Monadic Second-Order Model Theory

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Monadic Second-Order Model Theory Achim Blumensath

Monadic Second-Order Model Theory Achim Blumensath

ab BRNO 2023

Achim Blumensath [email protected]

This document was last updated 2023-12-19. The latest version can be found at www.fi.muni.cz/~blumens

Copyright 2023 Achim Blumensath This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Contents A. Fundamentals

1

I Logics and Their Expressive Powers 1. Structures and logics . . . . . . . . . . . . . 2. Simple translations between logics . . . . . 3. Theories and back-and-forth arguments . . 4. Operations for monadic second-order logic 5. Operations for first-order logic . . . . . . .

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3 4 11 15 23 51

II Finite Words 1. Words and languages . . . . . . . 2. Semigroups and Green’s relations 3. Simon’s Lemma . . . . . . . . . . 4. Regular languages of finite words . 5. First-order logic . . . . . . . . . .

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61 61 63 73 86 93

III Infinite Words 1. Ramsey theory . 2. The theory of ω 3. ω-semigroups . 4. ω-automata . . .

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107 107 117 120 130

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IV Parity Games 145 1. Positional games . . . . . . . . . . . . . . . . . . . . . . . . . . 145 2. Reachability games . . . . . . . . . . . . . . . . . . . . . . . . . 147

monadic second -order model theory 2023-12-19

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©achim blumensath

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3. Gale-Steward games . . . . . . . . . . . . . . . . . . . . . . . . 4. Regular games and parity games . . . . . . . . . . . . . . . . . 5. The modal µ-calculus . . . . . . . . . . . . . . . . . . . . . . . V Trees 1. Composition theorems . . . . 2. Tree automata . . . . . . . . . 3. The Muchnik Iteration . . . . 4. Löwenheim-Skolem theorems 5. The Cantor Topology . . . . . 6. Counting quantifiers . . . . . .

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157 159 179

193 . 193 . 206 . 226 . 233 . 238 . 253

B. Structure Theory

275

VI Linear Orders 1. Dense and scattered orders 2. Partition theorems . . . . . 3. Interpretations . . . . . . . 4. Regular linear orders . . . 5. Choice functions . . . . . . 6. Uniformisation . . . . . . . 7. First-order logic . . . . . .

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277 277 292 301 306 316 337 364

VII Sparse Structures 1. Spanning forests . . . . . . 2. Sparse hypergraphs . . . . 3. Translating GSO into MSO 4. Sparse distributions . . . .

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383 . 383 . 403 . 412 . 419

VIII Tree-Width and Graph Minors 453 1. Tree-decompositions . . . . . . . . . . . . . . . . . . . . . . . 453 2. Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 3. Brambles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

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4. The Excluded Grid Theorem . . . . . . . . . . . . . . . . . . . 482 5. Branch-decompositions and tangles . . . . . . . . . . . . . . . 500 6. Well-quasi-orderings . . . . . . . . . . . . . . . . . . . . . . . 514 IX Crossing-Width 1. Partitions and ranks . . . . . . 2. Decompositions . . . . . . . . 3. Term . . . . . . . . . . . . . . 4. Tree-width and crossing-width 5. Interpretations . . . . . . . . . 6. Non-standard crossing-width .

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521 . 521 . 532 . 537 . 544 . 552 . 560

X Guarded Second-Order Transductions 1. Transductions . . . . . . . . . . . . . . . 2. Tree-decompositions . . . . . . . . . . . 3. Trees of bounded height . . . . . . . . . . 4. The Transduction Hierarchy . . . . . . . 5. Defining tree-decompositions . . . . . .

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567 . 567 . 578 . 579 . 597 . 604

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C. Applications XI Automatic Structures 1. Automatic presentations 2. Interpretations . . . . . . 3. Closure properties . . . . 4. Undecidability . . . . . . 5. Injective presentations . . 6. Partition theorems . . . . 7. Counting quantifiers . . . 8. Proving non-automaticity 9. Automatic groups . . . . 10. Automatic semirings . . . 11. Automatic partial orders

625 . . . . . . . . . . .

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627 . 627 . 635 . 643 . 647 . 651 . 658 . 677 . 686 . 708 . 735 . 749

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Bibliography Symbol Index Index

viii

765 777 783

Part A

Fundamentals

I Logics and Their Expressive Powers he two central topics of this book are (i) the model checking problem for specific structures and (ii) the study of the expressive power of various logics. To this end we will develop techniques to compute and compare the theories of given structures. This obviously solves the model checking problem since, if we know the theory of a structure A, we can decide whether a formula is satisfied by it. But this also helps us to prove that certain things are not expressible in a given logic L. If we can find two structures A and B with the same L-theory such that A has a given property P, but B does not, then the property P cannot be expressed in L.

