪ α k−2 )β k−1 ∪ x(k − 2) α k−2 β k−1 ∪ y(k − 1)β k−1 , ⎪ ⎪ ⎩ if k ≥ 2 and this is to be compared with (37.3.5)page 365 : when ∀k ≥ 2, α k−2 β k−1 = 1, the input values u(k) are also reproduced by the output y with a pure delay of 2 time units.

37.4.5 Circuit with two NOT gates and feedback Definition 151. The system is defined by one of (a) the generator function  : B2 → B2 , (x(k), y(k)) = (y(k), x(k)), (b) the equation ∀k ∈ N,  x(k + 1) = y(k)α k ∪ x(k)α k , y(k + 1) = x(k)β k ∪ y(k)β k ,

(37.4.5)

x(0) = x 0 ∈ B, y(0) = y 0 ∈ B, α ∈ , β ∈ , (c) the state portrait of  (0, 0) 

? (0, 1)

-

(1, 1)

- ? (1, 0)

There are two symbols of the system, given in Figs. 10, 11. Remark 270. Eq. (37.4.5) results from (37.4.4), for u = y (the feedback). Remark 271. The analysis of the state portrait shows the existence of two sinks (0, 1), (1, 0) and two transient points (0, 0), (1, 1).

372

Boolean Systems

FIGURE 10 Two NOT gates with feedback.

FIGURE 11 Two NOT gates with feedback.

Remark 272. The system is similar2 with the one from Definition 146, page 365. The next theorem represents the adaptation of Theorem 270 with (37.3.6)page 365 replaced by (37.4.5). Theorem 272. In Eq. (37.4.5), (a) the following statements are equivalent: (a.1) ∃k  ∈ N, x(k  ) = y(k  ), (a.2) one of lim (x(k), y(k)) = (0, 1), lim (x(k), y(k)) = (1, 0) is true, k→∞

k→∞

(b) the following statements are equivalent too: (b.1) ∀k ∈ N, x(k) = y(k), (b.2) x 0 = y 0 and α = β, (b.3) lim (x(k), y(k)) does not exist. k→∞

37.4.6 Circuit with a delay element and a NOT gate and feedback Definition 152. The system is defined by one of (a) the generator function  : B2 → B2 , (x(k), y(k)) = (y(k), x(k)), (b) the equation ∀k ∈ N,  x(k + 1) = y(k)α k ∪ x(k)α k , y(k + 1) = x(k)β k ∪ y(k)β k with the parameters x(0) = x 0 ∈ B, y(0) = y 0 ∈ B, α ∈  and β ∈ , 2

The word ‘similar’ is to be interpreted as: (θ (0,1) , 1B2 ) is isomorphism.

(37.4.6)

Chapter 37 • Wires, gates, and flip flops

373

FIGURE 12 Delay element and NOT gate with feedback.

(c) the state portrait (0, 0) 

? (1, 0)

(0, 1) 6 - (1, 1)

The symbol of the circuit is drawn in Fig. 12. Remark 273. By analyzing the state portrait, we notice the fact that all the points μ ∈ B2 are transient, thus the system is unstable under the form: ∀μ ∈ B2 , ∀μ ∈ B2 , ∀α ∈ 2 , ωα (μ ) = {μ}. This is stronger than ∃μ ∈ B2 , ∀μ ∈ B2 , ∀α ∈ 2 , ωα (μ ) = {μ}, the negation of the weakest stability property (29.1.1)page 275 . Theorem 273. In Eq. (37.4.6), we fix an arbitrary k  ∈ N and we define kx = min{1 + p|p ≥ k  , α p = 1}, ky = min{1 + p|p ≥ k  , β p = 1}. (a) If (x(k  ), y(k  )) = (0, 0), then  (x(k), y(k)) =

(0, 0), if k ∈ {k  , ..., kx − 1}, (1, 0), if k = kx ,

(b) if (x(k  ), y(k  )) = (1, 0), then  (x(k), y(k)) =

(1, 0), if k ∈ {k  , ..., ky − 1}, (1, 1), if k = ky ,

374

Boolean Systems

FIGURE 13 The NAND gate.

(c) if (x(k  ), y(k  )) = (1, 1), then  (x(k), y(k)) =

(1, 1), if k ∈ {k  , ..., kx − 1}, (0, 1), if k = kx ,

(d) if (x(k  ), y(k  )) = (0, 1), then  (x(k), y(k)) =

(0, 1), if k ∈ {k  , ..., ky − 1}, (0, 0), if k = ky .

Proof. In case (a), (37.4.6) gives for k = k  , k = k  + 1, ... 

x(k + 1) = α k , y(k + 1) = 0

until k = kx when (x(k), y(k)) = (1, 0), etc.

37.4.7 The NAND gate Definition 153. The NAND gate is defined by any of (a) the generator function  : B × B2 → B, (x(k), (u(k), v(k))) = u(k)v(k), (b) the equation of evolution ∀k ∈ N, x(k + 1) = u(k)v(k)α k ∪ x(k)α k ,

(37.4.7)

where x(0) = x 0 ∈ B, α ∈ , (c) the state portrait family, presented under a condensed form as 0

1

(u, v) = (1, 1)

0

- 1 (u, v) = (1, 1)

The symbol of the NAND gate is drawn in Fig. 13. Remark 274. We notice in Fig. 13 the existence of three wires, labeled u, v, x, and also the asymmetry of the figure, suggesting the fact that the information flow is from u, v, which are inputs, to x, which is output.

Chapter 37 • Wires, gates, and flip flops

375

Remark 275. The input-output function of  is called the function of Sheffer, or the Sheffer stroke and it is defined as | : B2 → B, ∀μ ∈ B, ∀ν ∈ B, μ|ν = μν. The importance of the Sheffer stroke is given by the fact that any Boolean function can be expressed with it, for example ∀μ ∈ B, ∀ν ∈ B, μ = μ|μ,

μν = μν = μ|ν = (μ|ν)|(μ|ν), and so on.

37.5 The SR latch Definition 154. The SR-latch is the system defined by any of (a) the generator function  : B2 × B2 → B2 , ((z(k), w(k)), (x(k), y(k))) = (x(k)w(k), y(k)z(k)), (b) the equation ∀k ∈ N, 

z(k + 1) = x(k)w(k)α k ∪ z(k)α k ,

(37.5.1)

w(k + 1) = y(k)z(k)β k ∪ w(k)β k , with z(0) = z0 ∈ B, w(0) = w 0 ∈ B, α ∈ , β ∈  parameters, (c) the state portrait family (z, w) (0, 0)

? (0, 1)

(z, w) - (1, 1) - 6

(1, 0)

(x, y) = (0, 0)

(0, 0)

- (1, 1) -

? (0, 1)

- ? (1, 0)

(x, y) = (0, 1)

376

Boolean Systems

FIGURE 14 The SR latch.

(z, w) (0, 0)

(z, w) - (1, 1) 6

(1, 0)

? (0, 1)

(x, y) = (1, 0)

(0, 0) 

- (1, 1)

? (0, 1)

- ? (1, 0)

(x, y) = (1, 1)

which was drawn in the plane (z, w). The symbol of the circuit is drawn in Fig. 14. Remark 276. In the state portrait family of the SR-latch, we can see that the situation (x, y) = (1, 1) coincides with the state portrait from Definition 151, page 371. Remark 277. In Theorem 274 that follows, we could have stated the usual hypothesis ∀k ∈ N, (x(k), y(k)) = (x(0), y(0)). We prefer asking that ∀k ≥ k  , (x(k), y(k)) = (x(k  ), y(k  )), where k  is arbitrary, because this allows using the theorem later more conveniently. Theorem 274. In Eq. (37.5.1), for any k  ∈ N, the following implications ∀k ≥ k  , (x(k), y(k)) = (0, 0) =⇒ ∃k  ≥ k  , ∀k ≥ k  , (z(k), w(k)) = (1, 1),

(37.5.2)

∀k ≥ k  , (x(k), y(k)) = (0, 1) ≥ k  , ∀k ≥ k  , (z(k), w(k)) = (1, 0),

(37.5.3)

∀k ≥ k  , (x(k), y(k)) = (1, 0) ≥ k  , ∀k ≥ k  , (z(k), w(k)) = (0, 1),

(37.5.4)

=⇒ ∃k  =⇒ ∃k 

((z(k  ), w(k  )) = (0, 1) and ∀k ≥ k  , x(k) = 1) =⇒ ∀k ≥ k  , (z(k), w(k)) = (0, 1),

(37.5.5)

Chapter 37 • Wires, gates, and flip flops

((z(k  ), w(k  )) = (1, 0) and ∀k ≥ k  , y(k) = 1) =⇒ ∀k ≥ k  , (z(k), w(k)) = (1, 0)

377

(37.5.6)

hold. Proof. (37.5.2): Eq. (37.5.1) becomes ∀k ≥ k  ,  z(k + 1) = α k ∪ z(k)α k , w(k + 1) = β k ∪ w(k)β k . For k1 and k2 defined by: k1 = min{1 + k|k ≥ k  , α k = 1},

(37.5.7)

k2 = min{1 + k|k ≥ k  , β k = 1},

(37.5.8)

we have

 z(k) =  w(k) =

z(k  ), if k ∈ {k  , ..., k1 − 1}, 1, if k ≥ k1 , w(k  ), if k ∈ {k  , ..., k2 − 1}, 1, if k ≥ k2 .

(37.5.3): We get from (37.5.1) that ∀k ≥ k  ,  z(k + 1) = α k ∪ z(k)α k , w(k + 1) = z(k)β k ∪ w(k)β k . We define k1 , k2 , k3 by (37.5.7), (37.5.8) and k3 = min{1 + k|k ≥ k1 , β k = 1}, and the possibility k2 = k3 exists. We infer  z(k) =

z(k  ), if k ∈ {k  , ..., k1 − 1}, 1, if k ≥ k1 ,

⎧   ⎨ w(k ), if k ∈ {k , ..., k2 − 1}, w(k) = z(k2 ), if k ∈ {k2 , ..., k3 − 1}, ⎩ 0, if k ≥ k3 . (37.5.5): We have (z(k  ), w(k  )) = (0, 1), ∀k ≥ k  , x(k) = 1, and we prove by induction on k ≥ k  + 1, taking into account (37.5.1), that  z(k) =

w(k − 1) = 1 = 0, if α k−1 = 1, z(k − 1) = 0, if α k−1 = 0,

378

Boolean Systems  w(k) =

y(k − 1)z(k − 1) = 0 = 1, if β k−1 = 1, w(k − 1) = 1, if β k−1 = 0.

Definition 155. The signal x is called the set input of the SR latch, and the signal y is called the reset input of the SR latch. Remark 278. (a) Functionally, it is convenient to think of z, w as complementary signals, and the final values of these signals are complementary at (37.5.3)–(37.5.6). We say that z is ‘set’ to 1 and it is ‘reset’ to 0. At Definition 155, x and y have been called set and reset inputs, because the 0 final value of x sets z to 1 at (37.5.3) and the 0 final value of y resets z to 0 at (37.5.4). (b) The undesired situation is considered to be the one from (37.5.2), where we have lim z(k) = lim w(k) = 1, the final values of z, w are not complementary.

k→∞

k→∞

Theorem 275. Eq. (37.5.1) is fulfilled and we suppose that k  ∈ N exists such that ∀k ≥ k  , (x(k), y(k)) = (x(k  ), y(k  )). The instability property lim (z(k), w(k)) does not exist

k→∞

is true if and only if (x(k  ), y(k  )) = (1, 1),

(37.5.9)

z(k  ) = w(k  ),

(37.5.10)

∀k ≥ k  , α k = β k

(37.5.11)

hold.3 Proof. If. We obtain from (37.5.1): 





z(k  + 1) = w(k  )α k ∪ z(k  )α k = z(k  )α k ∪ z(k  )α k = z(k  ) ⊕ α k , 









w(k  + 1) = z(k  )β k ∪ w(k  )β k = w(k  )β k ∪ w(k  )β k = w(k  ) ⊕ β k . 



On one hand, we can prove by induction on k ≥ k  that z(k) = w(k), z(k + 1) = z(k) ⊕ α k . 3

(37.5.12)

See the last state portrait of the state portrait family: the instability properties represented by the negations of (29.1.5)page 275 –(29.1.8)page 276 are all true for this system if (37.5.9), (37.5.10), (37.5.11) hold.

Chapter 37 • Wires, gates, and flip flops

379

On the other hand, Eq. (37.5.12) shows that z is omega-limit equivalent with z(k  ), z(k  ),

z(k  ), z(k  ), ... instability.

Only if. In case that (x(k  ), y(k  )) = (0, 0), (37.5.2) shows that lim (z(k), w(k)) = (1, 1). k→∞

If (x(k  ), y(k  )) = (0, 1), then (37.5.3) implies lim (z(k), w(k)) = (1, 0). And if (x(k  ), y(k  )) = k→∞

(1, 0), we have from (37.5.4) that lim (z(k), w(k)) = (0, 1). These show the validity of (37.5.9). k→∞

We suppose against all reason the falsity of (37.5.10), i.e. z(k  ) = w(k  ). We obtain 

z(k  + 1) = z(k  )α k ∪ z(k  )α k = z(k  ), 



w(k  + 1) = w(k  )β k ∪ w(k  )β k = w(k  ), 

and we can prove by induction on k that ∀k ≥ k  , (z(k), w(k)) = (z(k  ), w(k  )), contradiction. (37.5.10) takes place. We suppose against all reason that (37.5.11) is false, i.e. k ≥ k  exists with α k = β k , for example, without losing the generality, k1 = min{1 + k|k ≥ k  , α k = 1} < min{1 + k|k ≥ k  , β k = 1}. We infer ∀k ∈ {k  , ..., k1 − 1}, z(k) = w(k) and z(k1 ) = w(k1 − 1)α k1 −1 ∪ z(k1 − 1)α k1 −1 = z(k1 − 1)

= w(k1 − 1) = z(k1 − 1)β k1 −1 ∪ w(k1 − 1)β k1 −1 = w(k1 ). We can prove, like before, by induction on k, that ∀k ≥ k1 , (z(k), w(k)) = (z(k1 ), w(k1 )), contradiction. (37.5.11) is true. Remark 279. This type of reasoning concerning instability has been previously made under the form Theorem 270 (b), page 366, and a similar result was stated at Theorem 272 (b), page 372. Remark 280. The stability analysis of the SR latch, which started from the assumption of autonomy ∀k ≥ k  , (x(k), y(k)) = (x(k  ), y(k  )), was made in the following cases, that cover all the possibilities4 : (i) ∀k ≥ k  , (x(k), y(k)) = (0, 0) : (37.5.2), (ii) ∀k ≥ k  , (x(k), y(k)) = (0, 1) : (37.5.3), (iii) ∀k ≥ k  , (x(k), y(k)) = (1, 0) : (37.5.4), (iv) ∀k ≥ k  , (x(k), y(k)) = (1, 1), (z(k  ), w(k  )) = (0, 1) : (37.5.5), (v) ∀k ≥ k  , (x(k), y(k)) = (1, 1), (z(k  ), w(k  )) = (1, 0) : (37.5.6), (vi) ∀k ≥ k  , (x(k), y(k)) = (1, 1), z(k  ) = w(k  ), ∃k ≥ k  , α k = β k , when lim (z(k), w(k)) exk→∞

ists, Theorem 275, 4

In this reasoning, we take advantage in (iv) of the fact that the hypothesis of (37.5.5) is independent on y and in (v) that the hypothesis of (37.5.6) is independent on x.

