Fuzzy Relational Calculus: Theory, Applications And Software (Advances in Fuzzy Systems) 9812560769, 9789812560766, 9789812701336

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Fuzzy Relational Calculus Theory, Applications and Software

ADVANCES IN FUZZY SYSTEMS — APPLICATIONS AND THEORY Honorary Editor: Lotfi A. Zadeh (Univ. of California, Berkeley) Series Editors: Kaoru Hirota (Tokyo Inst. of Tech.), George J. Klir (Binghamton Univ.-SUNY), Elie Sanchez (Neuhnfo), Pei-Zhuang Wang (West Texas A&M Univ.), Ronald R. Yager (lona College) Vol. 8:

Foundations and Applications of Possibility Theory (Eds. G. de Cooman, D. Ruan and E. E. Kerre)

Vol. 9:

Fuzzy Topology (Y. M. Liu and M. K. Luo)

Vol. 10: Fuzzy Algorithms: With Applications to Image Processing and Pattern Recognition (Z Chi, H. Yan and T. D. Pham) Vol. 11: Hybrid Intelligent Engineering Systems (Eds. L. C. Jain and R. K. Jain) Vol. 12: Fuzzy Logic for Business, Finance, and Management (G. Bojadziev and M. Bojadziev) Vol. 13: Fuzzy and Uncertain Object-Oriented Databases: Concepts and Models (Ed. R. de Caluwe) Vol. 14: Automatic Generation of Neural Network Architecture Using Evolutionary Computing (Eds. E. Vonk, L C. Jain and R. P. Johnson) Vol. 15: Fuzzy-Logic-Based Programming (Chin-Liang Chang) Vol. 16: Computational Intelligence in Software Engineering (W. Pedrycz and J. F. Peters) Vol. 17: Non-additive Set Functions and Nonlinear Integrals (Forthcoming) (Z. Y. Wang) Vol. 18: Factor Space, Fuzzy Statistics, and Uncertainty Inference (Forthcoming) (P. Z. Wang and X. H. Zhang) Vol. 19: Genetic Fuzzy Systems, Evolutionary Tuning and Learning of Fuzzy Knowledge Bases (O. Cordon, F. Herrera, F. Hoffmann and L Magdalena) Vol. 20: Uncertainty in Intelligent and Information Systems (Eds. B. Bouchon-Meunier, R. R. Yager and L. A. Zadeh) Vol. 21: Machine Intelligence: Quo Vadis? (Eds. P. Sincak, J. VaScak and K. Hirota) Vol. 22: Fuzzy Relational Calculus: Theory, Applications and Software (With CD-ROM) (K. Peeva and Y. Kyosev)

Advances in Fuzzy Systems - Applications and Theory - Vol. 22

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Fuzzy Relational Calculus Theory, Applications and Software (With CD-Rom)

Ketty Peeva & Yordan Kyosev Technical University of Sofia, Bulgaria

\Hp World Scientific NEW JERSEY • LONDON

• SINGAPORE

• BEIJING • SHANGHAI • H O N G K O N G

• TAIPEI • CHENNA!

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Advances in Fuzzy Systems — Applications and Theory, Vol. 22 FUZZY RELATIONAL CALCULUS Theory, Applications and Software (With CD-ROM) Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

Fuzzy relations appear as a natural generalization of crisp relations. While a crisp relation determines the presence or absence of interconnectedness between the elements of two or more sets, fuzzy relations supply additional information for degrees of membership, strengths of associations, interaction between elements. Advantage of fuzzy relations is also that they permit to manipulate values that can be specified in linguistic terms. As [Zadeh and Desoer (1963)] show, in the general study of systems the relationship between input and output parameters can be modelled by fuzzy relation between input and output spaces. Investigating behavior of such systems presumes a powerful fuzzy relational calculus. The importance of the theory of fuzzy relational equations is best described by Zadeh in the preface of the monograph by [Di Nola et al. (1989)]: "Human knowledge may be viewed as a collection of facts and rules, each of which may be represented as the assignment of a fuzzy relation to the unconditional or conditional possibility distribution of a variable. What this implies is that the knowledge may be viewed as a system of fuzzy relational equations. In this perspective, then, inference from a body of knowledge reduces to the solution of a system of fuzzy relational equations."

Fuzzy relations and fuzzy relational calculus have many reasonable applications in pure and applied mathematics. The basic operations with fuzzy relations correspond to the key operations in fuzzy logic. They are implemented in all inference forward or backward chain reasoning schemes as described for instance in [Bezdek (1999), Bien and Min (1995), Dubois and Prade (2000), Dubois et al. (1999), Klir and Yuan (1995), Zadeh et al. (1996)]. The most valuable implementations are in expert systems and in artificial intelligence areas - approximate reasoning, inference systems, knowledge representation, knowledge acquisition and validation, learning, in information processing, in pattern analysis and classification, in fuzzy system science for fuzzy control and modelling, in decision making, in engineering for fault detection and diagnosis, in management, etc. Implementing fuzziness requires developing fuzzy relational calculus, special mathematical skills, ability to operate with modern mathematics and software. V

vi

Preface

In Part 1 we propose methodology and universal algorithms for direct and inverse problem resolution in fuzzy relational calculus. They include in unified frame fuzzy linear systems of equations, fuzzy relational equations, fuzzy relational inequalities and fuzzy relational inclusions upon various compositions in bounded chain. Then intuitionistic fuzzy relational calculus is reasonably developed. Computational complexity of the problems is investigated, numerical estimations are obtained. Based on these methods and algorithms, in Part 2 we solve open problems in the theory of fuzzy machines, fuzzy languages and syntactic fuzzy pattern recognition, we propose applications in modules of expert systems. The corresponding software for fuzzy relational calculus is described in Part 3. The Appendix and CD complete the exposition. Theoretical background is presented in Part 1, Chapters 1-6. In Chapter 1 we include in unified frame the recent results in fuzzy relational calculus for compositions of fuzzy relations and solving fuzzy relational equations and also outline the place of investigations in this book. Compositions of fuzzy relations and direct problem resolution are studied in Chapter 2. Attention is paid on their properties, the interconnection between fuzzy relation and its representative membership matrix, as well as between compositions of fuzzy relations and matrix multiplications. Models based on fuzzy logic require methods and algorithms for solving fuzzy relational equations. Fuzzy relational equations stay in the heart of fuzzy relational calculus. They are subject of Chapters 3 - 6 . Chapter 3 is devoted to fuzzy linear systems of equations and fuzzy relational equations over a bounded chain, when the composition is the standard one (in particular max —min). Methods and algorithms are proposed for finding the complete solution set. Solvability criterion is proved. Analytical expressions are given for determining the solutions, if the system is consistent. In case of inconsistency, the connections, that can not be satisfied simultaneously with the other connections, are marked. We also investigate algorithmical solvability and computational complexity of the problems in this subject. Chapter 4 covers inverse problem resolution for fuzzy linear systems of inequalities and fuzzy relational inclusions. Solvability condition is proved and analytical expressions are given for the solutions. Algorithms are proposed for solving fuzzy linear systems of inequalities and fuzzy relational inclusions. Applications in fuzzy linear programming are described. Chapters 5 and 6 are reasonable extension of the previous two chapters. We study inverse problem resolution for co-standard (in particular min - max) composition of fuzzy relational equations and for intuitionistic fuzzy relational equations thereby solving open problems in fuzzy relational calculus. In Chapter 5 we investigate fuzzy relational equations over a bounded chain, if the composition is the co-standard one. We solve the inverse problem developing a conventional approach, based on the methodology of Chapter 3. Specially for