T

Notation. The following basic notation will be used throughout the book. For n < ω, we set [n] ∶= {0, . . . , n − 1}. We tacitly identify a tuple a¯ = ⟨a 0 , . . . , a n−1 ⟩ with the set {a 0 , . . . , a n−1 } of its components. This allows us to write a¯ ⊆ C or c ∈ a¯. The empty tuple is ⟨⟩. ℘(A) denotes the power set of A, and A + B is the disjoint union of A and B. For a function f ∶ A → B, we denote the domain by dom f ∶= A and its range by rng f ⊆ B. We write f ↾ X ∶ X → B for the restriction of f to the set X. For a partial order ⟨A, ≤⟩ and a subset X ⊆ A, we set ⇑X ∶= { a ∈ A ∣ a ≥ x for some x ∈ X } , ⇓X ∶= { a ∈ A ∣ a ≤ x for some x ∈ X } . We denote the infimum and the supremum of two elements x and y by, respectively, x ⊓ y and x ⊔ y.

monadic second -order model theory 2023-12-19

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©achim blumensath

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I. Logics and Their Expressive Powers

1 Structures and Logics Logics are formal languages designed to talk about mathematical objects. As we will deal with several different logics in the course of this book it is useful to adopt an abstract point of view. In general a logic consists of (i) a class of objects to talk about; (ii) a set of statements we can make about them; and (iii) a relation telling us which statements hold for a given object. Definition 1.1. A logic is a triple ⟨L, M, ⊧⟩ consisting of a set L of formulae, a class M of models, and a satisfaction relation ⊧ ⊆ M × L. To keep notation ⌟ light, we usually identify a logic with its set of formulae L. For instance, we can define first-order logic as a triple ⟨FO[Σ], STR[Σ], ⊧⟩ where FO[Σ] is the set of all first-order formulae (without free variables) over the signature Σ and STR[Σ] is the class of all Σ-structures. For formulae with free variables we can use the logic ⟨FO[Σ, X], STR[Σ, X], ⊧⟩ where FO[Σ, X] is the set of all first-order formulae with free variables in the set X and STR[Σ, X] is the class of all pairs ⟨A, β⟩ consisting of a Σ-structure A and a variable assignment β ∶ X → A. The logics we consider in this book are mostly variants of first-order logic and monadic second-order logic. Let us quickly recall their definitions. A signature Σ is a set of relation symbols and function symbols, each of which has an associated arity. A Σ-structure A = ⟨A, (ξA ) ξ∈Σ ⟩ consists of a set A together with ◆ one n-ary relation RA ⊆ An , for every relation symbol R ∈ Σ of arity n, ◆ one n-ary function f A ∶ An → A, for every function symbol f ∈ Σ of arity n. Note that we allow functions of arity 0, which correspond to constants. Most of the time in this book we assume that all signatures are finite and purely relational. Most of the time we will only work with 1-sorted structures, but sometimes many-sorted ones are more convenient. An S-sorted structure A = ⟨(A s )s∈S , (ξA ) ξ∈Σ ⟩