380

Boolean Systems

FIGURE 15 The gated SR flip flop.

(vii) ∀k ≥ k  , (x(k), y(k)) = (1, 1), z(k  ) = w(k  ), ∀k ≥ k  , α k = β k , when lim (z(k), w(k)) k→∞

does not exist, Theorem 275.

37.6 The gated SR flip flop Definition 156. The gated SR flip flop is the system defined by one of (a) the generator function : B4 × B3 → B4 , ((x(k), y(k), z(k), w(k)), (u(k), v(k), t (k))) = (u(k)v(k), t (k)v(k), x(k)w(k), y(k)z(k)), (b) the equations ∀k ∈ N, 

x(k + 1) = u(k)v(k)γ k ∪ x(k)γ k , y(k + 1) = t (k)v(k)δ k ∪ y(k)δ k ,

(37.6.1)

and (37.5.1), with (x(0), y(0), z(0), w(0)) = (x 0 , y 0 , z0 , w 0 ) ∈ B4 and (γ , δ, α, β) ∈ 4 parameters. The symbol of the circuit is drawn in Fig. 15. Remark 281. We take profit of the fact that accepts the following state space decomposition ∀i ∈ {1, ..., 4},  =

i ((x(k), y(k), z(k), w(k)), (u(k), v(k), t (k))) i ((x(k), y(k)), (u(k), v(k), t (k))), if i ∈ {1, 2}, i−2 ((z(k), w(k)), (x(k), y(k), u(k), v(k), t (k))), if i ∈ {3, 4},

where : B2 × B3 → B2 is 1-combinational, ((x(k), y(k)), (u(k), v(k), t (k))) = (u(k)v(k), t (k)v(k)), and its equation of evolution is (37.6.1), while  : B2 × B5 → B2 , given by ((z(k), w(k)), (x(k), y(k), u(k), v(k), t (k))) = (x(k)w(k), y(k)z(k)), may be identified with 

Chapter 37 • Wires, gates, and flip flops

381

from Definition 154, page 375, and its equation of evolution is (37.5.1). Therefore, the last two coordinates z, w of were already analyzed as states of the SR latch. In addition, we can identify the gated SR flip flop with the state portrait families from Definition 154 (c) and (x, y)

(x, y)

- (1, 1) - 6

(0, 0)

(0, 0) 6

-

-

? (0, 1)

(1, 0)

v=0

u=t =0

or

- ? - (1, 0)

(0, 1)

(u, v, t) = (0, 1, 1)

(x, y) (0, 0)  ? (0, 1)  (u, v, t) = (1, 1, 0)

(1, 1)

(x, y) (1, 1) 6

 (0, 0)  6

(1, 0)

(0, 1)



(1, 1)

? (1, 0)

(u, v, t) = (1, 1, 1)

which were drawn in the plane (x, y). We have just avoided working with state portrait families in B4 . Theorem 276. In Eqs. (37.6.1), (37.5.1), for any k  ∈ N, the following implications hold: =⇒ ∃k 

≥ k  , ∀k

∀k ≥ k  , (u(k), v(k), t (k)) = (1, 1, 1) ≥ k  , (x(k), y(k)) = (0, 0) and (z(k), w(k)) = (1, 1),

(37.6.2)

=⇒ ∃k 

≥ k  , ∀k

∀k ≥ k  , (u(k), v(k), t (k)) = (1, 1, 0) ≥ k  , (x(k), y(k)) = (0, 1) and (z(k), w(k)) = (1, 0),

(37.6.3)

∀k ≥ k  , (u(k), v(k), t (k)) = (0, 1, 1) =⇒ ∃k  ≥ k  , ∀k ≥ k  , (x(k), y(k)) = (1, 0) and (z(k), w(k)) = (0, 1),

(37.6.4)

  =⇒

(z(k  ), w(k  )) = (0, 1) and x(k  ) = 1 and ∀k ≥ k  , (v(k) = 0 or u(k) = t (k) = 0) ∀k ≥ k  , (z(k), w(k)) = (0, 1) and ∃k  ≥ k  , ∀k ≥ k  , (x(k), y(k)) = (1, 1),

(37.6.5)

382

Boolean Systems   =⇒

(z(k  ), w(k  )) = (1, 0) and y(k  ) = 1 and ∀k ≥ k  , (v(k) = 0 or u(k) = t (k) = 0) ∀k ≥ k  , (z(k), w(k)) = (1, 0)  and ∃k ≥ k  , ∀k ≥ k  , (x(k), y(k)) = (1, 1).

(37.6.6)

Proof. Let k  ∈ N arbitrary. (37.6.2): We suppose that ∀k ≥ k  , (u(k), v(k), t (k)) = (1, 1, 1), and from (37.6.1) we infer ∀k ≥ k  ,  x(k + 1) = x(k)γ k , y(k + 1) = y(k)δ k . For k1 , k2 defined by k1 = min{1 + k|k ≥ k  , γ k = 1},

(37.6.7)

k2 = min{1 + k|k ≥ k  , δ k = 1},

(37.6.8)

we get

 x(k) =  y(k) =

x(k  ), if k ∈ {k  , ..., k1 − 1}, 0, if k ≥ k1 , y(k  ), if k ∈ {k  , ..., k2 − 1}, 0, if k ≥ k2 .

If we replace in Theorem 274, page 376, (37.5.2) k  with max{k1 , k2 }, we get the existence of k  ≥ max{k1 , k2 } such that ∀k ≥ k  , (z(k), w(k)) = (1, 1). The proofs of (37.6.3) and (37.6.4) are similar. (37.6.5): The hypothesis states that (z(k  ), w(k  )) = (0, 1), x(k  ) = 1 and ∀k ≥ k  , v(k) = 0 or u(k) = t (k) = 0. We get from (37.6.1) that ∀k ≥ k  :  x(k + 1) = γ k ∪ x(k)γ k , y(k + 1) = δ k ∪ y(k)δ k , i.e.

 (x(k), y(k)) =

(1, y(k  )), if k ∈ {k  , ..., k2 − 1}, (1, 1), if k ≥ k2 .

(37.6.9)

We apply Theorem 274, page 376, (37.5.5). (37.6.5) is true for k  = k2 . (37.6.6) is proved similarly with (37.6.5). Remark 282. Once again (see Remark 278, page 378), in circuits that include SR latches it is convenient that, when the inputs u, v, t have final values, the outputs z, w have final values, which are complementary, and this happens at (37.6.3)–(37.6.6). From this point of view, Theorem 276 shows the existence of an undesired situation at (37.6.2).

Chapter 37 • Wires, gates, and flip flops

383

Remark 283. In the situation that ∀k ≥ k  , (v(k) = 0 or u(k) = t (k) = 0), the hypotheses of (37.6.5), (37.6.6) do not cover all the possibilities. If for example x(k  ) = y(k  ) = 0, then, with k1 , k2 defined by (37.6.7), (37.6.8), Eq. (37.6.1) gives  x(k) =  y(k) =

0, if k ∈ {k  , ..., k1 − 1}, 1, if k ≥ k1 , 0, if k ∈ {k  , ..., k2 − 1}, 1, if k ≥ k2 ,

and (37.5.1) must be analyzed for (a) k1 < k2 , (b) k1 > k2 , (c) k1 = k2 . In case (a), (37.5.1) becomes ⎧ ⎪ (α k ∪ z(k)α k , β k ∪ w(k)β k ), if k ∈ {k  , ..., k1 − 1}, ⎨ (z(k + 1), w(k + 1)) = (w(k)α k ∪ z(k)α k , β k ∪ w(k)β k ), if k ∈ {k1 , ..., k2 − 1}, ⎪ ⎩ (w(k)α k ∪ z(k)α k , z(k)β k ∪ w(k)β k ), if k ≥ k2 .

37.7 The D type flip flop Definition 157. The D type flip flop (also known as data latch, delay flip flop, or D type bistable) is the system defined by any of (a) the generator function  : B5 × B2 → B5 , ((t (k), x(k), y(k), z(k), w(k)), (u(k), v(k))) = (u(k), u(k)v(k), t (k)v(k), x(k)w(k), y(k)z(k)), (b) the equations ∀k ∈ N, ⎧ ⎪ ⎨

t (k + 1) = u(k)ξ k ∪ t (k)ξ k , x(k + 1) = u(k)v(k)γ k ∪ x(k)γ k , ⎪ ⎩ y(k + 1) = t (k)v(k)δ k ∪ y(k)δ k ,

(37.7.1)

and (37.5.1), with (t (0), x(0), y(0), z(0), w(0)) = (t 0 , x 0 , y 0 , z0 , w 0 ) ∈ B5 and (ξ, γ , δ, α, β) ∈ 5 parameters. The symbol of the circuit is drawn in Fig. 16. Remark 284. We see that  accepts the following state space decomposition: ∀i ∈ {1, ..., 5},

=

⎧ ⎨

i ((t (k), x(k), y(k), z(k), w(k)), (u(k), v(k)))

ϒ(t (k), (u(k), v(k))), if i = 1, i−1 ((x(k), y(k)), (t (k), u(k), v(k))), if i ∈ {2, 3}, ⎩ i−3 ((z(k), w(k)), (t (k), x(k), y(k), u(k), v(k))), if i ∈ {4, 5}.

384

Boolean Systems

FIGURE 16 The D type flip flop.

We have denoted with ϒ : B × B2 → B the 1-combinational subsystem ϒ(t (k), (u(k), v(k))) = u(k). In addition, : B2 × B3 → B2 may be identified with 1 , 2 in Definition 156, page 380, which was analyzed as (37.6.1), and  : B2 × B5 → B2 may be identified with  in Definition 154, page 375, which was analyzed as (37.5.1). At the same time, we avoid referring to state portrait families which are useless and difficult to draw. Theorem 277. Let in Eqs. (37.7.1), (37.5.1), k  ∈ N arbitrary. Then k  ≥ k  exists such that: ∀k ≥ k  , (u(k), v(k)) = (1, 1) =⇒ ∀k ≥ k  , (t (k), x(k), y(k)) = (0, 0, 1) and (z(k), w(k)) = (1, 0), ∀k ≥ k  , (u(k), v(k)) = (0, 1)

=⇒ ∀k

≥ k  , (t (k), x(k), y(k)) = (1, 1, 0) and (z(k), w(k)) = (0, 1),

(37.7.2)

(37.7.3)



(z(k  ), w(k  )) = (0, 1) and x(k  ) = 1 and ∀k ≥ k  , v(k) = 0  ∀k ≥ k  , (z(k), w(k)) = (0, 1) =⇒ and ∀k ≥ k  , (x(k), y(k)) = (1, 1),

(37.7.4)



(z(k  ), w(k  )) = (1, 0) and y(k  ) = 1 and ∀k ≥ k  , v(k) = 0  ∀k ≥ k  , (z(k), w(k)) = (1, 0) =⇒ and ∀k ≥ k  , (x(k), y(k)) = (1, 1).

(37.7.5)

Proof. We fix an arbitrary k  ∈ N. (37.7.2): We can write, from (37.7.1) and taking into account the hypothesis, that ∀k ≥ k  , ⎧ ⎪ t (k + 1) = t (k)ξ k , ⎨ (37.7.6) x(k + 1) = x(k)γ k , ⎪ ⎩ k k y(k + 1) = t (k)δ ∪ y(k)δ .

Chapter 37 • Wires, gates, and flip flops

385

We define k1 = min{1 + k|k ≥ k  , ξ k = 1}, k2 = min{1 + k|k ≥ k  , γ k = 1}, k3 = min{1 + k|k ≥ k1 , δ k = 1} and we have: t (k) x(k) y(k)

(37.7.6)

=

0, if k ≥ k1 ,

(37.7.6)

=

0, if k ≥ k2 ,

(37.7.6)

1, if k ≥ k3 .

=

Theorem 274, (37.5.3)page 376 shows the existence of k  ≥ max{k2 , k3 } such that the conclusion of (37.7.2) is fulfilled. (37.7.4): We notice that the equations ∀k ≥ k  , x(k + 1) = γ k ∪ x(k)γ k , y(k + 1) = δ k ∪ y(k)δ k deduced from (37.7.1) and the hypothesis show that ∀k ≥ k  , x(k) = 1, and k  ≥ k  exists with ∀k ≥ k  , y(k) = 1. The fact that ∀k ≥ k  , (z(k), w(k)) = (0, 1) is deduced from (37.5.5)page 376 .

Remark 285. The values of t, which are missing in (37.7.4), (37.7.5), are considered to be not important. Remark 286. In Theorem 277, the property lim z(k) = lim w(k) is true at all of (37.7.2)– k→∞

k→∞

(37.7.5), and undesired situations lim z(k) = lim w(k) like at the SR latch and the gated SR k→∞

k→∞

flip flop do not exist. In other words, if we compare Theorem 277 with Theorem 276, page 381, the introduction of the inverter has avoided the undesired possibility (37.6.2)page 381 , while (37.7.2) has its origin in (37.6.3)page 381 and (37.5.3)page 376 , ..., (37.7.5) has its origin in (37.6.6)page 382 and (37.5.6)page 377 . The critical situation expressed by Theorem 275, page 378 is present here also when ∀k ≥ k  , v(k) = 0 and therefore ∃k  ≥ k  , ∀k ≥ k  , (x(k), y(k)) = (1, 1) are true, but the request ∀k ≥ k  , α k = β k , i.e. the identity of two logical gates, is practically impossible.