Preface

vii

the min —max composition on the real closed interval [0, 1] we propose another approach, implementing duality. In Chapter 6 we introduce and investigate direct and inverse problems in intuitionistic fuzzy relational calculus. Their resolution is provided by the methods and algorithms developed for standard and for co-standard compositions in previous chapters. Fuzzy relational calculus, as presented in Part 1, provides a powerful theoretical background for dealing with fuzzy machines, fuzzy languages, pattern recognition, expert systems and other artificial intelligence areas, subject of Part 2, Chapters 7, 8 and 9. Several types of fuzzy finite machines are studied in Chapter 7, all of them over a bounded chain. We investigate fuzzy finite machines with behavior obtained upon the standard or upon the co-standard law of composition. Both classes are reasonably joined in the case of intuitionistic fuzzy finite machines. Investigation of behavior, reduction and minimization problems of all these classes of fuzzy finite machines is provided by the fuzzy algebra theory as developed in the first part of this book. We express the behavior and study various equivalences, reduction and minimization problems and their algorithmical solvability, applying direct and inverse problem resolutions, as well as the algorithms from Part 1. In Chapter 8 we propose how to use regular fuzzy languages for syntactic pattern recognition and classification of distorted images. We introduce intuitionistic fuzzy languages and implement them for pattern recognition and classification. In Chapter 9 we give engineering applications of direct and inverse problem resolution in modules of expert systems for diagnosis, testing, validation, learning, fault detection and monitoring. Chapter 10 in Part 3 concerns the implemented in MATLAB Fuzzy Relational Calculus Toolbox. It describes realization of functions and algorithms, as presented in Part 1, for the fuzzy algebra I = ([0,1], max, min) without any restrictions about the size of the instant and about right-hand side constants (some of them or all may be equal). Working examples are also included. Bibliographical notes at the end of each chapter include the most substantial theoretical and applied papers and monographs and also show the place of this investigation. The book is based on original authors' results following mainly [Peeva (2002b)] and the recent publications by Peeva and Kyosev. Available with this book is a CD, that contains functions, described in the book, as well as a lot of examples with solutions. The software has been tested on systems of dimension 50x50, but in principal the dimension is not limited. The examples are presented with input data, complete solution set and computational solution time. The CD contains also a restricted demo version of an alternative fast algorithm (which is not described in this book) for solving fuzzy linear systems. Tests are made on a PC with CPU 900 MHz, but some of them are also tested on machines

viii

Preface

with CPU 200 MHz up to 2400 MHz. The toolbox should be of use for both teaching and research. The toolbox is distributed under the terms of the GNU General Public License (http://www.gnu.org/copyleft/gpl.html, version 2 of the License, or any later version) as published by the Free Software Foundation. The tools used to prepare the book are the MikTEX (http://www.miktex.org/), including its excellent YAP previewer. The authors are grateful to MATLAB for modern computer aids with the MathWorks Book Program. We wish to thank all of the editorial and production staff of World Scientific publishing company. Special thanks are due to Prof. Elie Sanchez, Editor in the book series Advances in Fuzzy Systems. We are also grateful to Prof. Krassimir Atanassov and to Dr. Tony Croft for their valuable comments and proposals that help to improve the presentation. Ketty Peeva Yordan Kyosev

Contents

Preface

v

Chapter 1 Introduction 1.1 Basic Concepts 1.2 Images and Compositions 1.3 Basic Problems in Fuzzy Relational Calculus 1.4 Aspects in Artificial Intelligence 1.5 Fuzzy Finite Machines and Fuzzy Algebras 1.6 Fuzzy Grammars in Syntactic Pattern Recognition 1.7 Bibliographical Notes

3 3 8 12 14 17 19 22

Chapter 2 Fuzzy Relations. Direct Problem Resolution 2.1 Basic Notions 2.2 Fuzzy Relations - Compositions and Properties 2.3 Fuzzy Relations and Membership Matrices 2.4 Bibliographical Notes

25 25 31 41 47

Chapter 3 Fuzzy Relational Equations 3.1 Inverse Problem Formulation 3.2 Fuzzy Linear Equations 3.3 Fuzzy Linear Systems of Equations 3.3.1 Basic Notions 3.3.2 Simplifications 3.3.3 Lower Solutions 3.3.4 Universal algorithm 3.4 Solving Fuzzy Relational Equations 3.5 Bibliographical Notes

49 50 51 59 60 61 75 82 88 91

Chapter 4

95

4.1

Fuzzy Relational Inclusions

Preliminaries

95 ix

x

4.2 4.3 4.4 4.5

Contents

Fuzzy Linear Systems of Inequalities Fuzzy Relational Inclusions Applications in Fuzzy Linear Programming Bibliographical Notes

Chapter 5 Fuzzy Linear Systems - Dual Approach 5.1 Basic Concepts 5.2 Solving Fuzzy Linear Systems 5.3 Fuzzy Relational Equations 5.4 Dual Approach to Inverse Problem Resolution 5.5 Bibliographical Notes

98 108 109 112 113 113 117 126 128 130

Chapter 6 Direct and Inverse Problems in Intuitionistic Fuzzy Relational Calculus 131 6.1 Intuitionistic Fuzzy Relations. Compositions 131 6.2 Intuitionistic Fuzzy Matrices. Direct and Inverse Problems 137 6.3 Intuitionistic Fuzzy Relational Equations 139 6.4 Bibliographical Notes 140 Chapter 7 L-Fuzzy Finite Machines 7.1 L—Fuzzy Finite Machines. Behavior 7.2 Equivalences 7.3 Reduction and Minimization 7.4 Intuitionistic Fuzzy Finite Machines 7.5 Bibliographical Notes

143 143 158 162 163 167

C h a p t e r 8 F u z z y L a n g u a g e s in S y n t a c t i c P a t t e r n R e c o g n i t i o n 169 170 8.1 Finite L - F u z z y Acceptors and Regular L—Fuzzy Languages 8.2 Intuitionistic Fuzzy Languages in Syntactic P a t t e r n Recognition . . . . 177 8.3 Bibliographical Notes 183

Chapter 9 Applications as Inference Engine 9.1 Architecture of System with Artificial Intelligence 9.2 Fuzzy Linear System of Equations as Inference Engine 9.3 Intuitionistic Fuzzy Linear System as Inference Engine 9.4 Bibliographical Notes

185 185 187 196 198

Chapter 10 Software Description 10.1 Unary Matrix Operations 10.2 Binary Matrix Operations 10.3 Compositions 10.4 Inverse Problem 10.4.1 Max-Min Composition 10.4.2 Min-Max Composition

201 201 203 204 208 208 234

Contents

xi

10.5 Intuitionistic Fuzzy Relational Calculus 10.6 Engineering Examples

235 237

Appendix A

Solved Samples

247

Appendix B

List of Symbols

257

Appendix C

List of Abbreviations

259

Bibliography

261

Index

287

PART 1

FUZZY RELATIONAL CALCULUS

Chapter 1

Introduction

In this chapter we outline the most important subjects in fuzzy relational calculus and point out the place of the problems that we solve in the book. The presentation in Sections 1.1, 1.2, 1.3 and 1.4 follows in essence the wellestablished monographs or chapters in monographs by [De Baets (2000), Di Nola et al. (1989), Klir et al. (1997), Klir and Yuan (1995)]. Di Nola and co-authors in 1989 published the first monograph for fuzzy relational equations that is still actual. De Baets presents in a unified frame the basic problems for various relational compositions, fundamental results, applications and modern aspects. Klir and Yuan in their monograph give fundamental concepts, enriched with conceptual treatments, motivations, when and why the problems have appeared and mention various applied subjects specially for fuzzy relational calculus. 1.1

Basic Concepts

Here we recall some notions from algebra, lattice theory and fuzzy set theory. The terminology for algebra and lattice theory is according to [Gratzer (1978), MacLane and Birkhoff (1979), Davey and Priestley (1990)], for fuzzy sets and fuzzy relations we follow [Klir and Yuan (1995), De Baets (2000)]. The reader is supposed to be familiar with the basics of set theory. Nevertheless, some notations and notions follow. The crisp set of all elements a; of a given set B such that x satisfies the property P, is denoted by A, A= {x\x£B

A P(x)}.