4

1 Structures and logics

has one domain A s , for each sort s ∈ S, and each relation symbol and function symbol has an associated type. For an n-ary relation symbol R, this type is an n-tuple s¯ ∈ S n , for an n-ary function symbol f , it is an (n + 1)tuple s¯t ∈ S n+1 , which we will usually write at s¯ → t. If R has type s¯, the corresponding relation is of the form RA ⊆ ∏ i A s i . Similarly, if f has type s¯ → t, we are given a function f A ∶ ∏ i A s i → A t . Example. (a) The field of real numbers ⟨R, +, ⋅ , 0, 1, ≤⟩ is a structure with signature {+, ⋅ , 0, 1, ≤}, where + and ⋅ are binary function symbols, 0 and 1 are 0-ary function symbols, and ≤ is a binary relation symbol. (b) A graph is a structure ⟨V , E⟩ with a single binary relation E ⊆ V × V. (c) We can represent a vector space V over a field K either as a 1-sorted structure of the form V = ⟨V , +, 0, ( f a ) a∈K ⟩ where scalar multiplication is split into separate functions f a ∶ V → V, for each a ∈ K, or we can use a two-sorted structure V = ⟨V , K, +, 0, ⋅ ⟩ with + ∶ V × V → V and ⌟ ⋅ ∶ K × V → V. The main logics we are concerned with in this book are first-order logic and various variants of monadic second-order logic. Recall that first-order logic FO[Σ] consist of formulae that are built up from atomic formulae of the form s = t and Rt 0 . . . t n−1 , where R ∈ Σ is an n-ary relation symbol and s, t, t 0 , . . . , t n−1 are terms built up from variables and the function symbols in Σ. Such atomic formulae can be combined using boolean operations ∧ (conjunction), ∨ (disjunction), ¬ (negation), and first-order quantifiers ∃x and ∀x. Definition 1.2. Let Σ be a signature. The formulae of monadic second-order logic MSO[Σ] are built up from atomic formulae of the form s = t, Zt, and Rt 0 . . . t n−1 , where R ∈ Σ is an n-ary relation symbol, Z is a set variable, and s, t, t 0 , . . . , t n−1 are terms built up from first-order variables and the function symbols in Σ. Such atomic formulae can be combined using boolean operations ∧ (conjunction), ∨ (disjunction), ¬ (negation), and quantifiers ∃x, ∀x, ∃Z, and ∀Z, where x is a first-order variable and Z is a set variable. The semantics of such a formula is defined as follows. Given a formula ¯ ∈ MSO[Σ] with free first-order variables x¯ and free set variables Z¯ φ(x¯ , Z)

5

I. Logics and Their Expressive Powers

and given a Σ-structure A, a tuple of elements a¯ of A, and a tuple of subsets P¯ of A, we define the satisfaction relation ¯ A ⊧ φ(a¯, P) by induction on φ. The definition is analogous to that for first-order logic. An atomic formula Zt holds in A if the element denoted by the term t belongs to the set denoted by Z. A formula of the form ∃Zψ holds if there exists a set satisfying ψ, and ∀Zψ holds if every set satisfies ψ. Throughout we use lower case letters for first-order variables and upper case ones for set variable. For readability we will sometimes use common short-hands such as, s ≠ t instead of ¬(s = t), or t ∈ Z instead of Zt. As above we write MSO[Σ, X] for the set of MSO-formulae with free variables in a given set X. A model of such a formula consists of a Σ-structure A ¯ for a formula φ to and a variable assignment β. We usually write φ(x¯ , Z) ¯ This allows indicate that the free variables of φ are among the variables x¯ Z. us to use the more common notation ¯ A ⊧ φ(a¯, P)

iff

⟨A, β⟩ ⊧ φ ,

where β is the variable assignment mapping x i to a i and Z i to Pi . Since Σ and X can usually be inferred from the context, we will frequently simplify notation by writing MSO instead of MSO[Σ, X], and similarly for other ⌟ logics. Example. (a) For a linear order A = ⟨A, ≤, ⟩, we can say that y is the immediate successor of x by the FO-formula φ(x, y) ∶= x ≤ y ∧ x ≠ y ∧ ∀z[x ≤ z ∧ z ≤ y → (z = x ∨ z = y)] . (b) For a tree T = ⟨T, ≤⟩ where ≤ is the predecessor order, we can express that a set variable X contains an infinite branch by the MSO-formula ∃Z[Z ⊆ X ∧ Z ≠ ∅ ∧ ∀x∀y[Zx ∧ Zy → (x ≤ y ∨ y ≤ x)] ∧ ∀x∃y[Zx → x < y ∧ Zy]] .

6

1 Structures and logics

(c) Given a graph G = ⟨V , E⟩, the MSO-formula φ(x, y) ∶= ∀Z[Zx ∧ ∀u∀v(Zu ∧ Euv → Zv) → Zy] expresses that there exists a path from x to y. (d) We can say that a graph G = ⟨V , E⟩ is connected by the formula ∀x∀yφ(x, y) , where φ is the formula from (c).