A Continuous time We show in this appendix the way that the discrete time theory from the book may be rewritten as a real time theory. At least the following questions arise: (a) do other possibilities of going from the discrete time to the real time exist? (b), in connection with (a): is the real time ‘stronger’ than the discrete time?

A.1 Limits, signals, and computation functions Definition 158. For any t ∈ R, the left limit x(t − 0) ∈ Bn and the right limit x(t + 0) ∈ Bn of the function x : R → Bn are defined by: ∃ε > 0, ∀ξ ∈ (t − ε, t), x(ξ ) = x(t − 0), ∃ε > 0, ∀ξ ∈ (t, t + ε), x(ξ ) = x(t + 0). When t runs in R, we get the left limit function and the right limit function of x, R ∈ t −→ x(t − 0) ∈ Bn , R ∈ t −→ x(t + 0) ∈ Bn . Definition 159. The initial value x(−∞ + 0) ∈ Bn of x : R → Bn is defined in the following way: ∃t  ∈ R, ∀t < t  , x(t) = x(−∞ + 0). Definition 160. We use the notation χA : R → B, A ⊂ R, for the characteristic function of the set A : ∀t ∈ R,  1, if t ∈ A, χA (t) = 0, if t ∈ / A. Definition 161. A real time n-dimensional signal is a function x : R → Bn with the property that μ ∈ Bn and the unbounded strictly increasing real sequence t0 < t1 < t2 < ... exist, such that x(t) = μ · χ(−∞,t0 ) (t) ⊕ x(t0 ) · χ[t0 ,t1 ) (t) ⊕ x(t1 ) · χ[t1 ,t2 ) (t) ⊕ ...

(A.1.1)

The set of the real time n-dimensional signals is denoted with S˙ (n) . Definition 162. A real time n-dimensional computation function is by definition a function ρ : R → Bn having the property that the unbounded strictly increasing sequence t0 < t1 < t2 < ... and the discrete time computation function α ∈ n exist with ρ(t) = α 0 · χ{t0 } (t) ⊕ α 1 · χ{t1 } (t) ⊕ α 2 · χ{t2 } (t) ⊕ ...

(A.1.2) 387

388

Boolean Systems

˙ n for the set of the real time n-dimensional computation functions. We use the notation  Remark 287. Previously, for any x ∈ S˙ (n) , μ = x(−∞ + 0) is unique, but the sequence t0 < t1 < t2 < ... making (A.1.1) true is not unique, since some terms x(tk ) · χ[tk ,tk+1 ) (t) may always be rewritten as x(tk ) · χ[tk ,t  ) (t) ⊕ x(t  ) · χ[t  ,tk+1 ) (t), where t  ∈ (tk , tk+1 ). In a similar way, given ˙ n , the computation function α ∈ n and the sequence t0 < t1 < t2 < ... making (A.1.2) ρ∈ true are not unique, since some terms (0, ..., 0) · χ{t  } (t) may always be added. ˙ n the left limit functions, the right limit funcTheorem 278. For any x ∈ S˙ (n) and any ρ ∈  tions and the initial values exist. We have ∀t ∈ R, x(t + 0) = x(t), ρ(t − 0) = ρ(t + 0) = (0, ..., 0),

(A.1.3)

ρ(−∞ + 0) = (0, ..., 0) and if x satisfies (A.1.1), then x(−∞ + 0) = μ,

(A.1.4)

x(t − 0) = μ · χ(−∞,t0 ] (t) ⊕ x(t0 ) · χ(t0 ,t1 ] (t) ⊕ x(t1 ) · χ(t1 ,t2 ] (t) ⊕ ...

(A.1.5)

Proof. (A.1.5): We suppose that x satisfies (A.1.1) and let t ∈ R arbitrary, fixed. Case t ≤ t0 For any ε > 0 and any ξ ∈ (t − ε, t), we have t − ε < ξ < t ≤ t0 , thus x(ξ ) = μ = x(t − 0). Case ∃k ∈ N, t ∈ (tk , tk+1 ] For any ε ∈ (0, t − tk ) and any ξ ∈ (t − ε, t), we infer tk < t − ε < ξ < t ≤ tk+1 , thus x(ξ ) = x(tk ) = x(t − 0).

A.2 Systems, several perspectives ˙ n × Bn × R → Bn is defined in Definition 163. Let the function  : Bn → Bn . The flow φ˙ :  n ˙ n , μ ∈ B and t ∈ R, if ρ fulfills (A.1.2), then the following way: for any ρ ∈  φ˙ ρ (μ, t) = φ α (μ, 0) · χ(−∞,t0 ) (t) ⊕ φ α (μ, 1) · χ[t0 ,t1 ) (t) ⊕ φ α (μ, 2) · χ[t1 ,t2 ) (t) ⊕ ... ˙ The signal x ∈ S˙ (n) given by ∀t ∈ R,  is called the generator function of φ. x(t) = φ˙ ρ (μ, t) is called state (function). Theorem 279. In Definition 163, the flow φ˙ does not depend on the choice of α ∈ n and t0 < t1 < t2 < ... that make (A.1.2) true.

Appendix A • Continuous time

389

Proof. We sketch how the proof is made and for this we fix α ∈ n and t0 < t1 < t2 < ... unbounded, arbitrary both, such that (A.1.2) holds. We define the sequence k0 , k1 , k2 , ... ∈ N by: k0 = min{k|k ∈ N, α k = (0, ..., 0)}, ∀p ∈ N, kp+1 = min{k|k > kp , α k = (0, ..., 0)}. At this moment we define β 0 , β 1 , β 2 , ... ∈ Bn , t0 , t1 , t2 , ... ∈ R like this: ∀p ∈ N, β p = α kp , tp = tkp . Then β ∈ n , t0 < t1 < t2 < ... is unbounded strictly increasing, the function ρ  : R → Bn given by ∀t ∈ R, ρ  (t) = β 0 · χ{t0 } (t) ⊕ β 1 · χ{t1 } (t) ⊕ β 2 · χ{t2 } (t) ⊕ ... ˙ n and we have, in addition, that ρ  = ρ. We must prove that ∀μ ∈ Bn , ∀t ∈ R, fulfills ρ  ∈   φ˙ ρ (μ, t) = φ˙ ρ (μ, t).

The theorem is proved because β and t0 , t1 , t2 , ... do not depend on the choice of α and t0 , t1 , t2 , ... that make (A.1.2) true. Theorem 280. Let the equations1 ∀t ∈ R, x(t) = φ˙ ρ (μ, t),

(A.2.1)

y(t) = ρ(t) (y(t − 0)),

(A.2.2)

⎧ ⎨ z1 (t) = 1 (z(t − 0))ρ1 (t) ∪ z1 (t − 0)ρ1 (t), ... ⎩ zn (t) = n (z(t − 0))ρn (t) ∪ zn (t − 0)ρn (t).

(A.2.3)

˙ n and μ ∈ Bn , if y(−∞ + 0) = μ and z1 (−∞ + 0) = μ1 , ..., zn (−∞ + 0) = μn , then For any ρ ∈  x = y = z. Proof. Let the unbounded strictly increasing sequence of real numbers t0 < t1 < t2 < ... and α ∈ n . We suppose that ρ satisfies (A.1.2), thus the state function x from (A.2.1) is: x(t) = φ α (μ, 0) · χ(−∞,t0 ) (t) ⊕ φ α (μ, 1) · χ[t0 ,t1 ) (t) ⊕ φ α (μ, 2) · χ[t1 ,t2 ) (t) ⊕ ... 1

This is to be compared with Theorem 65, page 67.

390

Boolean Systems

Since y is a signal, it is of the form (A.1.1), i.e. μ ∈ Bn and the unbounded strictly increasing sequence t0 < t1 < t2 < ... exist such that y(t) = μ · χ(−∞,t0 ) (t) ⊕ y(t0 ) · χ[t0 ,t1 ) (t) ⊕ y(t1 ) · χ[t1 ,t2 ) (t) ⊕ ...

(A.2.4)

We prove that y(−∞ + 0) = μ and (A.2.2) are fulfilled if we take in (A.2.4) μ = μ and tk = tk , k ∈ N. This means that y(t) = μ · χ(−∞,t0 ) (t) ⊕ y(t0 ) · χ[t0 ,t1 ) (t) ⊕ y(t1 ) · χ[t1 ,t2 ) (t) ⊕ ... (A.1.5)

y(t − 0) = μ · χ(−∞,t0 ] (t) ⊕ y(t0 ) · χ(t0 ,t1 ] (t) ⊕ y(t1 ) · χ(t1 ,t2 ] (t) ⊕ ...

(A.2.5) (A.2.6)

), ... ∈ Bn

are to be determined. and y(t0 ), y(t1 ), y(t2 The initial value of y from (A.2.5) is μ, indeed, see Theorem 278. We take now t ∈ R arbitrary. Case t ∈ (−∞, t0 ) In this case we have ρ(t) = (0, ..., 0), (A.2.5)

(A.2.7)

(A.2.7) (A.2.6)

y(t) = μ = (0,...,0) (μ) = ρ(t) (μ) = ρ(t) (y(t − 0)). Case ∃k ∈ N, t ∈ (tk , tk+1 ) We get ρ(t) = (0, ..., 0), (A.2.5)

(A.2.8)

(A.2.8)

(A.2.6)

y(t) = y(tk ) = (0,...,0) (y(tk )) = ρ(t) (y(tk )) = ρ(t) (y(t − 0)). Case t = t0 We have ρ(t) = α 0 , (A.2.2)

(A.2.9)

0

(A.2.9) (A.2.6)

0

y(t) = ρ(t) (y(t − 0)) = α (y(t − 0)) = α (μ) = φ α (μ, 1). We prove by induction on k that ∀k ∈ N, y(tk ) = φ α (μ, k + 1), and we conclude that y(t) = φ α (μ, 0) · χ(−∞,t0 ) (t) ⊕ φ α (μ, 1) · χ[t0 ,t1 ) (t) ⊕ φ α (μ, 2) · χ[t1 ,t2 ) (t) ⊕ ... = x(t). The equality y = z is obvious. Definition 164. Any of the equivalent equations (A.2.2) and (A.2.3) is called equation of evolution (of ) and is denoted by Eq˙  .

Appendix A • Continuous time

391

Remark 288. Remark 67, page 71 adapted to the real time continues to be true, and an autonomous system is defined as any of , G , φ˙ or Eq˙  . Remark 289. An apparent asymmetry exists here, since we have suggested an anchored discrete time theory, with 0 the initial time, and a floating real time theory, with some t0 the initial time.

B Theory of Cheng 

   1 0 , }, and Boolean 0 1 functions with logical matrices. Interesting and very important algebraical opportunities result, which can be used in systems theory. Our purpose in this appendix is to give a hint on the application of the theory of Cheng to asynchronicity. The theory of Daizhan Cheng [9] replaces B = {0, 1} with D = {

B.1 Semi-tensor product Notation 51. We use the notation Mm×n for the set of the matrices with binary entries that have m rows and n columns. Remark 290. In the following Definitions 165 and 166, the operations with matrices are induced by the field structure of B relative to ⊕, ·. Definition 165. The Kronecker product ⊗ of the matrices A ∈ Mm×n and B ∈ Mp×q is ⎛ A⊗B =⎝

a11 B

... ... ...

am1 B

a1n B

⎞ ⎠ ∈ Mmp×nq .

amn B

Definition 166. The semi-tensor product  of A ∈ Mm×n and B ∈ Mp×q is by definition A  B = (A ⊗ I nc )(B ⊗ I pc ) ∈ M mc × qc , n

p

where Ik is the k × k identity matrix and c is the least common multiple of n and p. Remark 291. At Definition 166, A ⊗ I nc has n nc columns and B ⊗ I pc has p pc rows, thus the product of the matrices A ⊗ I nc , B ⊗ I pc makes sense. Remark 292. If n = p, the semi-tensor product coincides with the usual product of the matrices. This happens because we get c = n = p, A ⊗ I1 = A, and B ⊗ I1 = B. Example 182. We have the following examples of Kronecker product 

1 0

 ⊗



1 1



 =

 1 1  0 1

   1 1 1 = , 0 0 1 393

394

Boolean Systems

and semi-tensor product 

1 0





  1 1 1  = 0 0 0 ⎛ ⎞ 1 0  ⎜ 0 1 ⎟ 1 1 ⎜ ⎟ =⎝ 0 0 ⎠ 0 0 0 0

 



1 0

⊗ I2

1 0



⎛

1 ⎜ 0 =⎜ ⎝ 0 0

1 1 1 0 0 0 ⎞ 1 1 0 0 ⎟ ⎟. 0 0 ⎠ 0



 ⊗ I1

0

Remark 293. The semi-tensor product is associative, and for this reason we shall omit writing brackets when it is used repeatedly.

B.2 Replacement of B with D Notation 52. We denote with δni ∈ Mn×1 the columns of the identity matrix of dimension n: ⎛ δni

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

0 ... 1 ... 0

⎞ ⎟ ⎟ ⎟ ⎟ − i, ⎟ ⎠

where n ≥ 1 and i ∈ {1, ..., n}. Notation 53. We use also the notations D = {δ21 , δ22 }, n

D(n) = {δ21n , ..., δ22n }. Remark 294. D and D(n) do not have a name and an algebraical structure1 of their own, but they will act as B and Bn in the following. Obviously, card(B) = card(D) = 2 and card(Bn ) = card(D(n) ) = 2n . Notation 54. We use the notations ζ : B → D, ζn : Bn → D(n) for the following functions: ∀μ ∈ B, ∀λ ∈ Bn ,   μ , ζ (μ) = μ 1

We can define algebraical structures on D and D(n) , but it is not our purpose to do so.