An arbitrary crisp subset A of a given universal set E may be introduced by assigning the number 1 to each element of E that is element of A and by assigning the number 0 to the remaining elements of E. This assignment is known as characteristic function of the set A. The numbers 1 and 0 are used here only as convenient symbols and they do not have any numerical significance. 3

4

Fuzzy Relational Calculus - Theory, Applications and Software

Universal sets are always supposed to be crisp, regardless of whether we deal with their crisp or fuzzy subsets. Each fuzzy set A is introduced in terms of a relevant crisp universal set £ b y a membership function. The membership function of a fuzzy set A is usually denoted by \iA and it has the form: HA : E-> [0, 1]. The value HA(X) is the degree of membership of the element x G E in A. If X and Y are crisp sets, we let X x Y denote the Cartesian product of X and Y: XxY

= {(x,y)

\xeX,yeY}.

A binary relation R from a set X into a set Y is defined as a subset of X x Y, written as R C X x Y. For (a;, y) G R we also write xRy. A binary fuzzy relation R from a set X into a set Y is defined as a fuzzy subset of X x Y. In this book we operate with binary relations only. Next 'relation' is used instead of 'binary relation' and 'fuzzy relation' is used instead of 'binary fuzzy relation' with 'binary' being always suppressed. If X = Y we call R a relation on X. When the set (relation, respectively) is taken in conventional sense, we call it crisp set (crisp relation, respectively). Let R C X x Y be a crisp relation. The domain of R is defined to be the set Bom(R) = {x | x G X A (3y£Y)

{xRy) }.

The image (or range) of R is defined to be the set lm(R) = {y\yeYA

(3xeX)

(xRy)}.

A crisp relation R on the set P is called: Reflexive: if aRa for all a £ P. Antisymmetric: if aRb and bRa implies a — b. Transitive: if aRb and bRc implies aRc. These three properties are extended for fuzzy relations as well, see for details [Klir and Yuan (1995)]. A reflexive, antisymmetric and transitive crisp relation on P is called partial order relation and the corresponding set P is called partially ordered set or poset. We use the symbol < for partial order relation on a poset P. When P is a poset with respect to [0, 1 ] is a map denned with f(a A b)= min (f(a), f(b))

and

f(a V b) = max (/(a), /(&)),

then / : L —> [ 0, 1 ] determines a morphism of lattices. For a complete lattice L = (L, V, A, 0, 1) the Boolean logical operations A, V, =4> are extended as follows. A conjunctor C : L x L —» L is a binary operation on L with order-preserving partial mappings such that C(0, 1) = C(l, 0) = 0 and C(l, 1) = 1. A disjunctor D : L x L —> L i s a binary operation on L with order-preserving partial mappings such that D(0, 1) = D(l, 0) = 1 and 13(0, 0) = 0. An implicator I : L x L —* L i s a binary operation on L with orderreversing first partial mapping and order-preserving second partial mapping such that 7(0, 1) = 1(1, 1) = 1 and / ( I , 0) = 0. A negator N : L —» L is an order-reversing unary operation on L such that N(0) = 1 and JV(1) = 0. Specially, the negator N : [0,1] —> [ 0 , 1 ] , defined for any x G [0, 1] with N(x) = 1 — x, is called standard. When additional requirements are imposed on these operators (see [Schweizer and Sklar (1960), (1961), (1983)]), for instance - if a conjunctor is required to satisfy at least the axioms 1.1- 1.4 below, we are in the setting of so called i-norms. Similarly, if a disjunctor is required at least to be commutative, associative, to have 0 as neutral element, to satisfy monotonicity, then we are in the setting of t—conorms, cf. Axioms 1.5- 1.8 below.

Introduction

7

Formally, a t—norm (triangular norm or fuzzy intersection) is defined as a binary operation C on L that satisfies the following axioms for all a, b, d £ L: Axiom 1.1

C(a, 1) = a (boundary condition).

Axiom 1.2

b < d => C(a, b) < C(a, d) (monotonicity).

Axiom 1.3

C(a, b) = C(b, a) (commutativity).

Axiom 1.4

C(a, C(b, d)) = C(C(a, b), d) (associativity).

Axioms 1.1 - 1.4 are called axiomatic skeleton for t—norms. Often additional requirements are imposed on a t—norm. For instance when C is continuous function that leads to continuous t—norm. A t-conorm (triangular conorm or fuzzy union) is denned as a binary operation D on L that satisfies the following axioms for all a, b, d £ L: Axiom 1.5

D(a, 0) = a (boundary condition).

Axiom 1.6

b < d => D(a, b) < D(a, d) (monotonicity).

Axiom 1.7

D(a, 6) = D(b, a) (commutativity).

Axiom 1.8

D(a, D(b, d)) = D(D(a, b), d) (associativity).

Axioms 1.5 - 1.8 are called axiomatic skeleton for t—conorms. Continuity of D as additional requirement on a t—conorm leads to continuous t—conorm. The following two axioms are called axiomatic skeleton for fuzzy complement N: Axiom 1.9 Axiom 1.10

iV(0) = 1 and iV(l) = 0 (boundary conditions). For all a, b € L, if a < b, then N(a) > N(b) (monotonicity).

The next two axioms are among the most desirable requirements on fuzzy complement: Axiom 1.11

N is continuous function (continuity).

Axiom 1.12

N is involutive, i. e. N (N(a)) = a, for each a £ L.

The requirement for continuity guarantees that infinitesimal changes in the argument do not result in large (discontinuous) changes in the function. Some frequently used continuous t—norms on the real closed unit interval [0, 1 ], each defined for all a, b € [0, 1]), have specific names, as listed below. For more details the interested reader is referred to [Klir and Yuan (1995)]. Standard intersection (minimum): M(a, b) = min(a, b). Algebraic product: P(a, b) = ab. Bounded difference (Lukasiewicz t—norm): W(a,b) — max(a + 6—1, 0). It is easy to verify that for these t-norms the following chain of inequalities is satisfied for all a, b S [0, 1]: W(a, b) < P(a, b) < M(a, b).

(1.1)

8

Fuzzy Relational Calculus - Theory, Applications and Software

Frequently used continuous £-conorms on the real closed unit interval [0, 1], each denned for all a, b G [0, 1], have specific names, as listed below, see also [Klir and Yuan (1995)]. Standard union (maximum): U(a, b) = max(a, b). Algebraic sum: S(a, b) = a + b — ab. Bounded sum: V(a, b) = min(l, a + b). For all a, b € [0, 1] these t—conorms are ranked as U(a, b) < S(a, b) < V(a, b).