⌟

We will also study the following variants of monadic second-order logic. Definition 1.3. Let Σ be a signature. (a) Weak monadic-second order logic WMSO[Σ] has the same syntax as MSO[Σ], but all set variables range over finite sets only. (b) Monadic-second order logic with first-order counting CMSO[Σ], or counting monadic-second order logic for short, is the extension of MSO[Σ] by statements of the form ∣X∣ < ℵ0

and

∣X∣ ≡ k (mod m) ,

for a set variable X and finite numbers k, m < ω. A statement of the form ∣X∣ < ℵ0 holds if X is a finite set, and ∣X∣ ≡ k (mod m) is true if, X is finite and its size is congruent k modulo m. We write MSO[inf ] if we only allow predicates of the first form. (c) Let A be a Σ-structure. A tuple a¯ ∈ An is guarded if there exists a relation R of A containing a tuple c¯ ∈ R with a¯ ⊆ c¯. Here, we allow R to be the equality relation =, even though it is not present in the signature. A relation S ⊆ An is guarded if every tuple a¯ ∈ S is guarded. (d) Guarded second-order logic GSO[Σ] extends first-order logic by atomic formulae of the form Zt 0 . . . t n−1 , where t 0 , . . . , t n−1 are terms and Z is a relation variable of arity n, and by quantifiers ∃Z and ∀Z over relation variables. A formula of the form ∃Zψ holds if there exists a guarded relation ⌟ satisfying ψ, and ∀Zψ holds if every guarded relation satisfies ψ.

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I. Logics and Their Expressive Powers

Example. We consider undirected graphs G = ⟨V , E⟩ as structures over the signature {E} consisting of one binary edge relation (irreflexive and symmetric). (a) To express that a graph has a Hamiltonian cycle we can write down a GSO-formula stating that there is a guarded binary relation Z (i.e., a set of edges) such that ◆ for every vertex x there are unique vertices y and z with (y, x) ∈ Z and (x, z) ∈ Z, ◆ every two vertices are connected by a sequence of Z-edges. (b) A minor of a graph G is a graph H obtained from the first graph by deleting vertices and edges and by contracting edges. To say that a fixed finite graph H is a minor of the given graph, we can use an MSO-formula stating that, for each vertex v of H, there exists a set Xv such that ◆ the subgraph induced by Xv is connected and ◆ for every edge ⟨u, v⟩ of H there is an edge connecting some vertex of X u ⌟ with some vertex of Xv . As defined above the logic MSO is not always convenient to use in proofs. Therefore, we introduce a simplified version that still has the same expressive power. Definition 1.4. Let Σ be a relational signature. The logic MSO0 [Σ] has atomic formulae of the form X⊆Y,

sing(X) ,

X ∩Y = ∅,

cover(X 0 , . . . , X n−1 ) ,

RX 0 . . . X n−1 ,

where R ∈ Σ is an n-ary relation symbol and X, Y, X 0 , . . . , X n−1 are set variables. The logic is closed under boolean operations and set quantifiers. The formulae X ⊆ Y and X ∩ Y = ∅ have the obvious meaning. sing(X) states that ∣X∣ = 1. An atomic formula of the form cover(X 0 , . . . , X n−1 ) holds if the union X 0 ∪ ⋅ ⋅ ⋅ ∪ X n−1 contains the whole universe, while a formula of the form RX 0 . . . X n−1 holds if each set X i is a singleton {a i } and the tuple ⌟ ⟨a 0 , . . . , a n−1 ⟩ of elements belongs to R.

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1 Structures and logics

Remark. (a) We frequently use abbreviations like (X = Y) ∶= (X ⊆ Y) ∧ (Y ⊆ X) , (X ⊂ Y) ∶= (X ⊆ Y) ∧ ¬(Y ⊆ X) , (X = ∅) ∶= (X ∩ X = ∅) . (b) Note that every MSO0 -formula is equivalent to one that does not ¯ contain atomic formulae of the form X ∩ X ′ = ∅, sing(X), or cover(X) since we can define these in terms of ⊆. X ∩ X ′ = ∅ ≡ ∀Y[Y ⊆ X ∧ Y ⊆ X ′ → ∀Z(Y ⊆ Z)] , sing(X) ≡ X ≠ ∅ ∧ ∀Y[Y ⊂ X → Y = ∅] , ¯ ≡ ∀Z[sing(Z) → ⋁ Z ⊆ X i ] . cover(X) i

But note that this translation does increase the quantifier rank.