Appendix B • Theory of Cheng 395 ⎛ ⎜ ⎜ ⎜ ζn (λ) = ⎜ ⎜ ⎝

λ1 ...λn−1 λn λ1 ...λn−1 λn λ1 ...λn−1 λn ... λ1 ...λn−1 λn

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

We denote in general μ = ζ (μ) and λ = ζn (λ). Remark 295. We notice that for any μ ∈ B, respectively λ ∈ Bn , exactly one of μ, μ is 1, respectively exactly one of λ1 ...λn−1 λn , λ1 ...λn−1 λn , λ1 ...λn−1 λn , ..., λ1 ...λn−1 λn is 1, meaning that μ ∈ D, respectively that λ ∈ D(n) indeed. Theorem 281. (a) ζ and ζn are bijections; (b) ∀λ ∈ Bn , λ = λ1  ...  λn . Proof. (a) When λ ∈ Bn takes the distinct 2n values (1, ..., 1, 1), (1, ..., 1, 0), (1, ..., 0, 1), ..., n (0, ..., 0, 0), λ takes the distinct 2n values δ21n , δ22n , δ23n , ..., δ22n . (b) For n = 2 and arbitrary λ ∈ B2 , we obtain  λ1  λ2 =

      λ1 λ2 λ1 1  =( ⊗ λ1 λ2 λ1 0 ⎛ ⎞ ⎛ λ1 λ2 λ1 0   ⎜ λ1 λ2 ⎜ 0 λ1 ⎟ λ2 ⎜ ⎟ =⎜ ⎝ λ1 0 ⎠ λ2 = ⎝ λ1 λ2 λ1 λ2 0 λ1

0 1 ⎞

   λ2 ⊗ 1) )( λ2

⎟ ⎟ = λ. ⎠

The property is supposed to be true for n and the proof is made for n + 1.

B.3 Structure matrix Notation 55. The notation of the i − th column of an arbitrary binary matrix A is coli (A). Definition 167. A matrix A with n rows and m columns is called logical if ∀j ∈ {1, ..., m}, colj (A) ∈ {δn1 , ..., δnn }. The set of the logical matrices with n rows and m columns is denoted with Ln×m . : Bn × Bn → Bn is defined by ∀μ ∈ Definition 168. Given  : Bn → Bn , the function  n n B , ∀λ ∈ B , (μ, λ) = λ (μ). 

(B.3.1)

396

Boolean Systems

: Bn × Bn → Bn . We denote with Mf ∈ Definition 169. Let f : Bn → B,  : Bn → Bn and  L2×2n the matrix   f (1, ..., 1, 1), f (1, ..., 1, 0), f (1, ..., 0, 1), ... f (0, ..., 0, 0) , Mf = f (1, ..., 1, 1), f (1, ..., 1, 0), f (1, ..., 0, 1), ... f (0, ..., 0, 0) with M ∈ L2n ×2n the matrix whose columns are ⎛ 1 (1, ..., 1, 1)...n−1 (1, ..., 1, 1)n (1, ..., 1, 1) ⎜  (1, ..., 1, 1)... (1, ..., 1, 1) (1, ..., 1, 1) ⎜ 1 n−1 n ⎜ col1 (M ) = ⎜ 1 (1, ..., 1, 1)...n−1 (1, ..., 1, 1)n (1, ..., 1, 1) ⎜ ⎝ ... 1 (1, ..., 1, 1)...n−1 (1, ..., 1, 1) n (1, ..., 1, 1) ⎛ ⎜ ⎜ ⎜ col2 (M ) = ⎜ ⎜ ⎝

1 (1, ..., 1, 0)...n−1 (1, ..., 1, 0)n (1, ..., 1, 0) 1 (1, ..., 1, 0)...n−1 (1, ..., 1, 0)n (1, ..., 1, 0) 1 (1, ..., 1, 0)...n−1 (1, ..., 1, 0)n (1, ..., 1, 0) ... 1 (1, ..., 1, 0)...n−1 (1, ..., 1, 0) n (1, ..., 1, 0)

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

... ⎛ ⎜ ⎜ ⎜ col2n (M ) = ⎜ ⎜ ⎝

1 (0, ..., 0, 0)...n−1 (0, ..., 0, 0)n (0, ..., 0, 0) 1 (0, ..., 0, 0)...n−1 (0, ..., 0, 0)n (0, ..., 0, 0) 1 (0, ..., 0, 0)...n−1 (0, ..., 0, 0)n (0, ..., 0, 0) ... 1 (0, ..., 0, 0)...n−1 (0, ..., 0, 0) n (0, ..., 0, 0)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and with M ∈ L2n ×22n the matrix ⎛

(1,...,1,1)

1

(1,...,1,1)

(1, ..., 1)...n−1

(1,...,1,1)

(1, ..., 1)n

(1, ..., 1)

⎜ (1,...,1,1) (1,...,1,1) ⎜ 1 (1, ..., 1)...n−1 (1, ..., 1)n(1,...,1,1) (1, ..., 1) ⎜ ⎜ (1,...,1,1) (1, ..., 1)...(1,...,1,1) (1, ..., 1)(1,...,1,1) (1, ..., 1) col1 (M ) = ⎜  n n−1 ⎜ 1 ... ⎝ (1,...,1,1)

1 ⎛

(1,...,1,1)

(1, ..., 1)...n−1

(1,...,1,0)

1

(1,...,1,1)

(1, ..., 1) n

(1,...,1,0)

(1, ..., 1)...n−1

(1,...,1,0)

(1, ..., 1)n

(1,...,1,0)

(1,...,1,0)

(1, ..., 1)...n−1

(1, ..., 1)

...

(1,...,1,0)

(1, ..., 1) n

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(1, ..., 1)

⎜ (1,...,1,0) (1,...,1,0) (1,...,1,0) ⎜ 1 (1, ..., 1)...n−1 (1, ..., 1)n (1, ..., 1) ⎜ (1,...,1,0) (1,...,1,0) (1,...,1,0) ⎜ col2 (M ) = ⎜  (1, ..., 1)...n−1 (1, ..., 1)n (1, ..., 1) ⎜ 1 ... ⎝ 1

⎞

(1, ..., 1)

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

Appendix B • Theory of Cheng 397 ⎛

(0,...,0,0)

1

(0,...,0,0)

(1, ..., 1)...n−1

(0,...,0,0)

(1, ..., 1)n

(1, ..., 1)

⎜ (0,...,0,0) (0,...,0,0) (0,...,0,0) ⎜ 1 (1, ..., 1)...n−1 (1, ..., 1)n (1, ..., 1) ⎜ (0,...,0,0) (0,...,0,0) (0,...,0,0) ⎜ n col2 (M ) = ⎜  (1, ..., 1)...n−1 (1, ..., 1)n (1, ..., 1) ⎜ 1 ... ⎝ (0,...,0,0)

1

(0,...,0,0)

(1, ..., 1)...n−1

(0,...,0,0)

(1, ..., 1) n

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(1, ..., 1)

... ⎛

(0,...,0,0)

1

(0,...,0,0)

(0, ..., 0)...n−1

(0,...,0,0)

(0, ..., 0)n

(0, ..., 0)

⎜ (0,...,0,0) (0,...,0,0) (0,...,0,0) ⎜ 1 (0, ..., 0)...n−1 (0, ..., 0)n (0, ..., 0) ⎜ (0,...,0,0) (0,...,0,0) (0,...,0,0) ⎜ col22n (M ) = ⎜  (0, ..., 0)...n−1 (0, ..., 0)n (0, ..., 0) ⎜ 1 ... ⎝ (0,...,0,0)

1

(0,...,0,0)

(0, ..., 0)...n−1

(0,...,0,0)

(0, ..., 0) n

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(0, ..., 0)

. Mf , M  , M  are called the structure matrices of f, ,  : Bn × Bn → Bn like previously. Theorem 282. We consider f : Bn → B,  : Bn → Bn and  The assignments μ → Mf  μ, μ → M  μ, (μ, λ) → M  μ  λ, ) : D(n) × with μ ∈ Bn , λ ∈ Bn , define the functions M(f ) : D(n) → D, M() : D(n) → D(n) , M( (n) (n) (n) (n) D → D in the following way: ∀μ ∈ D , ∀λ ∈ D , M(f )(μ) = Mf  μ,

(B.3.2)

M()(μ) = M  μ,

(B.3.3)

)(μ, λ) = M M(  μ  λ.

(B.3.4)

Mf  μ = Mf · μ,

(B.3.5)

M  μ = M · μ,

(B.3.6)

M  μ  λ = M · (μ  λ),

(B.3.7)

We have

where ‘·’ is the product of the matrices.

398

Boolean Systems

Proof. We note first that D = L2×1 , D(n) = L2n ×1 are true. As far as μ ∈ L2n ×1 and Mf ∈ L2×2n , we infer from Remark 292, page 393 that (B.3.5) holds. On the other hand, μ ∈ L2n ×1 makes Mf · μ coincide with one of col1 (Mf ), ..., col2n (Mf ) and we know that col1 (Mf ), ..., col2n (Mf ) ∈ L2×1 , thus we can define M(f ) as D(n) μ → M(f )(μ) = Mf  μ ∈ D. The other statements are proved similarly, since col1 (M ), ..., col2n (M ) ∈ L2n ×1 and col1 (M ), ..., col22n (M ) ∈ L22n ×1 . Theorem 283. (a) ∀μ ∈ Bn , ∀λ ∈ Bn , we have f (μ) = Mf · μ,

(B.3.8)

(μ) = M · μ,

(B.3.9)

(μ, λ) = M  · (μ  λ).

(B.3.10)

(b) The following diagrams f

Bn ζn

?

D(n)

ζ M(f ) - ? D 

Bn ζn

?

D(n)

- Bn ζn ?

M()- (n) D

Bn × Bn ζn × ζn

- B



- Bn

ζn ? ? ) M( - D(n) D(n) × D(n)

commute.

Appendix B • Theory of Cheng 399

−→ (c) The assignments Fn,1 f −→ Mf ∈ L2×2n , Fn,n  −→ M ∈ L2n ×2n , F2n,n  ∈ L are bijective. M n 2n 2 ×2 Proof. We fix μ ∈ Bn and λ ∈ Bn arbitrary. (a) In order to prove (B.3.8), we use the fact that f (μ) = f (1, ..., 1, 1)μ1 ...μn−1 μn ⊕ f (1, ..., 1, 0)μ1 ...μn−1 μn ⊕f (1, ..., 0, 1)μ1 ...μn−1 μn ⊕ ... ⊕ f (0, ..., 0, 0)μ1 ...μn−1 μn , f (μ) = f (1, ..., 1, 1)μ1 ...μn−1 μn ⊕ f (1, ..., 1, 0)μ1 ...μn−1 μn ⊕f (1, ..., 0, 1)μ1 ...μn−1 μn ⊕ ... ⊕ f (0, ..., 0, 0)μ1 ...μn−1 μn , wherefrom

 f (μ) =

f (μ) f (μ)

 = Mf · μ.

(B.3.11)

And in (B.3.10) we have ⎛ ⎜ ⎜ (μ, λ) = ⎜  ⎜ ⎜ ⎝

λ1 (μ)...λn−1 (μ)λn (μ) λ1 (μ)...λn−1 (μ)λn (μ) λ1 (μ)...λn−1 (μ)λn (μ) ... λ1 (μ)...λn−1 (μ) λn (μ)

⎛

μ1 ...μn λ1 ...λn−1 λn μ1 ...μn λ1 ...λn−1 λn μ1 ...μn λ1 ...λn−1 λn ... μ1 ...μn λ1 ...λn−1 λn

⎜ ⎜ ⎜ μλ=⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

For example, the second row in (B.3.10) is proved like this: λ1 (μ)...λn−1 (μ)λn (μ) (1,...,1,1)

= 1

(1,...,1,1)

(1, ..., 1)...n−1

(1,...,1,1)

(1, ..., 1)n

(1, ..., 1)μ1 ...μn λ1 ...λn−1 λn

(1,...,1,0) ⊕1(1,...,1,0) (1, ..., 1)...n−1 (1, ..., 1)n(1,...,1,0) (1, ..., 1)μ1 ...μn λ1 ...λn−1 λn

... (0,...,0,0)

⊕1

(0,...,0,0)

(1, ..., 1)...n−1

(0,...,0,0)

(1, ..., 1)n

(1, ..., 1)μ1 ...μn λ1 ...λn−1 λn

... (0,...,0,0) ⊕1(0,...,0,0) (0, ..., 0)...n−1 (0, ..., 0)n(0,...,0,0) (0, ..., 0)μ1 ...μn λ1 ...λn−1 λn .

400

Boolean Systems

(b) The commutativity of the first diagram results from (B.3.2)

(M(f ) ◦ ζn )(μ) = M(f )(ζn (μ)) = M(f )(μ) = Mf  μ (B.3.5)

(B.3.8)

= Mf · μ = f (μ) = ζ (f (μ)) = (ζ ◦ f )(μ),

the situation for the second diagram is similar, and for the third diagram also )(ζn (μ), ζn (λ)) = M( )(μ, λ) (B.3.4) ) ◦ (ζn × ζn ))(μ, λ) = M( = M (M( μλ (B.3.7)

= M · (μ  λ)

(B.3.10)

=

(μ, λ) = ζn ( (μ, λ)) = (ζn ◦  )(μ, λ). 

(c) For example we suppose against all reason that f, f  ∈ Fn,1 exist, f = f  , with the property that Mf = Mf  . The hypothesis states the existence of μ ∈ Bn such that f (μ) = f  (μ) thus, from Theorem 281, page 395, f (μ) = f  (μ). We have: f (μ) = ζ (f (μ)) = (ζ ◦ f )(μ) = (M(f ) ◦ ζn )(μ) = M(f )(ζn (μ)) = M(f )(μ) = Mf  μ = Mf   μ = M(f  )(μ) = M(f  )(ζn (μ)) = (M(f  ) ◦ ζn )(μ) = (ζ ◦ f  )(μ) = ζ (f  (μ)) = f  (μ), contradiction, showing that the assignment Fn,1 f −→ Mf ∈ L2×2n is injective. Due to the n fact that card(Fn,1 ) = card(L2×2n ) = 22 , injectivity and bijectivity coincide.