(1.2)

The last two chains of inequalities (1.1) and (1.2) have the following interconnectedness: W(a, b) < P(a, b) < M(a, b) < U(a, b) < S(a, b) < V(a, b). The operations standard negator, standard intersection and standard union are called standard fuzzy operations. They generalize the classical crisp set operations when the membership degrees belong to the real closed interval [0, 1]. A t—norm C and a t—conorm D are called dual triplet with respect to the fuzzy negator JV, written (C,D,N)), if and only if N(C(a, b)) = D(N(a), N(b)) and N(D(a, 6)) = C(N(a), N(b)). The following triplets are dual triplets with respect to the standard negator: (M, U, N) = (min(a,6), max (a, 6), N), {P,S, N) = (ab, a + b-ab,

N),

(W, V, N) = (max(a + 6 - l , 0), min(a+6, 1), JV). 1.2

Images and Compositions

The material in this section is essentially from [De Baets (2000), Klir and Yuan (1995)]. Images and compositions are among the basic operations on crisp or fuzzy relations. We present them first for the crisp case and after that for the fuzzy one. Let X and Y be crisp sets and R C X x Y be a crisp relation. The afterset xR C Y for the element x € X is the set of all elements y £ Y such that (x,y) € R, cf. (1.3) and Fig. 1.1:

9

Introduction

Fig. 1.1 The afterset xR C Y for x 6 X

R

Fig. 1.2 The foreset

xR = {y\y j/, where a; and y are strings of symbols from V = VN U VT such that x contains at least one symbol of VN; s £ V/v is the starting symbol. The language C(G), generated by grammar G, is the set of all strings of symbols in VT that can be obtained starting from s and using the production rules in P. In syntactic pattern recognition, features of patterns are represented by the elements of the terminal vocabulary VT. They are usually called primitives. Each pattern is represented by a string of these primitives and each pattern class is defined by a grammar that generates strings representing patterns in that class. For background in this area of classical pattern recognition see [Fu (1974)], where syntactic methods for pattern recognition by stochastic languages are studied. In many applications, structural information is inherently vague. In such applications, it is desirable to increase the descriptive power of syntactic pattern recognition by fuzzifying the primitives involved or by fuzzifying the production rules of the grammar and, consequently, also the language denned by the grammar. A fuzzy grammar Gf is defined by the quintuple Gf = (VN, VT, P, s, A),

,

(1.17)

where V/v, VT, P, S 6 VN are the nonterminal vocabulary, the terminal vocabulary, the finite set of production rules and the starting symbol, respectively, as given in (1.16); A is a fuzzy set defined on P. Every fuzzy grammar defined by (1.17) has an associated crisp grammar defined by (1.16). The language generated by the fuzzy grammar is a fuzzy set C(Gf). For each string x £ C(Gf), we compute its membership degree C{Gf){x)=

max

min A(ptfc),

(1.18)

l [0, 1] } ,

HA • E

—> [0, 1] is called membership function and the value

£ [0, 1 ] is called degree of membership of x in A.

Fuzzy sets of the type introduced in Definition 2.1 are also called ordinary fuzzy sets. Reasonable generalizations of this definition concern so called L—fuzzy sets and B—fuzzy sets. [Goguen (1967)] enlarges the range of Definition 2.1, substituting [0, 1] by a suitable poset L. The most interesting L—fuzzy sets are considered when L is the underlying set of a lattice. Goguen defines an L— fuzzy set A on a universe E as A = {(x,

HA(X))

\xGE,

fiA:

E

->

L},

where L is the underlying set of a complete distributive lattice and HA '• E —> L is a map. L-fuzzy set is involved as more abstract concept. Calculus with them is much closer to the nature of symbolic computations. Operating with L—fuzzy sets is easier if we apply a (residuated or bijective) lattice morphism / : L —> [0, 1] that permits computing by f(fiA(x)) £ [0, 1]. Residuation in fuzzy algebra is subject of [Cuninghame-Green and Chehlarova (1995)]. J5-fuzzy sets are defined over Boolean lattice in [Brown (1971)]. A Boolean lattice is a distributive lattice such that every element of it has a complement. For Boolean lattices see [MacLane and Birkhoff (1979), Gratzer (1978)]. Remark 2.2 2.2

From here on for the (L—)fuzzy set A we write A instead of A.

Fuzzy Relations - Compositions and Properties

The most important problems in fuzzy relational algebra arise from applications, where various compositions of fuzzy relations as well as their properties are implemented.

32

Fuzzy Relational Calculus - Theory, Applications and Software

Applications in fuzzy logic, artificial intelligence, approximate reasoning, relational databases, fuzzy control, etc., need a powerful fuzzy relational calculus, as it is mentioned in the literature, see [Guu and Wu (2002), Di Nola et al. (1989), De Baets (2000), De Baets and Kerre (1993), (1994a), Dubois and Prade (2000), Klir et al. (1997), Pedrycz (2000)]. In this section we present and study various compositions of fuzzy relations and their properties. The analytical expressions that we obtain here are used in next chapters and parts to provide existence of extremal solutions of fuzzy relational equations and fuzzy relational inclusions.

Fuzzy relations Any fuzzy relation is a fuzzy set defined on a Cartesian product as universal set. Let X and Y be two nonempty sets. Definition 2.2 A fuzzy relation R between X and Y is a fuzzy set R on the Cartesian product X x Y. An L—fuzzy relation R between X and Y is an L—fuzzy set on the Cartesian product X xY. If R is (L—)fuzzy relation between X and Y, then the set X x Y is called support oiR. Definition 2.2 means that each fuzzy relation R has a description of the form: R={((x,y),

»R(x,y))\(x,y)eXxY,(xR:

X xY

-> [0, 1 ] }

(2.9)

and each L-fuzzy relation R has a description of the form: R={((x,y),

vm{x,y))\{z,y)eXxY,

fiR : X XY

- » L].

(2.10)

When computing with L—fuzzy relations, it is preferable to work with suitable (residuated or bijective) lattice morphism / : L —> [ 0, 1 ] that permits to calculate f(Hn(x, y)) S [0, 1] and to operate with it. Since fuzzy relations are special case of L—fuzzy relations, all results that are valid for L—fuzzy relations, are also valid for fuzzy relations. Remark 2.3 From here on for the (L-)fuzzy relation R from X to Y we simply write R C X x Y, when there is no danger of confusion. In what follows 'fuzzy relation' means L—fuzzy relation. The fuzzy relations R and S are called compatible if they are defined on the same support. Let RC X xY and 5 C X x Y be compatible fuzzy relations. Then i) R C S HR{X, y) < ns{x, y)

for all pairs

(x, y)eX

xY.

ii) R = S HR(X, y) - Hs{x, y)

for all pairs

(x, y) £ X

xY.

33

Fuzzy Relations. Direct Problem Resolution

Unary operations Some unary operations are specific for fuzzy relations, other unary operations are obtained as fuzzy extension of their crisp counterparts. The operation, called inverse (or transpose) of a fuzzy relation, is among the specific unary operations for fuzzy relations. Definition 2.3 The fuzzy relation R"1 C Y x X is called inverse of the fuzzy relation RCX xY ii: (2.11)

R-\y,x) = R(x,y) for all pairs (y, x) € Y x X. Note that for each (y, x) € Y x X, R~l{y,x) = R(x,y) A (R(x, y) a (R(x, y) A S(y, z))). Now, for the right hand side of the last inequality, according to the first expression aa(a A b) > b in (2.4) from Lemma 2.2 ii), if we take a = R(x,y),

b = S(y,z),

we have

R{x, y) a {R(x, y) A S(y, z)) > S(y, z), hence SCR~la{R»S). ii) For R C X xY and S C Y x Z, we denote by T the co-standard composition T = RoS with T C X x Z. Let P = fi"1 eT, where P CY x Z. From Definition 2.3 and Definition 2.7 we obtain P(y,z)= V (R-\y,x)eT(x,z))=

V (R(x,y)e{Ro S)(x,z))

= Jx (R(X, y) e ^ Ay (R(x, t) V S(t, *))) )

= V fJR(x,y)£f(JR(a;,2/)v5(2/,z))Af x€ X \

\

A (R(x,t)\/S(t,z))))) .