⌟

¯ ∈ Lemma 1.5. Let Σ be a relational signature. For every formula φ(x¯ , Z) 0 ○ ¯ ¯ MSO[Σ], there is a formula φ (X, Z) ∈ MSO [Σ] such that ¯ A ⊧ φ○ ({a 0 }, . . . , {a m−1 }, P)

iff

¯ , A ⊧ φ(a 0 , . . . , a m−1 , P)

¯ for every Σ-structure A and all parameters a¯ and P. Proof. We define φ○ by induction as follows. (x = y)○ ∶= sing(X) ∧ sing(Y) ∧ X ⊆ Y ∧ Y ⊆ X , (Rx 0 . . . x n−1 )○ ∶= RX 0 . . . X n−1 , (φ ∧ ψ)○ ∶= φ○ ∧ ψ ○ ,

(∃xψ)○ ∶= ∃X[sing(X) ∧ ψ ○ ] ,

(φ ∨ ψ)○ ∶= φ○ ∨ ψ ○ ,

(∀xψ)○ ∶= ∀X[sing(X) → ψ ○ ] ,

○

○

(¬φ) ∶= ¬φ ,

(∃Zψ)○ ∶= ∃Zψ ○ , (∀Zψ)○ ∶= ∀Zψ ○ .

Analogous statements hold for the other variants of MSO.

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I. Logics and Their Expressive Powers

Exercise 1.1. We consider coloured linear orders of the form ⟨A, ≤, P⟩ where P ⊆ A is a unary predicate. Find MSO-formulae expressing the following statements: (a) The set P is dense, i.e., it is non-empty and between any two elements of A there is an element of P. (b) The set P contains infinitely many elements. ⌟ (c) The set P is finite and it has an even number of elements. Exercise 1.2. An (m × n)-grid is a graph G = ⟨V , E⟩ where V ∶= [m] × [n] , E ∶= { ⟨⟨i, k⟩, ⟨ j, l⟩⟩ ∣ ∣i − j∣ + ∣k − l∣ = 1 } . (a) Construct an MSO-formula expressing that a graph is a grid. (b) For each of the following functions f ∶ ω → ω, find an MSO-formula stating that the given graph is an (n × f (n))-grid, for some n. (i)

f (n) = n ,

(ii)

f (n) = n 2 ,

(iii)

f (n) = 2n .

⌟

∗

Exercise 1.3. We can encode a finite word w = a 0 . . . a n−1 ∈ Σ over the alphabet Σ by a word structure wˆ ∶= ⟨[n], ≤, (Pa ) a∈Σ ⟩ , where the universe [n] = {0, . . . , n − 1} is the set of positions in the word w and the predicates Pa ∶= { i < n ∣ a i = a } contain all positions carrying the corresponding letter. Prove that, for every regular expression α, there exists an MSO-formula φ such that wˆ ⊧ φ

iff

w ∈ L(α) .

Hint. First construct, for each regular expression α, an MSO-formula φ(x, y) such that wˆ ⊧ φ(x, y)

iff

w[x, y] ∈ L(α) ,

where w[x, y] denotes the factor of w between positions x and y.

10

⌟

2 Simple translations between logics

2 Simple Tranªations Between Logics In this section we relate the various logics introduced above to each other, and we provide translations between them. We start with MSO and FO. Definition 2.1. Let Σ be a relational signature. (a) The power-set structure of a Σ-structure A is the structure ℘(A) with signature Σ ∪ {⊆} whose universe is the power set ℘(A) of the universe of A. The relation symbol ⊆ denotes the usual subset relation on ℘(A). For each n-ary relation symbol R ∈ Σ, ℘(A) has the relation R℘(A) ∶= { P¯ ∈ ℘(A)n ∣ each Pi = {a i } is a singleton and a¯ ∈ RA } . (b) The finite power-set structure of a Σ-structure A is the substructure ⌟ ℘fin (A) of ℘(A) consisting of all finite subsets of A. It is straightforward to check that MSO over Σ-structures corresponds to FO over their power-set structures.

Proposition 2.2. Let Σ be a relational signature. ¯ there exists an FO[Σ ∪{⊆}]-formula (a) For every MSO[Σ]-formula φ(X), φ′ (x¯) such that ¯ A ⊧ φ(P)

¯ , ℘(A) ⊧ φ′ (P)

iff

for all Σ-structures A and all sets P¯ in A. (b) For every FO[Σ ∪ {⊆}]-formula φ(x¯), there exists an MSO[Σ]-formula ′ ¯ φ (X) such that ¯ ℘(A) ⊧ φ(P)

iff

¯ , A ⊧ φ′ (P)

for all Σ-structures A and all sets P¯ in A. Proof. (a) By Lemma 1.5 and the remark after Definition 1.4, we may assume that φ is an MSO0 -formula without subformulae of the form sing(X), ¯ Then we obtain the desired formula φ′ from φ by X ∩ Y = ∅, or cover(X). replacing every set variable X by a corresponding first-order variable x. (b) It is sufficient to construct an MSO0 -formula. We obtain it from φ by replacing every first-order variable x by a corresponding set variable X.