B.4 Equations of evolution Remark 296. Daizhan Cheng’s theory adapted to asynchronicity replaces the equations of evolution k

x(k + 1) = α (x(k)), where  : Bn → Bn and α ∈ n , k ∈ N, with the equations k k (x(k), α k ) (B.3.10) x(k + 1) = α (x(k)) =  = M · (x(k)  α ),

(B.4.1)

which are easier to be studied. The price to pay is the increase of the dimension of these systems from n to 2n . Theorem 284. For any α ∈ n , μ ∈ Bn and k ∈ N,

φ α (μ, k) =

⎧ ⎪ ⎪ ⎪ ⎨

μ, if k = 0, 0 M · (μ  α ), if k = 1,

0 k−1 ), if k > 1. ⎪ M  ...  M ⎪ · (μ  α  ...  α ⎪ ⎩     k

Appendix B • Theory of Cheng 401

Proof. We fix α ∈ n , μ ∈ Bn arbitrary. For k = 0, φ α (μ, 0) = μ and k = 1 0 0 (μ, α 0 ) (B.3.10) = M φ α (μ, 1) = α (μ) =  · (μ  α )

things are obvious. We suppose that the result is true for k, and for k + 1 we infer: (φ α (μ, k), α k ) φ α (μ, k + 1) = α (φ α (μ, k)) =  k

(B.3.10)

=

α k M · (φ (μ, k)  α )

0 k−1 ))  α k = M  (M  ...  M  (μ  α  ...  α    k k 0 k = M  ...  M  (μ  α  ...  α ) = M  ...  M · (μ  α  ...  α ).       0

k+1

k+1

B.5 Example Example 183. We consider the system  : B2 → B2 , ∀μ ∈ B2 , (μ1 , μ2 ) = (μ2 , μ1 ), (0, 0) 6 (1, 0) 

- (0, 1)

? (1, 1)

which is described by the equations ∀k ∈ N,  x1 (k + 1) = x2 (k)α1k ∪ x1 (k)α1k , x2 (k + 1) = x1 (k)α2k ∪ x2 (k)α2k ,

(B.5.1)

where x ∈ S (2) fulfills x(0) = (0, 0) and α ∈ 2 is defined as α = (1, 0), (0, 1), (1, 1), (0, 1), (1, 0), ... We get 0

x(1) = α (x(0)) = (1,0) (0, 0) = (0, 0), 1

x(2) = α (x(1)) = (0,1) (0, 0) = (0, 1), 2

x(3) = α (x(2)) = (1,1) (0, 1) = (1, 1),

(B.5.2) (B.5.3) (B.5.4)

402

Boolean Systems

3

x(4) = α (x(3)) = (0,1) (1, 1) = (1, 0), 4

x(5) = α (x(4)) = (1,0) (1, 0) = (0, 0),

(B.5.5) (B.5.6)

... Function (μ1 , μ2 , λ1 , λ2 ) = (λ1 μ1 ∪ λ1 μ2 , λ2 μ2 ∪ λ2 μ1 )  defines the matrix ⎛  2 (1, 1, 1, 1) 1 (1, 1, 1, 1) ⎜  2 (1, 1, 1, 1) ⎜ 1 (1, 1, 1, 1) M =⎜ 1 (1, 1, 1, 1) 2 (1, 1, 1, 1) ⎝   2 (1, 1, 1, 1) 1 (1, 1, 1, 1)  ⎛ 0 1 0 1 0 0 0 0 ⎜ 1 0 1 0 0 0 1 1 =⎜ ⎝ 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

...

1 (0, 0, 0, 0) 2 (0, 0, 0, 0) ⎞  2 (0, 0, 0, 0) ⎟ 1 (0, 0, 0, 0)  ⎟ ⎟ 1 (0, 0, 0, 0) 2 (0, 0, 0, 0) ⎠  1 (0, 0, 0, 0)  2 (0, 0, 0, 0)  ⎞ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎟. 0 1 1 1 0 1 0 ⎠

1 0 0 0 0 0 0 0 1 0 1

Eq. (B.5.1) implies that ∀k ∈ N, (B.4.1) is true, with x, α : N → D(2) . We can see that, via (B.4.1), Eq. (B.5.2) becomes 14 , x(0)  α 0 = (0, 0)  (1, 0) = δ44  δ42 = δ16 14 4 x(1) = M · δ16 = δ4 = (0, 0),

while (B.5.3) becomes 15 , x(1)  α 1 = (0, 0)  (0, 1) = δ44  δ43 = δ16 15 3 x(2) = M · δ16 = δ4 = (0, 1),

(B.5.4) becomes 9 , x(2)  α 2 = (0, 1)  (1, 1) = δ43  δ41 = δ16 9 1 x(3) = M · δ16 = δ4 = (1, 1),

(B.5.5) becomes 3 , x(3)  α 3 = (1, 1)  (0, 1) = δ41  δ43 = δ16 3 2 x(4) = M · δ16 = δ4 = (1, 0),

Appendix B • Theory of Cheng 403

and (B.5.6) becomes 6 x(4)  α 4 = (1, 0)  (1, 0) = δ42  δ42 = δ16 , 6 4 x(5) = M · δ16 = δ4 = (0, 0),

...

C Symbolic dynamics Professor Boris Hasselblatt suggested the construction of a bridge between the asynchronous systems theory which is presented in our book and the symbolic dynamics theory from [31]. Following his idea, we introduce in this appendix the shift spaces, representing the fundamental concept that Douglas Lind and Brian Marcus operate with. We then present the timeless model, the unbounded delay model and the bounded delay model of computation of the Boolean functions  : Bn → Bn as spaces of states X ⊂ S (n) . And the main results are: the timeless model and the bounded delay model are shift spaces, while the unbounded delay model, i.e. the main concern of this book, is not. Does this conclusion have any consequences?

C.1 Blocks Definition 170. A block (or word) over Bn is an l-tuple u = (μ1 , μ2 , ..., μl ) ∈ (Bn )l , where l ≥ 1 is the length of the block. u will also be called an l-block. We identify for any μ ∈ Bn the 1-block (μ) with the point μ. Remark 297. We give an interpretation of the identification of (μ) with μ. The Hausdorff’s definition of (μ, ν, ..., λ) ∈ (Bn )l is (μ, ν, ..., λ) = {{μ, 1}, {ν, 2}, ..., {λ, l}}, and we agree that Bn ∧ {1, 2, ..., l} = ∅. Then {{μ, 1}} is identified with μ. Definition 171. The empty block ε over Bn is given. It has by definition the length 0. Definition 172. A subblock of u = (μ1 , μ2 , ..., μl ) is a block v = (μi , μi+1 , ..., μj ), with 1 ≤ i ≤ j ≤ l. We use the notation v  u. Definition 173. The empty block ε is by definition a subblock of any other block. Remark 298. The nonempty blocks represent the values in succession of the signals x ∈ S (n) , i.e. u = (x(i), x(i + 1), ..., x(j )), where 0 ≤ i ≤ j . The empty block is identified with any of (x(i), x(i + 1), ..., x(j )), where 0 ≤ j < i, and (μ1 , μ2 , ..., μl ), where l = 0. Remark 299. The set of the l-blocks over Bn , l ≥ 1 has no special notation, other than {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn } or (Bn )l . 405

406

Boolean Systems

Definition 174. The concatenation of the nonempty blocks u = (μ1 , μ2 , ..., μk ) and v = (λ1 , λ2 , ..., λl ), in this order, is the k + l-block defined as (u, v) = (μ1 , μ2 , ..., μk , λ1 , λ2 , ..., λl ). For any block u, we define (u, ε) = (ε, u) = u. Remark 300. The concatenation is associative, thus if we concatenate more than two blocks we can omit the order of the operations, for example (u, v, w) denotes any of ((u, v), w) and (u, (v, w)). The blocks over Bn are a unitary semigroup (i.e. a monoid) relative to the concatenation, where the unit is ε.

C.2 Shift spaces Definition 175. Let F ⊂

 {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn } a set of nonempty blocks over l≥1

Bn (F can be empty, finite nonempty, or infinite). The set / F} X = {x|x ∈ S (n) , ∀i ∈ N, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ (n)

is called Bn -shift space, shift space over Bn , or shortly shift space, and we denote X = SF . F is the set of forbidden blocks of X. Definition 176. If X, Y are shift spaces and X ⊂ Y , then X is called a subshift of Y . Example 184. The set S (n) is a shift space, with the set of forbidden blocks F = ∅, i.e. (n) S (n) = S∅ . S (n) is called the full Bn -shift space. Example 185. The empty set ∅ is a shift space also, for which F = Bn : (n)

/ Bn } = ∅. SBn = {x|x ∈ S (n) , ∀i ∈ N, x(i) ∈ This shift space has no 1-blocks since all are forbidden, see Definition 170 and Remark 297. It is called the empty Bn -shift space. Remark 301. Subsets X ⊂ S (n) exist that are not shift spaces. In this respect, we define the functions ε i ∈ S, i ∈ N, by ∀k ∈ N,  ε i (k) =

1, if k = i, 0, if k = i,

order to show that X ⊂ S is not a shift space, let us suppose and the set X = {ε i |i ∈ N}. In against all reason that F ⊂ {(μ1 , ..., μl )|μ1 ∈ B, ..., μl ∈ B} exists such that X = SF . The l≥1

Appendix C • Symbolic dynamics

407

fact that x = (0, 0, 0, ...) ∈ S is not an element of X means that 







∃i  ∈ N, ∃j  ≥ i  , (x(i  ), ..., x(j  )) = (0, ..., 0) ∈ F ∧ Bj −i +1 , while each ε i satisfies (ε i (i + 1), ..., ε i (i + j  − i  + 1)) = (0, ..., 0) ∈ F ∧ Bj −i +1 , / X, contradiction. thus ε i ∈ (n)

Theorem 285. If X = SF is a shift space, then ∀x ∈ X, σ 1 (x) ∈ X holds. Proof. Let x ∈ X arbitrary and we take i ∈ N, j ≥ i arbitrary themselves. We know from the / hypothesis that (x(i + 1), x(i + 2), ..., x(j + 1)) ∈ / F, thus (σ 1 (x)(i), σ 1 (x)(i + 1), ..., σ 1 (x)(j )) ∈ 1 (n) 1 F. As σ (x) ∈ S , we have that σ (x) ∈ X. Remark 302. The property expressed by Theorem 285 is the one that gives the name of the shift spaces. However if X ⊂ S (n) fulfills ∀x ∈ X, σ 1 (x) ∈ X, then it is possible that X is not a shift space, and Definition 181 to follow gives an important counterexample in this respect. (n)

(n)

Remark 303. We may have F1 = F2 and SF1 = SF2 . An example for this is given by F1 = {(μ, μ )|μ ∈ Bn , μ ∈ Bn , μ = μ }, F2 = {(μ, μ , μ )|μ ∈ Bn , μ ∈ Bn , μ ∈ Bn , μ = μ or μ = μ }, when F1 ∧ F2 = ∅ and (n) (n) = SF = {x|x ∈ S (n) , ∃μ ∈ Bn , ∀k ∈ N, x(k) = μ}. SF 1 2

Theorem 286. Let Fl , l ≥ 1 sets of blocks over Bn . We have (n)

(n)

(n)

(n)

F1 ⊂ F2 =⇒ SF1 ⊃ SF2 , (n)

(n)

(n)

(C.2.1) (n)

SF1 ∨F2 = SF1 ∧ SF2 ⊂ SF1 ∨ SF2 ⊂ SF1 ∧F2 , (n)

S ∨ Fl = l≥1

 (n) SFl . l≥1

(n)

Proof. (C.2.1). We take an arbitrary x ∈ SF2 , thus ∀i ∈ N, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / F2 . We have in particular that

(C.2.2) (C.2.3)

408

Boolean Systems

∀i ∈ N, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / F1 , (n)

i.e. x ∈ SF1 .

(n)

(n)

(C.2.2). The inclusion F1 ∨ F2 ⊃ F1 implies SF1 ∨F2 ⊂ SF1 , and similarly F1 ∨ F2 ⊃ F2

(n) (n) (n) (n) (n) implies SF ⊂ SF , wherefrom SF ⊂ SF ∧ SF . The inclusion F1 ∧ F2 ⊂ F1 shows 1 ∨F2 2 1 ∨F2 1 2 (n)

(n)

(n)

(n)

(n)

(n)

that SF1 ⊂ SF1 ∧F2 , and in a similar way F1 ∧ F2 ⊂ F2 implies SF2 ⊂ SF1 ∧F2 , thus SF1 ∨ SF2 ⊂ (n)

SF1 ∧F2 .

(n)

(n)

(n)

(n)

(n)

In order to prove the inclusion SF1 ∧ SF2 ⊂ SF1 ∨F2 we take an arbitrary x ∈ SF1 ∧ SF2 , meaning that x ∈ S (n) , and for arbitrary i ∈ N, j ≥ i, we have (x(i), ..., x(j )) ∈ / F1 and (x(i), ..., x(j )) ∈ / F2 . We infer non ((x(i), ..., x(j )) ∈ F1 or (x(i), ..., x(j )) ∈ F2 ), (n)

i.e. (x(i), ..., x(j )) ∈ / F1 ∨ F2 . It has resulted that x ∈ SF1 ∨F2 . (C.2.3): The inclusion  (n) (n) S ∨ Fl ⊂ SFl l≥1

(n) holds similarly with SF 1 ∨F2

(n)

⊂ SF1 ∧ SF2 , we prove  (n) (n) SFl ⊂ S ∨ Fl . l≥1

Let x ∈

l≥1

(n)

l≥1

 (n) SFl arbitrary, i.e. the following statements are true, in succession: l≥1

∀l ≥ 1, x ∈ S (n) and ∀i ∈ N, ∀j ≥ i, (x(i), ..., x(j )) ∈ / Fl , / Fl , x ∈ S (n) and ∀l ≥ 1, ∀i ∈ N, ∀j ≥ i, (x(i), ..., x(j )) ∈ x ∈ S (n) and ∀i ∈ N, ∀j ≥ i, ∀l ≥ 1, (x(i), ..., x(j )) ∈ / Fl , x ∈ S (n) and ∀i ∈ N, ∀j ≥ i, non ∃l ≥ 1, (x(i), ..., x(j )) ∈ Fl , x ∈ S (n) and ∀i ∈ N, ∀j ≥ i, (x(i), ..., x(j )) ∈ /

 Fl . l≥1

(n)

We infer that x ∈ S ∨ Fl . l≥1

(n)

(n)

(n)

Remark 304. The inclusion SF1 ∧F2 ⊂ SF1 ∨ SF2 is false in general. A counterexample for this was already met at Remark 303, where F1 ∧ F2 = ∅ and (n) (n) (n) (n) (n) SF = S∅ = S (n)  SF = SF ∨ SF . 1 ∧F2 1 1 2

Appendix C • Symbolic dynamics

409

Remark 305. We get from Theorem 286 that the intersection of shift spaces is a shift space, but it is possible that the union of shift spaces is not a shift space. Notation 56. We denote the complement of a set F of blocks over Bn relative to the set of all nonempty blocks over Bn by F c :  F c = {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn }  F. l≥1

C.3 Languages Definition 177. Let X ⊂ S (n) a nonempty subset. We denote with Bl (X), l ∈ N the set of the l-blocks that occur in the elements x ∈ X: B0 (X) = {ε}, and ∀l ≥ 1, Bl (X) = {(x(i), x(i + 1), ..., x(i + l − 1))|x ∈ X, i ∈ N}. The language of X is the set defined as B(X) =



Bl (X).

l∈N

Example 186. The full Bn -shift S (n) has the language:  B(S (n) ) = {ε} ∨ {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn } l≥1

= {ε, (0, ..., 0), (0, ..., 1), ..., (1, ..., 1), ((0, ..., 0), (0, ..., 0)), ((0, ..., 0), (0, ..., 1)), ...}. Theorem 287. Let F ⊂

 {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn } a set of nonempty blocks and the l≥1

(n)

shift space X = SF . Then B(X) ∧ F = ∅. Proof. We suppose against all reason that this is not true and let w ∈ B(X) ∧ F a nonempty word. A function x ∈ S (n) exists with the property ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / F,

(C.3.1)

∃i  ≥ 0, ∃j  ≥ i  , w = (x(i  ), x(i  + 1), ..., x(j  )),

(C.3.2)

so far x ∈ X, and

i.e. w ∈ B(X). The statements (C.3.1), (C.3.2) imply the fact that w ∈ / F, contradiction.