\t£Y, t^y

JJJ

Fuzzy Relations. Direct Problem Resolution

37

We apply to the last expression the first inequality from (2.8), Lemma 2.5, namely ae(b Ad) < aeb, when a = R{x,y), b=R(x,y)VS(y,z), A (R{x,t)V S(t,z)). d= t&Y, tjty

We obtain P(y, z) < v (R(x, y) e (R(x, y) V S(y, z))). Now, for the right hand side of the last inequality, according to the first expression ae(a V b) CLij)mXn with elements -.ay- = 1 - dij

(2.25)

for each i, 1 < i < m, and for each j , 1 < j < n, is called complementary to A = (flij)mxn-

If the matrix A = (ar,) m x n is on a complemented lattice L, then each element -i ay of the complementary matrix ->A = {~^o-ij)mxn is the complement of a^ in L. When a fuzzy relation R C X x Y on [0, 1] or on complemented lattice is represented by a matrix R = (r xy ), we obtain the matrix representation of the fuzzy complement -• R by replacing each element with its complement. The resulting matrix -i R is the complement of the given matrix R = (rXy)Simple binary operations For matrices of the same type we introduce two simple binary operations. Definition 2.11 For the matrices A = (a,j) mX7l and B = (b,j) mxn :

Fuzzy Relations. Direct Problem Resolution

i) the standard fuzzy union of A = ( a y ) m x n and B = (bij)mxn C = {cij)mxn with elements c ij = V(a b

(3.3)

53

Fuzzy Relational Equations

Expression (3.3) means that the coefficients in (3.2) are classified in three groups - greater, equal or smaller than b, depending on their contribution to solve the equation. We often use a linguistic description of the coefficients in A*. In A* a coefficient a* — 0 is called S—type coefficient (from smaller, because aj < b), a coefficient a* = b is called E—type coefficient (from equal, because aj = 6,), a coefficient a*j = 1 is called G—type coefficient (from greater, because aj > b). The equation A*»X = b with A* computed by (3.3) is called associated equation to the equation A • X = b. We denote by (A* : b) the matrix with elements a* determined by (3.3) and b as given in (3.1). The matrix (A* : b) is called augmented matrix for the associated equation A* • X = b. Obviously (A* : b) depends on both A and b. In what follows, whenever A • X = b is given, we suppose its augmented matrix (A* : b) is determined. Solving fuzzy linear equations Two fuzzy linear equations are called equivalent, if each solution of the first equation is a solution of the second equation and vice versa. Lemma 3.1

Any fuzzy linear equation is equivalent with its associated equation.

The proof follows from the properties of the operations in L. Since the proof is not difficult, we omit it. Lemma 3.2

Let the equation ( b aj = b aj < b

Xj = b Xj G [ b, 1 ] Xj e 0

Xj for aj A Xj < b

= b

Xj e [0, b] Xj 6 L Xj e L

Table 3.1 illustrates Lemma 3.2. Corollary 3.1 given.

Let the equation A • X = b and its associated A* • X = b be

i) A • X = b is solvable iff there exists at least one coefficient a*j > b, 1 < j < n, in A* • X = b. ii) A • X = b does not have solution iff a* = 0 for each j , 1 < j < n, in A* • X = b. Hi) Time complexity function for establishing solvability of A • X = b is O(n). Proof. i) Let the equation (ai A xx) V . . . V (a n A xn) = b be solvable. It means that aj AXJ < b for each j , 1 < j < n, and hence V

(a, Aij)

< 6,

but there exists at least one index t, 1 < £ < n, such that at Axt — b. The last implies that there exists at least one coefficient at > b, i.e., a^ is of G— or E- type. The converse implication is obvious. ii) Follows from Lemma 3.2 iii). iii) Concerning the time complexity function, according to Corollary 3.1 i) we search for one E— or G—type coefficient among the elements of the set { CL*J | 1 < j < n } which cardinality does not exceed n. Corollary 3.2

Any solvable equation A • X = b:

i) has greatest point solution XgT = (xj)nxl,

Xj = {h; [_ 1,

i f

where

"i>b otherwise

.

(3.4)

55

Fuzzy Relational Equations

ii) The greatest solution XgT = {Xj)nxl has representations:

with components computed by (3-4)

XgI = Atab=(A*)tab.

(3.5)

Hi) The time complexity function for computing XgT is 0(n). Proof. i) For any solvable equation the existence of X g r follows from Theorem 2.7 ii). In order to find the components Xj of the greatest solution, we distinguish two cases for any term aj A Xj, namely: a,j > b and aj < b. 1. aj > b and Xj G [0, 6] imply aj A Xj < b. In this case Xj = b is the greatest value for XJ. 2. If aj < b and Xj € [0, 1} then a,j A Xj < b holds. The greatest value for Xj is Xj = 1. ii) The expression (3.4) for the components of the greatest point solution equals X g r = A*a b. The validity of formulae in (3.5) follows from Theorem 2.7 ii) and expression (3.3). We denote by (.4* : 6) the set of all matrices (A : B) that have the same augmented matrix (^4* : b). All equations with augmented matrix (A* : b) G (A* : b) are equivalent. The interpretation of (A* : b) leads to so called solution-set invariance. From practical point of view (^4* : b) describes the range of admissible changes in input data with respect to the output that do not influence on the model description of the process. If the input data of a process are described by the matrix A and the output data - by b, then the solution of the initial problem is invariant for any process that has the same augmented matrix (A* : b). In order to compute the lower solutions of the equation A • X = b we introduce a suitable help matrix H = (hj) iXn with elements: h j

f 0, - \ b ,

if if

aj aj

b •

(3 6)

'

The equality A • X = b should be satisfied by any term aj A Xj when aj > 6, i.e, with hj = b and thus each hj = b marks a potential lower solution. Hence H is used twofold: to indicate all coefficients that contribute A • X = b to be solved and to drop all coefficients that do not contribute the equation to be satisfied. The help matrix obeys some interesting properties as described in the corollaries below. Corollary 3.3

For any fuzzy linear equation A • X = b:

56

Fuzzy Relational Calculus - Theory, Applications and Software

i) The matrices (A : b) and (A* : b) have the same H. ii) The time complexity function for computing H is 0(n). The proof follows from constructions of (A* : b) and H. In what follows the property density of ordering (introduced for I by [Butkovic et al. (1987)]) is used: (Va, b G L) (a < b) => (3c G L) (a < c < b). Let the number ( | G | + |£?|) denote the sum of all G- and iJ-type coefficients in A • X = b. Corollary 3.4

/ / the equation A • X = b is solvable, then:

i) Each lower solution X\OVI has one component equal to the right-hand side constant b and all other components are equal to 0. ii) The number of all its lower solutions is (| G \ + \ E | ) . Hi) The number of all its maximal interval solutions is (| G \ + \ E | ) . iv) The components of each maximal interval solution X m a x are among L, [ 0 , 6 ] , [ 6 , 1 ] , b.

v) The greatest, lower and maximal interval solutions are computable in polynomial time.