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I. Logics and Their Expressive Powers

We obtain the analogous result for finite power-sets and weak MSO. The proof is identical to the one above. Proposition 2.3. Let Σ be a relational signature. ¯ there exists an FO[Σ ∪ {⊆}](a) For every WMSO[Σ]-formula φ(X), formula φ′ (x¯) such that ¯ A ⊧ φ(P)

iff

¯ , ℘fin (A) ⊧ φ′ (P)

for all Σ-structures A and all finite sets P¯ in A. (b) For every FO[Σ ∪ {⊆}]-formula φ(x¯), there exists an WMSO[Σ]¯ such that formula φ′ (X) ¯ ℘fin (A) ⊧ φ(P)

iff

¯ , A ⊧ φ′ (P)

for all Σ-structures A and all finite sets P¯ in A. Finally, we can also relate MSO to GSO via a suitable operation. Definition 2.4. Let Σ be a relational signature. The incidence structure of a Σ-structure A is the 2-sorted Σ in -structure Ain ∶= ⟨A, E, (Pc )c , in0 , in1 , . . . ⟩ with domains A and E ∶= { c¯ ∈ A 0. Again it is sufficient to check the Forth Property. We distinguish two cases, depending on whether we deal with a first-order parameter or with a monadic one. First, consider a monadic parameter P ⊆ A. If P = ∅, we choose Q ∶= ∅. If P = A, we choose Q ∶= B. In both cases it follows by inductive hypothesis that m A, P ≡MSO B, Q .

Hence, we may assume that P is neither empty, nor all of A. If ∣P∣ ≤ 2m−1 , choose a subset Q ⊆ B of size ∣Q∣ = ∣P∣. Otherwise, choose a subset Q ⊆ B with ∣B ∖ Q∣ = ∣A ∖ P∣. Let A0 and B0 be the substructures of A and B induced by P and Q, and let A1 and B1 be the substructures induced by A ∖ P and B ∖ Q. It follows that

28

4 Operations for monadic second-order logic

◆ ∣P∣ = ∣Q∣ or ∣P∣, ∣Q∣ ≥ 2m−1 ; ◆ ∣A ∖ P∣ = ∣B ∖ Q∣ or ∣A ∖ P∣, ∣B ∖ Q∣ ≥ 2m−1 . By inductive hypothesis, this implies that m A0 ≡MSO B0

and

m A1 ≡MSO B1 .

By Proposition 4.2, it follows that m A, P ≅ A0 , P ⊕ A1 , ∅ ≡MSO B0 , Q ⊕ B1 , ∅ ≅ B, Q .

(Again, we have to omit the relations Left and Right.) For a first-order parameter a ∈ A, we choose an arbitrary element b ∈ B. We denote by A0 and B0 the substructures of A and B induced by {a} and {b}, and we write A1 and B1 for the substructures induced by A ∖ {a} and B ∖ {b}. Then A0 , a ≅ B0 , b

m implies A0 , a ≡MSO B0 , b .

Furthermore, it follows by inductive hypothesis that m A1 ≡MSO B1 .

By Proposition 4.2, this implies that m A, a ≅ A0 , a ⊕ A1 ≡MSO B0 , b ⊕ B1 ≅ B, b .

Example. There is no MSO[Σ]-formula φ such that, for every finite Σstructure A, A⊧φ

iff

∣A∣ is even.

For the proof, let m ∶= qr(φ) and let A and B be Σ-structures of size 2m and 2m + 1, respectively, where every relation is empty. By Proposition 4.3, m+1 we have A ≡MSO B. Consequently, A⊧φ

iff

A contradiction.

B ⊧ φ. ⌟

29

I. Logics and Their Expressive Powers

Exercise 4.1. We consider structures of the form A = ⟨A, E⟩ where E is an equivalence relation. For an equivalence relation E, we denote by N =k (E) the number of E-classes [a]E of size ∣[a]E ∣ = k and N >k (E) denotes the number of classes of size ∣[a]E ∣ > k. We write m = k n iff m = n or m, n ≥ k. Let E and F be equivalence relations on the sets A and B, respectively. m Prove that ⟨A, E⟩ ≡FO ⟨B, F⟩ if, and only if, for all k ≤ m, N =k (E) =m−k N =k (F)

and

N >k (E) =m−k N >k (F) .