410

Boolean Systems

(n)

Theorem 288. Let X = SF a shift space. If X is nonempty, then B(X) has nonempty blocks and ∀w ∈ B(X), (a) every subblock w   w fulfills w  ∈ B(X), (b) v ∈ B(X) nonempty exists such that (w, v) ∈ B(X). Proof. We prove the first statement of the theorem and let x ∈ X arbitrary, for which we get ∀i ≥ 0, ∀l ≥ 1, (x(i), x(i + 1), ..., x(i + l − 1)) ∈ Bl (X). We infer from here that ∀l ≥ 1, Bl (X) = ∅, wherefrom B(X)  {ε}. We take now an arbitrary w ∈ B(X) and two possibilities exist. Case w = ε (a) is obvious, because ε is the only subblock of w, and ε ∈ B(X). (b) Nonempty blocks v ∈ B(X) exist. For any such v, we get (w, v) = v ∈ B(X). Case w = ε We have the existence of x ∈ X with the property that i ≥ 0, j ≥ i exist such that w = (x(i), x(i + 1), ..., x(j )) ∈ Bj −i+1 (X). (a) The case when w  = ε is the empty subblock of w is clear, so let us take i  , j  arbitrary such that i ≤ i  ≤ j  ≤ j . The block w  = (x(i  ), x(i  + 1), ..., x(j  )) ∈ Bj  −i  +1 (X) is an arbitrary nonempty subblock of w and w  ∈ B(X). (b) We take l ≥ j + 1 arbitrary. We obtain that v = (x(j + 1), x(j + 2), ..., x(l)) ∈ Bl−j (X) satisfies (w, v) = (x(i), x(i + 1), ..., x(j ), x(j + 1), ..., x(l)) ∈ Bl−i+1 (X), therefore (w, v) ∈ B(X). Theorem 289. Let L ⊂ {ε} ∨

 {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ Bn } a set of blocks that fulfills l≥1

ε ∈ L and ∀w ∈ L, (a) for any subblock w   w we have w  ∈ L, (b) v ∈ L nonempty exists such that (w, v) ∈ L. (n) Then X = SLc is a shift space with the property that B(X) = L. Proof. X is a shift space indeed, and we prove that B(X) ⊂ L.

Appendix C • Symbolic dynamics

411

Let w ∈ B(X) an arbitrary block. Case w = ε Obviously w ∈ L. Case w = ε Then x ∈ S (n) and i ≥ 0, j ≥ i exist with the property that w = (x(i), x(i + 1), ..., x(j )) and w∈ / Lc , in other words w ∈ (Lc )c = L. We prove L ⊂ B(X), and let us take an arbitrary w ∈ L. When w = ε, we obviously have ε ∈ B(X), thus we can suppose that w is nonempty. We get the existence of μ0 , μ1 , ..., μl−1 ∈ Bn such that w = (μ0 , μ1 , ..., μl−1 ). We apply (b) and we have the existence of μl , μl+1 , ..., μp−1 ∈ Bn such that ((μ0 , ..., μl−1 ), l (μ , ..., μp−1 )) = (μ0 , ..., μl−1 , μl , ..., μp−1 ) ∈ L. By a repeated use of (b), we obtain the existence of the sequence μ0 , μ1 , ... ∈ Bn that defines the function x ∈ S (n) in the following way: ∀i ∈ N, x(i) = μi . In addition, we infer from the previous construction that ∀k ∈ N, ∃k  ≥ k, (x(0), x(1), ..., x(k  )) ∈ L is true thus, by making use of (a), ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ L. This means that ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / Lc , i.e. x ∈ X. We have obtained that w = (x(0), ..., x(l − 1)) ∈ B(X). Theorem 290. For any shift space X ⊂ S (n) , we have (n)

X = SB(X)c . Proof. If X = ∅, then B(X) = {ε}, B(X)c contains all the nonempty blocks over Bn and (n) SB(X) c = ∅, therefore the statement of the theorem is true. We shall suppose in the rest of the proof that X = ∅. We prove that (n)

X ⊂ SB(X)c and we take an arbitrary x ∈ X. We have that ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ B(X), (n) thus ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / B(X)c , in other words x ∈ SB(X)c . We show that (n) SB(X) c ⊂ X.

412

Boolean Systems

The fact that X is a shift space implies the existence of F ⊂

 {(μ1 , ..., μl )|μ1 ∈ Bn , ..., μl ∈ l≥1

Bn }

(n)

(n) ∈ SB(X)c

for which X = SF . If the function x is arbitrary, then ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ B(X), i.e. since B(X) ∧ F = ∅, from Theorem 287, page 409 we infer ∀i ≥ (n) 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / F. The conclusion is that x ∈ SF .

Theorem 291. We suppose that X is a subset of S (n) . The following statements are equivalent: (i) X is a shift space, (ii) ∀x ∈ S (n) , ∀i ∈ N, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ B(X) implies x ∈ X. Proof. If X = ∅, then (i) is true and (ii) is also true with B(X) = {ε} (the false statement implies any statement), thus we shall suppose in the following that X = ∅. (n) (i)=⇒(ii) The hypothesis states that X is a shift space, therefore X = SB(X) c from Theo(n) rem 290. Let x ∈ S arbitrary with the property that ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ B(X), i.e. ∀i ≥ 0, ∀j ≥ i, (x(i), x(i + 1), ..., x(j )) ∈ / B(X)c holds. We infer that x ∈ X. (n) (ii)=⇒(i) We show that X = SB(X)c , and this implies that X is a shift space. Let us suppose against all reason that this is not true under the form (n)

SB(X)c  X = ∅, / B(X)c and meaning that x ∈ S (n) and i ≥ 0, j ≥ i exist such that (x(i), x(i + 1), ..., x(j )) ∈ x∈ / X. We have obtained a contradiction with (ii). And if (n)

X  SB(X)c = ∅, then x ∈ X and i ≥ 0, j ≥ i exist such that (x(i), x(i + 1), ..., x(j )) ∈ B(X)c , i.e. (x(i), x(i + 1), ..., x(j )) ∈ / B(X). The last statement represents a contradiction with the definition of B(X).

C.4 The timeless model of computation f

Definition 178. The full state portrait1 of  : Bn → Bn is the directed graph G = f f f f (V , E ), where the set of vertices (or points) V and the set of arrows (or edges) E are defined as f

V = Bn , E = {(μ, μ )|μ ∈ Bn , μ ∈ Bn , ∃λ ∈ Bn , μ = λ (μ)}. f

Definition 179. The generalized flow2 of  is the function φ : (Bn )N × Bn × N → Bn defined in the following way. Let α : N → Bn be an arbitrary function, whose values are denoted 1 2

To be compared with the Definition 28, page 32 of the state portraits. To be compared with the Definition 59, page 59 of the flows.

Appendix C • Symbolic dynamics

413

∀k ∈ N with α k instead of α(k). We take μ ∈ Bn and k ∈ N arbitrary also. Then  μ, if k = 0, α φ (μ, k) = k−1 α α  (φ (μ, k − 1)), if k ≥ 1. Theorem 292. Let  : Bn → Bn and the set of forbidden blocks F = {(μ, μ )|μ ∈ Bn , μ ∈ Bn , ∀λ ∈ Bn , μ = λ (μ)}.

(C.4.1)

(n)

The shift SF is equal with any of X1 = {x|x ∈ S (n) , ∀k ∈ N, ∃λ ∈ Bn , x(k + 1) = λ (x(k))}, f

(C.4.2)

X2 = {x|x ∈ S (n) , ∀k ∈ N, (x(k), x(k + 1)) ∈ E },

(C.4.3)

X3 = {φ α (μ, ·)|μ ∈ Bn , α : N → Bn }.

(C.4.4)

Proof. In order to prove the inclusion (n)

SF ⊂ X1 , (n)

we take an arbitrary x ∈ SF . This means that ∀k ∈ N, ∀l ≥ k, (x(k), ..., x(l)) ∈ / F, where F is given by (C.4.1). We fix k and l ≥ k arbitrary. If l = k + 1 the previous property holds trivially, thus we can consider the case l = k + 1, when we have non ∀λ ∈ Bn , x(k + 1) = λ (x(k)), ∃λ ∈ Bn , x(k + 1) = λ (x(k)).

(C.4.5)

As k ∈ N was arbitrary, we have obtained that x ∈ X1 . We prove X1 ⊂ X2 , so we take x ∈ X1 arbitrary. We get that x ∈ S (n) and, for k ∈ N arbitrary, fixed, (C.4.5) is true. f The conclusion is (x(k), x(k + 1)) ∈ E , i.e. x ∈ X2 . The next inclusion to be proved is X2 ⊂ X3 , so we take an arbitrary x ∈ X2 , thus x ∈ S (n) . We see that x(0) ∈ Bn and, on the other hand, if for any k ∈ N we denote with α k ∈ Bn a point whose existence is guaranteed by the fact that k f (x(k), x(k + 1)) ∈ E , then x(k + 1) = α (x(k)) holds and we have obtained the existence of a function α : N → Bn . The conclusion is

414

Boolean Systems

∀k ∈ N, x(k) = φ α (x(0), k), hence x ∈ X3 . We finally prove (n)

X3 ⊂ SF , so we take μ ∈ Bn and α : N → Bn arbitrary. Obviously φ α (μ, ·) ∈ S (n) and we consider k ∈ N, l ≥ k arbitrary too. Two possibilities exist. If l = k +1, the property (φ α (μ, k), ..., φ α (μ, l)) ∈ / α α F is clear. In the case that l = k + 1, the couple (φ (μ, k), φ (μ, k + 1)) satisfies k

φ α (μ, k + 1) = α (φ α (μ, k)), thus ∃λ ∈ Bn , φ α (μ, k + 1) = λ (φ α (μ, k)), non ∀λ ∈ Bn , φ α (μ, k + 1) = λ (φ α (μ, k)), (n)

/ F. It has resulted that φ α (μ, ·) ∈ SF . therefore (φ α (μ, k), φ α (μ, k + 1)) ∈ (n)

Definition 180. The shift space SF is called the timeless model of computation of the Boolean function  and its usual notation is XGf . The set XGf is also called the edge shift 

f



of G . Example 187. We consider the function  : B2 → B2 with the following state portrait (0, 1) 

(0, 0)

- (1, 0)

? (1, 1) The shift space XGf consists in the signals: 

x 1 = (0, 0), (0, 0), (0, 0), ... x 2 = (0, 1), (0, 1), (0, 1), ... x 3 = (1, 0), (1, 0), (1, 0), ... x 4 = (1, 1), (1, 1), (1, 1), ... x 5 = (0, 0), (0, 1), (0, 1), (0, 1), ... x 6 = (0, 0), (1, 0), (1, 0), (1, 0), ...

Appendix C • Symbolic dynamics

415

x 7 = (0, 0), (1, 1), (1, 1), (1, 1), ... x 8 = (1, 0), (1, 1), (1, 1), (1, 1), ... x 9 = (0, 0), (1, 0), (1, 1), (1, 1), (1, 1), ... together with these x ∈ S (2) that are orbitally equivalent with them. Remark 306. The way that the signals x 1 , x 3 , x 6 reflect reality is that of indicating possibilities. For example x 1 shows that ∀k ∈ N, x(k) = (0, 0) is possible if the system is slow enough such that none of 1 (0, 0), 2 (0, 0) is computed at time instances less or equal with k. The situation that either of 1 (0, 0), 2 (0, 0) is never computed reflects the possibility of computing , not the necessity.3 All the other signals mentioned at Example 187 indicate possibilities, this is the most general model of computation of the Boolean functions. Remark 307. The existence of x 1 , x 3 , x 6 ∈ XGf for which the computation of  is not com

pleted makes us say that there is no progress of time, and this is indicated by the presence of computation functions α : N → Bn that are not progressive. This remark gives XGf the name of timeless model of computation of .



Lemma 7. If we use, for l ∈ N, the following notations Al = {(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α ∈ n , i ∈ N}, Bl = {(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α ∈ n }, Cl = {(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α : N → Bn }, Dl = {(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α : N → Bn , i ∈ N}, then we have Al = Bl = Cl = Dl . Proof. For l = 0, we see that A0 = {φ α (μ, i)|μ ∈ Bn , α ∈ n , i ∈ N} = Bn , B0 = {μ|μ ∈ Bn , α ∈ n } = Bn , C0 = {μ|μ ∈ Bn , α : N → Bn } = Bn , 3

Or perhaps the fact that the computation should have normally happened, but it did not, due to the occurrence of an error; or two ‘stuck at 0’ errors, in the case of x 1 .