Proof. i) Since A • X = b is solvable, there are only terms with a, AXj < b in the left side of (3.2). Equality is possible only for G— and E—type coefficients. In this case any o* = 1 or a* = b determines exactly one lower point solution with Xj = b and all other components equal zero. We use the matrix H to produce the set of all lower solutions of the equation A • X = b. To achieve this aim, we introduce an index s, 1 < s < (\G\ + \E\) and let Hs = (hj(s))ixn be a matrix, obtained from H as described below: . h {s)=

J

\

( b, 0,

if hj = b and j = s otherwise

In this manner we determine (| G | + | E \) matrices Hs from H and each matrix Hs generates a lower point solution Xi o w s = (XJ) = (ffs)* with components / b, if j = s ii) Follows from Corollary 3.4 i) and (3.7). iii) Follows from Definition 3.3 and Corollary 3.4 i).

(3.7)

57

Fuzzy Relational Equations

iv) Follows from expressions (3.4), (3.7) and from Definition 3.3. v) Follows from the computability of analytic expression (3.4) for the greatest solution and according to (3.7) for the (\G\ + \E\) lower point solutions. The time complexity function for computing these solutions is O(n). Corollary 3.5 Proof.

There exists a polynomial time algorithm for solving A • X = b.

Follows from Corollaries 3.1, 3.2, 3.3 and 3.4.

•

These results are summarized in the next Algorithm 3.1. Algorithm 3.1

for solving A • X = b.

1. Enter the matrix A\xn and the right-hand side constant 6. 2. Obtain (A* : b): determine a* for each j , 1 < j < n, according to (3.3). 3. If a* = 0 for each j = 1,..., n, then the equation A • X = b has no solution. Go to Step 5. 4. Determine XST according to (3.4), H according to (3.6), the lower solutions according to (3.7). From the set of the lower solutions and from X gr determine the set of the maximal interval solutions according to Definition 3.3. 5. End. Example 3.1

In the fuzzy algebra I the equation (0.3 A xi) V (0.1 A x2) V (0.5 A x3) = 0.3

has solution, because there exist coefficients that are greater than the right-hand side constant b = 0.3 (a3 = 0.5) or equal to it (ai = 0.3) and: (A* : b) = ( 0.3

0

1 : 0.3 ) .

The greatest solution XgT according to (3.4) is:

Xgr

/0.3\ (1 = A a b= 0.1 \ a 0.3 = 1

\

1

V 0.5 7

V 0.3 /

The help-matrix is H = (0.3 0 0.3) and it splits in two matrices Hi = (0.3

0 0 )

and H2 = ( 0 0 0.3 ) ,

.

58

Fuzzy Relational Calculus - Theory, Applications and Software

V Xlow2W, 0,0.3)

xlmil(0.3,o,o) /

i I

/ f

/ >

/ >

/

y

y

'

g&> L °-3)

X

^ /

y-

*•

-™-low2

6

j

j

—,

-j

X

Xmaxl

™"*>

-

i

Complete solution set Fig. 3.1

Solutions of (0.3 A x i ) V (0.1 A x2) V (0.5 A x3) = 0.3

giving two lower solutions X\owi =

/0.3\ 0

V0 /

/0 and X l o w 2 =

\ 0

\ 0.3 /

and two maximal interval solutions / [0.3, 1] \ / [0, 1] \ ^maxi= [0, 1] andXmax2= [0, 1] . V [0, 0.3] / \ 0.3 / Figure 3.1 illustrates XgI, Xi ow i, XiOW2, X m a x i and X m a x 2, the complete solution set. Density of ordering means that if Xi and X2 are solutions of the equation and if X\ < X2 then any X with X\ < X < X2 is also a solution. On this base

59

Fuzzy Relational Equations

maximal interval solutions are determined by a lower and the greatest solution. The overlap of the solutions is also visible, see the complete solution set in Figure 3.1. 3.3

Fuzzy Linear Systems of Equations

We study fuzzy linear systems of equations (FLSE) ( a n A xi) V---V ( a ] B A xn)

= bx

( a m i A x{) V - - - V (amn

=

, A xn)

(3-8)

bm

simply written as A»X = B, where A = (aij)mXn stands for the matrix of coefficients, X = (XJ) n x i stands for the matrix of unknowns, B — (bi) m x I is the right-hand side of the system. The matrices A, X, B are on L, i.e., for each i £ { l , . . . , m } and for each j £ {!,-• • ,n}, the elements aij, Xj, bi £ L and the composition is the standard one. The purpose of this section is to develop universal method for solving FLSE, resulting in: 1. A necessary and sufficient condition for consistency of FLSE as analogue of the general test for consistency of a system of linear equations in linear algebra. 2. Universal and exact method and algorithm for obtaining complete solution set of the FLSE. We remind that the general test for consistency of a system of linear equations in linear algebra [MacLane and Birkhoff (1979)] asserts that a system of linear equations is consistent, iff the rank of the coefficient matrix is equal to the rank of the augmented matrix. In this chapter 'system' means FLSE. We develop universal and exact method providing algorithm for solving the system A • X = B. To achieve this aim, we simplify the system A • X = B to obtain it in so called normal form. Rather than work with the actual system A • X = B, we work with its associated system and introduce a matrix, whose elements capture all the properties of the original system. Manipulation of this new matrix is carried out by performing a sequence of operations that bring the matrix into a new form. Once in this form, the complete solution set is easily found. The main Theorem 3.1 provides efficient algorithm for solving fuzzy linear systems of equations. Corollary 3.6 gives a necessary and sufficient condition for consistency of the system.

60

Fuzzy Relational Calculus - Theory, Applications and Software

The first ideas were published in [Peeva (1985a), (1992)] and next developed in [Peeva (2002b)].

3.3.1

Basic

Notions

For X = {xj)nXi and Y = (j/j) nx i the inequality X < Y means that Xj < yj for each j , 1 < j < n. We define types of solutions of the system A • X = B, as well as a classification of the systems according to their solutions. Definition 3.4

Let the system A • X — B in n unknowns be given.

i) X° = (z°)nxi with x° £ L, when 1 < j < n, is called a (point) solution of the system A*X = B if A»X° = B holds. ii) The set of all point solutions of A • X = B is called complete solution set and it is denoted by X°. If X° ^ 0 then A • X = B is called consistent, otherwise it is called inconsistent. hi) A point solution X°ow e X° is called a lower (point) solution of (3.8) if for any X° € X° the relation X° < X ° w implies X° = X?ow, where < denotes the partial order, induced in X° by the order of L. Dually, a solution X° £ X° is called an upper (point) solution of A • X = B if for any I ° e X the relation X° < X° implies X° = X°. When the upper point solution is unique, it is called greatest (or maximum) point solution. iv) (Xi,... ,Xn) with Xj C L for each j , 1 < j < n, is called an interval solution of the system A • X = B if any X° = (x°)nXi with x° € Xj for each j , 1 < i < n, implies X° = (x°)n x l e X ° . In particular from Definition 3.4 for m = 1 we obtain Definition 3.2. If bi = 0 some i £ { 1 , . . . , m } then the i—th equation in A • X = S could be always satisfied. In accordance with traditional linear algebra, a FLSE is called homogeneous when 5 = 0, otherwise it is called inhomogeneous. Solving a homogeneous system is trivial. It has unique lower solution and unique upper (maximal) solution. Remark 3.1 Any consistent system A • X = B has unique upper point solution, i.e., greatest solution XgT = AtaB, according to Theorem 2.7 ii). The next example illustrates some of the notions from Definition 3.4. Example 3.2 (0.3 A xi) (0.7 A xi) (0.6 A xi) (0.8 A xt) (0.4 A Xi)