⌟

Interpretations Disjoint unions alone are not that interesting as they cannot be used to modify the relations of a structure. The next operation, called an interpretation, fills that hole. We will present the definition for of many-sorted structures since this more general case is what is needed in Chapter VII below. Definition 4.4. Let L be one of the logics FO, MSO, WMSO, or CMSO, let Σ and Γ be relational signatures, and assume that Γ is S-sorted. An Linterpretation from Σ to Γ is an operation τ transforming Σ-structures into Γ-structures that is defined by a list ⟨(δ s (x))s∈S , (φ R (x¯))R∈Γ ⟩ of L-formulae over the signature Σ as follows. We assume that the formulae δ s have one free variable, while the number of free variables of φ R matches the arity of R. Then τ maps a Σ-structure A to the Γ-structure τ(A) ∶= ⟨(δ As )s , (φAR )R∈Γ ⟩ whose domain of sort s is the set δ As ∶= { a ∈ A ∣ A ⊧ δ(a) } defined by δ s and whose relations are φAR ∶= { a¯ ∣ A ⊧ φ R (a¯) } ,

30

for R ∈ Σ .

4 Operations for monadic second-order logic

We call the list ⟨(δ s )s , (φ R )R∈Γ ⟩ the definition scheme of τ. The quantifier rank of τ is the maximal quantifier rank of a formula in its definition scheme. ⌟ Let us show that L-interpretations are L-compatible. Proposition 4.5. Let L be one of the logics FO, MSO, WMSO, or CMSO, and let τ = ⟨(δ s (x))s∈S , (φ R (x¯))R∈Γ ⟩ be an L-interpretation from Σ to Γ with ¯ there exists an L[Σ]-formula quantifier rank m. For every L[Γ]-formula ψ(X), τ ¯ ψ (X) with quantifier rank at most qr(ψ) + m such that τ(A) ⊧ ψ(α¯ )

iff

A ⊧ ψ τ (α¯ ) ,

for all Σ-structures A and all parameters α¯ in τ(A). Proof. We define ψ τ by induction on ψ as follows. (x = y)τ ∶= x = y ,

(φ ∧ ψ)τ ∶= φ τ ∧ ψ τ ,

(Xy)τ ∶= Xy , (R x¯)τ ∶= φ R (x¯) ,

(φ ∨ ψ)τ ∶= φ τ ∨ ψ τ , (¬φ)τ ∶= ¬φ τ ,

(∃yψ)τ ∶= ∃y[δ s (y) ∧ ψ τ ] , τ

τ

(∀yψ) ∶= ∀y[δ s (y) → ψ ] ,

(∃Yψ)τ ∶= ∃Yψ τ , (∀Yψ)τ ∶= ∀Yψ τ ,

where s is the sort of the variable y. Remark. Note that this statement fails for L = GSO since guarded tuples in ⌟ τ(A) are not necessarily guarded in A. Corollary 4.6. Let τ be an L-interpretation from Σ to Γ with quantifier rank m. A ≡Lk+m A′

implies

τ(A) ≡Lk τ(A′ ) .

for all Σ-structures A, A′ .

Proof. By symmetry, it is sufficient to prove that τ(A) ⊧ φ

implies

τ(A′ ) ⊧ φ ,

for all φ with qr(φ) ≤ k .

31

I. Logics and Their Expressive Powers

Hence, suppose that τ(A) ⊧ φ and let φ τ be the formula from Proposition 4.5. Then A ⊧ φτ

and

qr(φ τ ) ≤ k + m .

Thus, A ≡Lk+m A′ implies that A′ ⊧ φ τ . It follows that τ(A′ ) ⊧ φ. Lemma 4.7. Let σ and τ be L-interpretations. Then so is τ ○ σ. Proof. Suppose that τ = ⟨(δ s (x))s∈S , (φ R (x¯))R∈Γ ⟩. We claim that τ ○ σ has the definition scheme ⟨(δ sσ (x))s∈S , (φ σR (x¯))R∈Γ ⟩ . Note that, given a structure A, the elements τ(σ(A)) of sort s are exactly those a ∈ A satisfying σ(A) ⊧ δ s (a) . By Proposition 4.5, this condition is equivalent to A ⊧ δ sσ (a) . Similarly, a tuple a¯ belongs to a relation R if, and only if, σ(A) ⊧ φ R ( a¯)

iff

A ⊧ φ σR ( a¯) .