416

Boolean Systems

D0 = {φ α (μ, i)|μ ∈ Bn , α : N → Bn , i ∈ N} = Bn , thus we can consider from this moment that l ≥ 1. We prove Al ⊂ Bl and let w ∈ Al arbitrary. This means the existence of μ ∈ Bn , α ∈ n , i ∈ N with the property w = (φ α (μ, i), ..., φ α (μ, i + l)). For μ = φ α (μ, i), α  = σ i (α) we can write 



w = (φ α (μ , 0), ..., φ α (μ , l)) ∈ Bl . The inclusions Bl ⊂ Cl ⊂ Dl are obvious, so we prove Dl ⊂ Al and let w ∈ Dl arbitrary. This means the existence of μ ∈ Bn , α : N → Bn , i ∈ N with w = (φ α (μ, i), ..., φ α (μ, i + l)). We consider β ∈ n arbitrary and we define γ : N → Bn in the following way:  k α , if k ∈ {0, ..., i + l − 1}, k γ = β k , if k ≥ i + l. Then γ ∈ n and on the other hand w = (φ γ (μ, i), ..., φ γ (μ, i + l)) ∈ Al . Theorem 293. The language of XGf is B(XGf ) = {ε} ∨ 





{(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α : N → Bn }.

(C.4.6)

l∈N

Proof. We use Theorem 292, page 413 and X3 which was defined at (C.4.4). We have  B(X3 ) = {ε} ∨ {(x(i), x(i + 1), ..., x(i + l))|x ∈ X3 , i ∈ N} = {ε} ∨



l∈N

{(x(i), x(i + 1), ..., x(i + l))|x ∈ {φ α (μ, ·)|μ ∈ Bn , α : N → Bn }, i ∈ N}

l∈N

= {ε} ∨



{(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α : N → Bn , i ∈ N}

l∈N Lemma 7

=

{ε} ∨



{(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α : N → Bn }.

l∈N

Appendix C • Symbolic dynamics

417

Remark 308. Relating B(XGf ) with X3 from (C.4.4) like in the previous theorem is probably 

the most tempting solution of characterizing the language of XGf systemically. We can 

make use of X2 from (C.4.3) also and get B(XGf ) = {ε} ∨ Bn ∨ 

 {(x(0), ..., x(l))|x ∈ S (n) , l≥1

f (x(0), x(1)) ∈ E

f

and ... and (x(l − 1), x(l)) ∈ E }.

C.5 The unbounded delay model of computation Definition 181. Let  : Bn → Bn . The set of states X u f = {φ α (μ, ·)|μ ∈ Bn , α ∈ n } G

is called the unbounded delay model of computation of the function . Example 188. We reconsider the function  from Example 187, page 414. (0, 1) 

(0, 0)

- (1, 0)

? (1, 1) The unbounded delay model of computation X u f consists in the signals: G

x 2 = (0, 1), (0, 1), (0, 1), ... x 4 = (1, 1), (1, 1), (1, 1), ... x 5 = (0, 0), (0, 1), (0, 1), (0, 1), ... x 7 = (0, 0), (1, 1), (1, 1), (1, 1), ... x 8 = (1, 0), (1, 1), (1, 1), (1, 1), ... x 9 = (0, 0), (1, 0), (1, 1), (1, 1), (1, 1), ... together with the signals x ∈ S (2) which are orbitally equivalent with them. From the timeless model, we have eliminated x 1 = (0, 0), (0, 0), (0, 0), ...

418

Boolean Systems

x 3 = (1, 0), (1, 0), (1, 0), ... x 6 = (0, 0), (1, 0), (1, 0), (1, 0), ... as they contain computations that are never completed. Remark 309. The model from the previous example indicates the form of the states x ∈ S (2) of a system in the situation that the coordinates 1 , 2 are eventually computed, with unknown delays. Theorem 294. (a) We have X u f ⊂ XGf , G



(b) the language of X u f fulfills G

B(X u f ) = B(XGf ), G



(c) X u f is not a shift space, in general. G

Proof. (a) Taking into account Definition 180, page 414 of XGf , together with Theorem 292, 

Eq. (C.4.4), on one hand, as well as Definition 181, page 417 of X u f on the other hand, the inclusion is equivalent with the obvious one

G

{φ α (μ, ·)|μ ∈ Bn , α ∈ n } ⊂ {φ α (μ, ·)|μ ∈ Bn , α : N → Bn }. (b) We have B(X u f ) = {ε} ∨ G

 l∈N

= {ε} ∨



{(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α ∈ n , i ∈ N}

l∈N Lemma 7, page 415

=

= {ε} ∨

{(x(i), ..., x(i + l))|x ∈ {φ α (μ, ·)|μ ∈ Bn , α ∈ n }, i ∈ N}



{ε} ∨



{(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α : N → Bn , i ∈ N}

l∈N

{(x(i), ..., x(i + l))|x ∈ {φ α (μ, ·)|μ ∈ Bn , α : N → Bn }, i ∈ N} = B(XGf ). 

l∈N

(c) We suppose the existence of μ ∈ Bn with (μ) = μ, when the constant state x ∈ ∈ N,

S (n) , ∀k

x(k) = μ

Appendix C • Symbolic dynamics

419

/ X u f , in other words the inclusion from (a) is strict. We suppose also fulfills x ∈ XGf , x ∈ G



against all reason that X u f is a shift space. Then G

Xu f

G

Theorem 290, page 411

=

(b)

(n) SB(X u

f ) G

c

(n) = SB(X

Theorem 290 f ) G

c

=

XGf , 

contradiction with the fact that X u f  XGf . G



Remark 310. We underline the importance of the ‘mismatch’ stated at Theorem 294 (c), since X u f is the most important concept from this monograph, while the shift spaces G

represent the most important concept from [31]. Which are the consequences of this mismatch?

C.6 The bounded delay model of computation Theorem 295. Let  : Bn → Bn and we define for N ≥ 1 the sets F0 = {(μ, μ )|μ ∈ Bn , μ ∈ Bn , ∀λ ∈ Bn , μ = λ (μ)}, F1 = {(μ1 , ..., μN , μN +1 )|μ1 ∈ Bn , ..., μN ∈ Bn , μN +1 ∈ Bn , +1 = 1 (μN )}, μ21 = 1 (μ1 ) and ... and μN 1

... Fn = {(μ1 , ..., μN , μN +1 )|μ1 ∈ Bn , ..., μN ∈ Bn , μN +1 ∈ Bn , +1 μ2n = n (μ1 ) and ... and μN = n (μN )}, n k+N −1 n k = 1}. ≤N n = {α|α : N → B , ∀i ∈ {1, ..., n}, ∀k ∈ N, αi ∪ ... ∪ αi (n)

The shift space SF0 ∨F1 ∨...∨Fn is equal with any of Y1 = {x|x ∈ S (n) , ∀k ∈ N, ∃λ ∈ Bn , x(k + 1) = λ (x(k))} ∧{x|x ∈ S (n) , ∀k ∈ N, ∃j ∈ {1, ..., N }, x1 (k + j ) = 1 (x(k + j − 1))}∧ ... ∧ {x|x ∈ S (n) , ∀k ∈ N, ∃j ∈ {1, ..., N }, xn (k + j ) = n (x(k + j − 1))},

(C.6.1)

f Y2 = {x|x ∈ S (n) , ∀k ∈ N, (x(k), x(k + 1)) ∈ E } (n) ∧{x|x ∈ S , ∀k ∈ N, ∃j ∈ {1, ..., N }, x1 (k + j ) = 1 (x(k + j − 1))}∧ ... ∧ {x|x ∈ S (n) , ∀k ∈ N, ∃j ∈ {1, ..., N }, xn (k + j ) = n (x(k + j − 1))},

(C.6.2)

420

Boolean Systems

Y3 = {φ α (μ, ·)|μ ∈ Bn , α : N → Bn } ∈ N, ∃j ∈ {1, ..., N }, x1 (k + j ) = 1 (x(k + j − 1))}∧ ∧{x|x ... ∧ {x|x ∈ S (n) , ∀k ∈ N, ∃j ∈ {1, ..., N }, xn (k + j ) = n (x(k + j − 1))}, ∈ S (n) , ∀k

(C.6.3)

Y4 = {φ α (μ, ·)|μ ∈ Bn , α ∈ ≤N n }.

(C.6.4)

Proof. We know from Theorem 292, page 413, that (n)

SF0 = {x|x ∈ S (n) , ∀k ∈ N, ∃λ ∈ Bn , x(k + 1) = λ (x(k))} f

= {x|x ∈ S (n) , ∀k ∈ N, (x(k), x(k + 1)) ∈ E } = {φ α (μ, ·)|μ ∈ Bn , α : N → Bn }.

(C.6.5)

On the other hand ∀i ∈ {1, ..., n}, (n)

SFi = {x|x ∈ S (n) , ∀k ∈ N, ∀l ∈ N, (x(k), ..., x(k + l)) ∈ / Fi } = {x|x ∈ S (n) , ∀k ∈ N, ∀l ∈ N, (x(k), ..., x(k + l)) ∈ / {(μ1 , ..., μN , μN +1 )|μ1 ∈ Bn , +1 = i (μN )}} ..., μN ∈ Bn , μN +1 ∈ Bn , μ2i = i (μ1 ) and ... and μN i

= {x|x ∈ S (n) , ∀k ∈ N, not (x(k), ..., x(k + N )) ∈ {(μ1 , ..., μN , μN +1 )|μ1 ∈ Bn , +1 = i (μN )}} ..., μN ∈ Bn , μN +1 ∈ Bn , μ2i = i (μ1 ) and ... and μN i

= {x|x ∈ S (n) , ∀k ∈ N, not (xi (k + 1) = i (x(k)) and ... and xi (k + N ) = i (x(k + N − 1)))} = {x|x ∈ S (n) , ∀k ∈ N, xi (k + 1) = i (x(k)) or ... or xi (k + N ) = i (x(k + N − 1))} = {x|x ∈ S (n) , ∀k ∈ N, ∃j ∈ {1, ..., N }, xi (k + j ) = i (x(k + j − 1))}. We have: (n) (n) (n) (n) SF = SF ∧ SF ∧ ... ∧ SF n 0 ∨F1 ∨...∨Fn 0 1 (C.6.5)

(n)

(n)

= {x|x ∈ S (n) , ∀k ∈ N, ∃λ ∈ Bn , x(k + 1) = λ (x(k))} ∧ SF1 ∧ ... ∧ SFn (C.6.5)

f

(n)

(n)

= {x|x ∈ S (n) , ∀k ∈ N, (x(k), x(k + 1)) ∈ E } ∧ SF1 ∧ ... ∧ SFn (C.6.5)

(n)

(n)

= {φ α (μ, ·)|μ ∈ Bn , α : N → Bn } ∧ SF1 ∧ ... ∧ SFn ,

(n)

therefore SF0 ∨F1 ∨...∨Fn is equal with any of Y1 , Y2 , Y3 . We prove Y3 ⊂ Y4 , and let x ∈ Y3 arbitrary. Then μ ∈ Bn and α : N → Bn exist such that ∀k ∈ N, x(k) = φ α (μ, k).

(C.6.6)

Appendix C • Symbolic dynamics

421

We fix k ∈ N and i ∈ {1, ..., n} arbitrary. We know about x that j ∈ {1, ...., N } exists with the property xi (k + j ) = i (x(k + j − 1)),

(C.6.7)

thus we can define for l ∈ {k, ..., k + N − 1},  βil =

αil , if l = k + j − 1, 1, if l = k + j − 1,

(C.6.8)

and we have βik ∪ ... ∪ βik+N −1 = 1.

(C.6.9)

When k runs in N and i runs in {1, ..., n}, (C.6.6), (C.6.7) and (C.6.8) show that x(k) = φ β (μ, k), and (C.6.9) implies β ∈ ≤N n . We have obtained that x ∈ Y4 . We prove Y4 ⊂ Y3 , so we take x ∈ Y4 arbitrary, wherefrom μ ∈ Bn and α ∈ ≤N n exist such that ∀k ∈ N, (C.6.6) holds. The fact that x ∈ {φ β (ν, ·)|ν ∈ Bn , β : N → Bn } is obvious, and we fix now arbitrarily k ∈ N, i ∈ {1, ..., n}. From αik ∪ ... ∪ αik+N −1 = 1 k+j −1

we get the existence of j ∈ {1, ..., N } with the property αi xi (k + j ) = αi

k+j −1

= 1, thus

(x(k + j − 1)) = i (x(k + j − 1)).

(n)

We infer from here that x ∈ SFi . As k and i were chosen arbitrarily, we get x ∈ Y3 . (n)

Definition 182. The shift space SF0 ∨F1 ∨...∨Fn is called the bounded delay model of computation of the Boolean function  and its notation is X ≤N f . The parameter N ≥ 1 is the G

upper bound of the delay of computation (of the computation time) of . Remark 311. The states of X ≤N f have the property that 1 , ..., n are computed, indepenG

dently on each other, in at most N time units. In Theorem 295, F0 shows the possible trajectories of the system, while F1 , ..., Fn indicate the speed of the system along these trajectories. We can consider the case that the speeds of computation of 1 , ..., n differ, and the system has the upper bounds of the computation times N1 , ..., Nn .

422

Boolean Systems

Example 189. We get back once again to  : B2 → B2 from Example 187, page 414, for which we consider F = F 0 ∨ F1 ∨ F2 like in Theorem 295. We obtain for N = 1,  is computed in 1 time units, that F0 = {((0, 1), (0, 0)), ((0, 1), (1, 0)), ((0, 1), (1, 1)), ((1, 0), (0, 0)), ((1, 0), (0, 1)), ((1, 1), (0, 0)), ((1, 1), (0, 1)), ((1, 1), (1, 0))}, F1 = {((0, 0), (0, 0)), ((0, 0), (0, 1)), ((0, 1), (1, 0)), ((0, 1), (1, 1)), ((1, 0), (0, 0)), ((1, 0), (0, 1)), ((1, 1), (0, 0)), ((1, 1), (0, 1))}, F2 = {((0, 0), (0, 0)), ((0, 0), (1, 0)), ((0, 1), (0, 0)), ((0, 1), (1, 0)), ((1, 0), (0, 0)), ((1, 0), (1, 0)), ((1, 1), (0, 0)), ((1, 1), (1, 0))}, F = B2 × B2  {((0, 0), (1, 1)), ((0, 1), (0, 1)), ((1, 0), (1, 1)), ((1, 1), (1, 1))}, and (2) X = SF = {x|x ∈ S (2) , ∃j ∈ {2, 4, 7, 8}, x ∼ x j },

where x 2 = (0, 1), (0, 1), (0, 1), ... x 4 = (1, 1), (1, 1), (1, 1), ... x 7 = (0, 0), (1, 1), (1, 1), (1, 1), ... x 8 = (1, 0), (1, 1), (1, 1), (1, 1), ... This is normal, since in this special case the state functions are all orbitally equivalent with μ, (μ), (2) (μ), (3) (μ), ... μ ∈ B2 , and we have exactly card(B2 ) = 4 representatives of these equivalence classes, which are identified with their initial values. The computation function α ∈ 2 that produces this effect is ∀k ∈ N, α k = (1, 1). Lemma 8. With the notations α α n ≤N AN l = {(φ (μ, i), ..., φ (μ, i + l))|μ ∈ B , α ∈ n , i ∈ N},

BlN = {(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α ∈ ≤N n }, where l ∈ N and N ≥ 1, we have N AN l = Bl .