In I the system V (0.9 A x2) V (0.0 A x2) V (0.1 A x2) V (0.7 A x2) V (0.1 A x2)

V (0.9 A x3) V (0.9 A x3) V (0.0 A x3) V (0.4 A x3) V (0.2 A x 3 )

V (0.4 A x4) V (0.4 A x 4 ) V (0.8 A x4) V (0.2 A xA) V (0.5 A x 4 )

V (0.2 A V (0.7 A V (0.5 A V (0.7 A V (0.1 A

x5) x5) x5) x5) x5)

= = = = =

0.9 0.9 0.7 0.7 0.5

(3.9)

61

Fuzzy Relational Equations

is consistent. Its greatest solution is Xgr, it has a point solution X°: / 0.7 \ / 0.7 \ 1 1 0.3 XgT = 1 , X° = 0.95 , 0.7 0.7

\ 0.34 /

V1 /

and three lower point solutions (see Example 3.11 for details)

^lowi =

/ 0.7 \

/ ° \

0

0.7

0.9

,

X[ov/2 —

0.9

/ ° \ 0 ,

X[OW3 =

0.9

0.7

0.7

0.7

\ 0 /

V0 /

\ 0.7 /

Lemma 3.3 [Di Nola et al. (1989)] The complete solutions set X° of A»X = B is an upper semi-lattice (X°, ...>bm,

is said to be in a normal form. The notion normal form of a FLSE is introduced in [Cechlarova (1995)] for FLSE in the fuzzy algebra I. Any FLSE could be rearranged to obtain a FLSE in a normal form. According to Lemma 3.5 both FLSEs are equivalent. Lemma 3.6 Let the system A* X = B be given. The time complexity function for obtaining its equivalent system in normal form is polynomial in the number m of the equations in the system. Proof. According to [Aho et al. (1976)] the time complexity function for obtaining the sequence b\ > 62 > ... > bm is the solution of the recurrence equation T( \ = / °' 1 ' \ (m-l)+T(m-l),

if m = 1 if m > l

for the number of comparisons made between the elements to be sorted. The solution is T(m) = 7;Tn(m - 1), which is O(m2). • Various sorting algorithms for the set {61, ..., bm} as a set with linear order are presented in the literature, see for instance [Aho et al. (1976)]. Depending on the method, the time complexity for sorting could be O(mlogm), O(m2), etc. The software realization as described in Part 3 for FLSE in n unknowns and with m equations in the fuzzy algebra I needs O(mn) operations, because it is based on physical rearrangement of the equations. Remark 3.2 In this book, without loss of generality, we always assume that the FLSE is in its normal form.

Fuzzy Relational Equations

63

Solving the system in normal form instead of the original one is the first step that provides reduction of the amount of computations. When the right-hand side constants are ranked non-increasingly some of the equations of the system may be automatically satisfied. The next example illustrates the advantages of reasoning when the FLSE is in normal form. Example 3.3

Solve in I the system (0.4 A xi) V (0.5 A x2) = 0.4 (0.8 A si) V (0.6Ai 2 ) = 0.7 . (0.5 A xi) V (0.9 A x2) = 0.5

(3.10)

If we do not rearrange the equations in the system (3.10), the conventional reasoning requires to solve each equation and after that to investigate equations simultaneously. First equation could be satisfied by any of its two terms, namely (0.4 A xi)

or (0.5 A x 2 ).

If we satisfy the equation by the term (0.4 A X\), for the components of the solution (xi, a?2 ) it implies that zi G [0.4, 1] and x2 G [0, 0.4].

(3.11)

When we satisfy the equation by the term (0.5 A x2), the components of the solution (xi, x2) should be zi G [0, 1] and x2 = 0.4.

(3.12)

For the second equation only the term (0.8 A i i ) contributes to solve the equation, because 0.6 A x2 < 0.7 for each x2 € [0, 1]. Hence components of the solution of second equation are xx =0.7

and x2 € [0, 1].

(3.13)

The third equation could be satisfied by any of its two terms, (0.5 A xi)

or (0.9 A x 2 ).

If we satisfy the equation by (0.5 A i i ) , then xi £ [0.5, 1] and x2 G [0, 0.5].

(3.14)

If the equation is satisfied by (0.9 A x2), then X! e [0, 1] and x2 = 0.5.

(3.15)

64

Fuzzy Relational Calculus - Theory, Applications and Software

In order to obtain the complete solution set of the system (3.10), we consider equations simultaneously. The analysis results in the following branches: 1. (3.12), (3.13) and (3.15) do not lead to a solution. First component should be Xi = 0.7 but we do not have choice for the second component because x2 = 0.4 according to (3.12) differs from x2 = 0.5 according to (3.15). 2. (3.12), (3.13) and (3.14) lead to a point solution xi = 0.7, x2 = 0.4. 3. From (3.11), (3.13) and (3.15) we do not obtain a solution. The first component should be x\ = 0.7, but we do not have choice for the second component because x2 = 0.5 £ [0, 0.4]. 4. If (3.11), (3.13) and (3.14) hold we have xi = 0 . 7

and x2 e [0, 0.4].

(3.16)

Now we form the complete solution set: the point solution in case 2 with components Xi = 0.7, x2 = 0.4 is included in the interval solution (3.16) obtained in case 4 and (3.16) is the maximal interval solution of the system, X =

- ([0°0 7 4])-

< 3 - 17 >

It is not difficult to see that in principal this method of reasoning should not lead to efficient procedure. If we rearrange the equations in (3.10) as stipulated in Remark 3.2, we obtain the equivalent to (3.10) system in normal form

(0.8 A xi) V (0.6 A x 2 ) = 0.7 (0.5 A xi) V (0.9 A x2) = 0.5

(3.18)

(0.4 A x / ) V (0.5 A x 2 ) = 0.4

In (3.18) the analyzes of the first equation (0.8 A xi) V (0.6 A x 2 ) = 0.7 leads to (3.13) for the components of the solution (on this step). The choice Xi = 0.7 satisfies not only of the first, but the second and third equations by the terms 0.5 A 0.7 = 0.5 and 0.4 A 0.7 = 0.4, respectively. Thus the system is consistent, and rearrangement of the equations simplifies reasoning. Concerning the second component of the solution in (3.13), the reasoning leads to x2 € [0, 0.4]: the term 0.6 A x2 does not contribute to satisfy the first equation in (3.18). If we select x2 > 0.4, the third equation should not be satisfied. But x2 G [0, 0.4] does not produce any contradiction in the system. The maximal interval solution is (3.17).

65

Fuzzy Relational Equations

B. Augmented matrix of the associated system For the FLSE (3.8) any coefficient ay > 6, corresponds to a term a^ A Xj that may contribute to satisfy the i—th equation, while any coefficient ay < bt does not contribute to solvability of this equation because ay A Xj < bi for each Xj € L. For this reason we distinguish coefficients that may contribute to solvability of the system from these coefficients that do not contribute to its solvability. We assign to the system A* X = B a, new system A* • X = B with coefficient matrix A* = (ay), determined from A and B according to the next expression:

{

0, b^

if if

ay < bi ai3=bi .

(3.19)

1, if ay > bi In A* a coefficient ay = 0 is called 5—type coefficient, a coefficient a*j = 6j is called E—type coefficient, a coefficient a^ = 1 is called G—type coefficient. The reasons for notations are linguistic: 5 comes from smaller, because ay < bi, E from equal, because ay = 6, and G - from greater, because ay > bi. To any FLSE we assign a new FLSE called associated system and a matrix called augmented matrix, as described below. The system A* • X = B

(3.20)

where A* = (ay) has elements ay determined by (3.19) is said to be associated FLSE to the FLSE A • X = B. The matrix (A* : B) having A* = (a*j) and with last column B from (3.20) is called augmented matrix of the system A* • X = B. The elements of (A* : B) depends both on A, and on B. Lemma 3.7

Each FLSE is equivalent with its associated FLSE: A*X

Proof.

= B^A**X

(3.21)

= B.

For the systems A • X — B and A* • X = B let the matrices A = (aij)mxn:

A* = (a*j)mXn,

X = {Xj)nxl,

B = (fcj)TOxl

be of the given type. Let / = { 1 , . . . , m} and J = { 1 , . . . , n} be index sets. For consistent systems we shall prove that any solution X° of A*X = B is also a solution of A* • X = B and vice versa:

A • X° = B^A* >X° = B.

66

Fuzzy Relational Calculus - Theory, Applications and Software

For the 'if part: if X° is a point solution of A • X = B, then A • X° = B holds. It means that for each i £ I, the i—th equation of the system is satisfied, namely V (ay A X*}) = bt.

For each term in the i—th equation we have ay A i ° < bi, and supplementary there exists at least one term, such that ay A x® = bi. Hence, for each i € I: ( V (ay Aa$) f V7 ( a y A a;° ) < 6i J & (3 j G J)

( ( ( a i j = hi) A ( 1° > bi) ) V ( K - > bi) A ( ^ = 6i)))

^ (Vjia*! A x*) < bA k (3j G J)

{( (a*, = h) A(x°j > h)) V ((ay = 1) A (i? = M ) )

4^ ( . ^ ( 4 A xoj ) < bt ) & (3j G J) (ay A 1° = 6i) . For the inverse implication, we let assume that the associated system A*»X = B has a point solution X°. Then (Vi G /) ( v (ay A a;°) < bt \ & (3 j G J) ( ay A x] = bi)

o (Vt G /) f.V^oy A iP) < 6i) & (3 j G J) ((ay A x°) = bi) , i.e., X° is a solution of A • X — B. When A • X = B (or A* • X = B) is inconsistent, the proof is trivial.

•

Lemma 3.7 permits to investigate the associated system instead of the original one. The associated system A* • X = B captures all the properties of the original system, but it is easier to analyze A* • X = B instead of A • X = B.

67

Fuzzy Relational Equations

Example 3.4

(A* : B) for the system (3.9) in Example 3.2 is: / 0 0 (A* : B) = 0 1 V0

0.9 0 0 0.7 0

0.9 0.9 0 0 0

0 0 0 0 1 0 0 0.7 0.5 0

: : : : :

0.9 \ 0.9 0.7 . 0.7 0.5 /

(3.22)

Augmented matrix is the next step for reduction of the computational effort. For any 5-type coefficient a*j = 0 and for each Xj € L, it holds a*j A Xj — 0 A Xj < bi.

It means that when searching lower solutions, we may ignore all S—type coefficients, because they do not contribute to solvability of the system. But each a* > 6j provides a way to satisfy the i—th equation and it may lead to a lower solution. Let |o*| stand for the number of a*j > &* in the i—th equation of A* • X — B. When in the i—th equation a*j = 0 for each j , 1 < j < n, we write \a*\ = 1 in the next product (3.23). Then the number of lower solutions of A* • X = B does not exceed the estimation m

PiVl = JJ|a*|.

(3.23)

t=i

Lemma 3.8 Let the system A* X = B in n unknowns and with m equations be given. The time complexity function for computing (A* : B) is polynomial, it is 0(mn). Proof. For computing A* with respect to B according to (3.19) we need mn comparisons between a,-j and 6,, when 1 < i < m and 1 < j < n. D Lemma 3.9 For each consistent FLSE A • X = B with associated A* • X = B we have for the greatest solution: Xgr = AtaB

= (A*)taB.

(3.24)

Proof. According to Theorem 2.7 ii) any consistent FLSE A»X = B has (unique) greatest solution Xgr = A* aB. Its associated A*»X = B also has (unique) greatest solution X*r = (A*)1 aB. Since both systems are equivalent, Xgx = X*T and hence XgT = AtaB = (A*)laB.

•

68

Fuzzy Relational Calculus - Theory, Applications and Software

Example 3.5 For the system (3.9) we have: / 0.7 \ 1 1 Xgr = AtaB = (A*)taB = 0.7 \ 1 / Solution-set invariance Let the matrices A\ and A2 be of the same type. The matrices A\ and A2 are called solution-set invariant with respect to B if for the FLSEs Ai»X

=B

and A2 • X = B

it holds (Al : B) = (A*2 : B). Example 3.6 The FLSEs Ax • X = B and A2 • X = B, given in I with 1

_ / 0.3 0.2 0.8 \ " V 0.7 0.4 0.2 ) '

2

_ / 0.4 0.1 0.9 \ ~ { 0.8 0.7 0.2 ) '

( 0.5 \ \ 0.2 ) '

have the same augmented matrix, namely {A

• B)

=

/ 0 0 1 : 0.5 \ [l 1 0.2 : 0.2 ) •

The systems Ai • X = B, A2 • X = B and 4* • X = B are equivalent, and A\, A2 and A* are solution-set invariant matrices with respect to B. For a system A • X — B, the set of all solution-set invariant matrices A with respect to B is denoted by (.4* : B). Main

simplifications

The next simplification steps over the FLSE, following after the simplification steps A. and B., are: C. Selection of these coefficients that contribute to solve the system. D. Creating a help matrix and a dominance matrix.

Fuzzy Relational Equations

69

C. Selection of coefficients In this book we suppose that the conventions given below are valid for the FLSE. Conventions 3.1 1. The system A»X = B has n unknowns and m equations, it is with coefficient matrix A = (aij)mxn, matrix of unknowns X = {xj)nx\ and right-hand side B = (6i)mxl2. The system A • X = B is in normal form. 3. The associated system A* • X = B for the system A • X = B is obtained. 4. For each j , j = 1, - • •, n, A*(j) = (a*j) m x i denotes the j—th column of A* and a*- denotes the i - t h element (1 < i < m) in A*(j). Theorem 3.1

Let the system A» X = B be given.

i) If A*(j) contains G—type coefficient a*kj = 1 and k (1 < k < m) is the greatest number of the row with a^ — 1 in A*(j), then (a) for each i, 1 < i < m, a^ in A*(j) and Xj G [0, 6/j] imply a*j AXj < bt. (b) a*j in A*{j) and Xj = bk imply a*j A Xj = 6j.• for each i, 1 < i < k with a*j > bi = bk, • for each i, k < i < m with a*j = bi. ii) If A*(j) does not contain any G—type coefficient, but it contains E—type coefficient a* • = br and r (1 < r < m) is the smallest number of the row with a*ri = br in A*(j), then (a) for each i, 1 < i < m, a*j in A*(j) and Xj e L, imply a*,- AXj < bt\

(b) for each i, r if bU•