Appendix C • Symbolic dynamics

Proof. We fix l ∈ N, N ≥ 1 arbitrary. We prove the inclusion N AN l ⊂ Bl n ≤N and let for this w ∈ AN l arbitrary, i.e. μ ∈ B , α ∈ n , i ∈ N exist with

w = (φ α (μ, i), ..., φ α (μ, i + l)). We define μ = φ α (μ, i), α  = σ i (α) with μ ∈ Bn , α  ∈ ≤N n and we infer 



w = (φ α (μ , 0), ..., φ α (μ , l)) ∈ BlN . The inclusion BlN ⊂ AN l is obvious. Theorem 296. The language of X ≤N f is G

B(X ≤N f ) = {ε} ∨ G



{(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α ∈ ≤N n }.

l∈N

Proof. We make use of Theorem 295, page 419. We get  B(Y4 ) = {ε} ∨ {(x(i), x(i + 1), ..., x(i + l))|x ∈ Y4 , i ∈ N} = {ε} ∨



l∈N

{(x(i), x(i + 1), ..., x(i + l))|x ∈ {φ α (μ, ·)|μ ∈ Bn , α ∈ ≤N n }, i ∈ N}

l∈N

= {ε} ∨



{(φ α (μ, i), ..., φ α (μ, i + l))|μ ∈ Bn , α ∈ ≤N n , i ∈ N}

l∈N Lemma 8

=

{ε} ∨



{(φ α (μ, 0), ..., φ α (μ, l))|μ ∈ Bn , α ∈ ≤N n }.

l∈N

423

Notations B, the binary Boole algebra, 1 ε i , the vectors of the canonical basis of Bn , 3  ai , the intersection of ai ∈ B, i ∈ I , 3 i∈I  ai , the union of ai ∈ B, i ∈ I , 3 i∈I

(  x j ), the modulo 2 sum of x j ∈ Bn , j ∈ J , 3 j ∈J

supp μ, the support of μ ∈ Bn , 3 μ  λ, 3 [μ, λ], the affine space defined by the points μ, λ ∈ Bn , 4 [μ, λ), (μ, λ], (μ, λ), 6 θ τ (μ), the translation of μ with τ , 6 ∗ , the dual of , 7 (k) , the k-th iterate of , 8, 296 λ , the λ-iterate of , 8, 296 (μ, ν), 10 ( × ), the Cartesian product of  and , 10, 297 μ+ , the set of the immediate successors of μ, 11 O + (μ), the set of the successors of μ, 11 μ− , the set of the immediate predecessors of μ, 11 O − (μ), the set of the predecessors of μ, 11 Af (Bn ), the set of the functions that are compatible with the affine structure of Bn , 15 d(μ, λ), the Hamming distance (between μ and λ), 18 H om(, ), the set of the morphisms from  to , 21, 302 I so(, ), the set of the isomorphisms from  to , 23 Aut (), the set of the automorphisms of , 23 P ({1, ..., n}) = {A|A ⊂ {1, ..., n}}, the set of the subsets of {1, ..., n}, 32 Spn , the set of the state portraits of dimension n, 32 G , the state portrait of , 32, the state portrait family of , 298 Fn , the set of the Bn → Bn functions, 33 μ → μ , the arrow (μ, μ ) ∈ E , 34 μi , unstable coordinate, 35 G |X , 39 S (n) , S, the set of the signals, 41 lim x(k), the final value of x, 42 k→∞

O(x), the orbit of x, 43 425

426

Notations

∼, orbital equivalence, 44 Txμ , 44 ω(x), (omega-)limit set of x, 45 ≈, omega-limit equivalence, 46  σ k , the forgetful function, 47  : S (n) −→ S (m) , the image of a signal via , 48  Px , the set of the periods of the eventually periodic signal x, 50 n , , sets of computation functions, 54  h, the morphism induced by h, 56

n,m , n , the sets of the functions that are compatible with the progressiveness of the computation functions, 56 ≤L , 57 ≤L n , L L n , , 57 φ, flow generated by , 59 F ln , the set of the flows with n-dimensional generator functions, 60 Eq , the equation of evolution of , 67, 309 Eqn , the set of the equations of evolution of the functions , 67  ⊂ ,  is a subsystem of , 75 (h, h ) : φ → ψ, morphism from φ to ψ, 79 H om(φ, ψ), the set of morphisms from φ to ψ, 79, 310  ) ∈ H om(φ, ψ), morphism induced by (h, h ) ∈ H om(, ), 81 (h, h I so(φ, ψ), the set of isomorphisms from φ to ψ, 84 Aut (φ), the set of automorphisms of φ, 84

, 88 (h, h ) : φ  ψ, pseudo-morphism from φ to ψ, 95 om(φ, ψ), the set of the pseudo-morphisms from φ to ψ, 95 H N Ci ⊂ Bn , i ∈ {1, ..., n}, 97 A0i , A1i , 100 lim Xk , limit of a sequence of sets, 111 k→∞ − O α (μ), 114

1 , ..., X 5 , maximal invariant subsets of X, 183 Xmax max O + (A), the set of the successors of A, 189 1 , ..., X 5 , minimal invariant supersets of X, 195 Xmin min ◦1

◦5

X min , ..., X min the minimal invariant subsets of X, 203 ω+ (μ), 221 ω+ (A), 221 W 1 (A), Wγ2 (A), ..., W 5 (A), basins of attraction of the set A, 221 W 1 (μ), Wγ2 (μ), ..., W 5 (μ), basins of attraction of the point μ, 231 W 1 (μ), Wγ2 (μ), ..., W 5 (μ), basins of attraction of the periodic point μ, 234 W 1 [φ α (μ, ·)], Wγ2 [φ α (μ, ·)], ..., W 5 [φ α (μ, ·)], basins of attraction of the state φ α (μ, ·), 237 W 1 (A, B), Wγ2 (A, B), ..., W 5 (A, B), local basins of attraction, 245

Notations

427

W 1 [φ α (μ, ·), A], Wγ2 [φ α (μ, ·), A], ..., W 5 [φ α (μ, ·), A], local basins of attraction of φ α (μ, ·), 253 ν , 296 Hν , 298 Spn,m , the set of the state portrait families, 298 Fn,m , the set of the functions with a parameter, 299 − O α (μ, u), O α (μ, u), 306 Eqn,m , the set of the equations of evolution of the functions , 309 χ , the input-output function of , 328, 343 O + (μ, u), O − (μ, u), 330   L 1 , L2 , ..., Lp , ..., 335 x(t − 0), left limit (function) of x, 387 x(t + 0), right limit (function) of x, 387 x(−∞ + 0) ∈ Bn , initial value of x, 387 χA , the characteristic function of the set A, 387 S˙ (n) , the set of the real time signals, 387 ˙ n , the set of the real time computation functions, 387 ˙ real time flow generated by , 388 φ, Eq˙  , equation of real time evolution of , 390 ⊗, Kronecker product, 393 , semi-tensor product, 393 δni , the columns of the identity matrix of dimension n, 394 D, D(n) , 394 ζ , ζn , μ, λ, 394 coli (A), the i − th column of an arbitrary binary matrix A, 395 Ln×m , the set of the logical matrices with n rows and m columns, 395 , 395 Mf , M , M   , the structure matrices of f, ,  n ε, the empty block over B , 405 , subblock, 405 (n) , Bn -shift space with the set of forbidden blocks F, 406 SF (u, v), the concatenation of the blocks u, v, 405 F c , the complement of a set F of blocks over Bn , 409 B(X), the language of X, 409 f f f G = (V , E ), full state portrait, 412 XGf , the timeless model of computation of , 414 

X u f , the unbounded delay model of computation of , 417 G

X ≤N f , the bounded delay model of computation of , 421 G

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Index A accessible point, 62 acyclic state portrait, 39 affine space defined by the points μ, λ, 4 asymptotically equivalent initial values of the states, 119 attractive set, 223 attractor, 259 automorphism of , 23 automorphism of φ, 84 B balanced state portrait, 38 basin of attraction of a periodic point, 234 basin of attraction of a set, 221 basin of attraction of a state, 237 bifurcation diagram, 300 bifurcation of , 300 binary Boole algebra, 1 block over Bn , 405 Boolean function, 6 bounded delay model of computation of , 421 C Cartesian product of  and , 10, 297 chaos, 146 characteristic function of a set, 387 closed path in a state portrait, 39 combinational function, 337 combinational function with one level, 328 compatible function with the affine structure of Bn , 15 compatible function with the progressiveness of the computation functions, 56 composition of the morphisms, 23, 84 computation function, 53, 387 concatenation, 405 conjugated flows, 84

conjugated generator functions, 23 connected components of a set, 218 connected set, 211 convergent sequence of sets, 111 coordinate-wise monotonic signal, 43 critical point of φ α (μ, ·), 105 cycle of a signal, 50 cyclic state portrait, 39 D D type flip flop, 383 data latch, 383 decreasing function, 172 decreasing sequence of sets, 111 decreasing signal, 43 delay buffer, 362 delay circuit, 362 delay element, 362 dependence on the initial conditions, 119 directed graph, 32 disabled coordinate, 34 disconnected set, 212 discrete topology, 2 double eventually periodic state, 127 double periodic state, 127 dual function, 7296 dual of a signal, 42 dual state portraits, 37 dual systems, 73 E f edge shift of G , 414 empty block, 405 empty shift space, 406 enabled coordinate, 34 equations of evolution, 67, 309, 390 equilibrium point of φ α (μ, ·), 105 Eulerian path, 40 eventually periodic point, 133

431

432

Index

eventually periodic signal, 50 evolution function, 59 excited coordinate, 34 F final time (instant) of a signal, 42 final value of a signal, 42 final value of φ α (μ, ·), 105 flow, 59, 305, 388 forbidden blocks, 406 forgetful function, 47 full shift space, 406 full state portrait, 412 fundamental (operating) mode, 325 G gate, 368 gated SR flip flop, 380 generalized flow, 412 global attractor, 259 H Hamiltonian path, 40 Hamming distance between a point and a set, 172 Hamming distance between μ and λ, 18 hazard, 346 I image of x via , 48 immediate predecessors of μ, 11 immediate successors of μ, 11 increasing function, 172 increasing sequence of sets, 111 increasing signal, 43 indecomposable set, 178 indegree of a point, 38 initial time (instant) of a signal, 42 initial value of a signal, 42, 387 input evolution function, 305 input flow, 305 input next state function, 305 input-output function, 328343 input state transition function, 305 input subsystem, 315

input system, 314 invariant set, 157 invariant subset, 160 isolated fixed point, 107 isolated set, 178 isomorphism of flows, 84 isomorphism of generator functions, 23 isomorphism of progressive computation functions, 57 isomorphism of state portraits, 37 k-th iterate, 8, 296 λ-iterate, 8, 296 K Kronecker product, 393 L language of X, 409 left limit of x, 387 length of a block, 405 level of , 341 limit cycle of a signal, 50 limit of a convergent sequence of sets, 111 limit of periodicity of a point, 133 limit of periodicity of a signal, 50 Lipschitz function, 19 local basin of attraction, 245 local basin of attraction of a state, 253 logical matrix, 395 M maximal invariant subset, 183 minimal invariant subset, 203 minimal invariant superset, 195 minimal set, 211 monotonic signal, 43 monotonically decreasing sequence of sets, 111 monotonically increasing sequence of sets, 111 morphism compatible with the subsystems, 88 morphism of flows, 79, 310 morphism of generator functions, 21, 302 morphism of progressive computation functions, 55

Index

N NAND gate, 374 next state function, 59 nonwandering point, 151 NOT gate, 369 nullcline, 97 O omega-limit equivalent signals, 46 omega-limit points of x, 45 omega-limit set of x, 45 orbit of a signal, 43 orbitally equivalent signals, 44 ordinary point, 104 outdegree of a point, 38 P parametric portrait of , 300 partially attractive set, 223 partially combinational function, 337 path-connected component of a set, 144 path-connected component of a state portrait, 39 path-connected points, 39, 135 path-connected set, 135 path-connected state portrait, 39 path in a state portrait, 39 path in X from μ to μ , 135 period of a point, 133, 233 period of a signal, 50 periodic point, 133 periodic signal, 50 Poisson stable point, 152 predecessors of μ, 11, 298 progressive computation function, 54 pseudo-morphism of flows, 95 R reachable point, 62 recurrent point, 151 relatively isolated sets, 177 repeller of X, 272 reset input of the SR latch, 378 rest position of φ α (μ, ·), 105

433

right limit of x, 387 S semi-tensor product, 393 separated set, 212 set input of the SR latch, 378 set of connecting orbits from R(X) to X, 272 shift space, 406 signal, 41, 387 simple path in a state portrait, 39 singular point of φ α (μ, ·), 105 sink, 107 source, 107 SR-latch, 375 stability, 277 stable coordinate, 34 state (function), 59, 305, 388 state portrait, 32 state portrait family of , 298 state portrait of , 32 state space decomposition, 316 state subportrait, 36 state transition function, 59 states that reach μ in k time units, 112 states which are reached, from μ, in k time units, 112 stationary point of φ α (μ, ·), 105 steady state of φ α (μ, ·), 105 stratum, 300 strictly increasing function, 172 structure matrix, 395 subblock, 405 subshift, 406 subsystem, 75, 315 successors of μ, 11, 298 support of μ, 3 symmetrical flows relative to the translation with τ , 86 symmetrical generator functions relative to the translation with τ , 26 synonymous generator functions, 25 system, 72, 314 T time-reversal symmetrical systems, 287

434

Index

timeless model of computation of , 414 topological transitivity, 136 topologically conjugated state portraits, 37 topologically equivalent flows, 84 topologically equivalent generator functions, 23 topologically equivalent state portraits, 37 totally attractive set, 223 transient point, 107 translation, 6

U unbounded delay model of computation of , 417 unstable coordinate, 34 W walk in a state portrait, 39 weak attractor, 274 weakly dissipative flow, 274 word over Bn , 405