Frequently, disjoint unions and interpretations are all one needs to compute a theory. As an example, let us show how to generalise Proposition 4.3 to structures with unary predicates. Proposition 4.8. Let Σ = {U 0 , . . . , U m−1 } be a signature consisting of unary ¯ x¯) predicates only. Over the class of all Σ-structures, every GSO-formula φ(X, is equivalent to an FO-formula. Proof. Since the only guarded tuples over a unary signature are singletons, every GSO-formula can trivially be translated to an MSO-formula. Hence,

32

4 Operations for monadic second-order logic

by Lemma 3.3, it is sufficient to prove that, for every quantifier-rank r < ω, there exists some p < ω such that A, P¯ a¯ ≡FO B, Q¯ b¯ p

implies

r A, P¯ a¯ ≡MSO B, Q¯ b¯ ,

¯ To simplify notation, we will for all A, B with parameters P¯ a¯ and Q¯ b. ¯ c¯⟩ where the not work with parameters but with structures A = ⟨A, P, parameters are part of the structure itself. Hence, let Um,n be the class of all such structures with m unary predicates P0 , . . . , Pm−1 ⊆ A and n constant symbols c 0 , . . . , c n−1 , Given A ∈ Um,n and a set θ ⊆ [m], we set Pi ,

Pθ ∶= ⋂ Pi ∖

⋃

i∈θ

i∈[m]∖θ

we denote by Aθ the substructure induced by Pθ ∖ c¯, and ⟪c¯⟫A is the substructure generated by c¯. Then we can write A as a disjoint union A ≅ ⟪c¯⟫A ⊕ ⊕ Aθ . θ⊆[m]

Let us make the following observations. (i) For every structure C of size at most k, there exists a first-order formula of quantifier-rank k + 1 that characterises C up to isomorphisms. Consequently, for a¯ ∈ Ak and b¯ ∈ B k , k+1 ¯ ⟪c¯⟫A ≡FO ⟪d⟫B

implies

¯ B. ⟪c¯⟫A ≅ ⟪d⟫

(ii) For every MSO-formula φ, we can use Proposition 4.3 to find a finite set H ⊆ ω and a number N < ω such that, or

C⊧φ

iff

∣C∣ ∈ H ,

C⊧φ

iff

∣C∣ ∈ H

or

∣C∣ ≥ N ,

for all C ∈ U0.0 . Since, for every k < ω, we can construct an FO-formula ψ k stating that the structure has at least k elements, it follows that there exists some number f (r) such that f (r)

A ≡FO B implies

r A ≡MSO B,

for all A, B ∈ U0,0 .

33

I. Logics and Their Expressive Powers

(iii) For every θ ⊆ [m], there exists a quantifier-free interpretation σθ mapping C ∈ U0.0 to a structure σθ (C) ∈ Um,0 with predicates ⎧ ⎪ ⎪C Pi ∶= ⎨ ⎪ ∅ ⎪ ⎩

if i ∈ θ , if i ∉ θ .

In particular, Aθ = σθ ((Aθ )∣∅ ) ,

for A ∈ Um,n and θ ⊆ [m] ,

where C∣∅ denotes the reduct to the empty signature. For A, B ∈ Um,n and θ ⊆ [m], it follows that r (Aθ )∣∅ ≡MSO (Bθ )∣∅

r implies Aθ ≡MSO Bθ .

(iv) There exists quantifier-free interpretations τ ○ and τ θ such that τ ○ (A) = ⟪a¯⟫A

and

τ θ (A) = (Aθ )∅ .

Hence, k A ≡FO B implies

k ¯ B ⟪c¯⟫A ≡FO ⟪d⟫

k and (Aθ )∣∅ ≡FO (Bθ )∣∅ .

We can conclude the proof as follows. Set p ∶= max { f (r), n + 1} and let A, B ∈ Um,n . Combining the above observations it follows by Proposition 4.2 that p

and (Aθ )∣∅ ≡FO (Bθ )∣∅ ,

⇒

A ≡FO B p ¯ B ¯ ⟪c ⟫A ≡FO ⟪d⟫ ¯ B ⟪c¯⟫A ≅ ⟪d⟫

and

⇒

r ¯ B ⟪c¯⟫A ≡MSO ⟪d⟫

and

⇒

⇒

p

for all θ,

r

(Aθ )∣∅ ≡MSO (Bθ )∣∅ ,

for all θ,

r Aθ ≡MSO Bθ ,

for all θ,

r

A ≡MSO B .

As a second example, let us give the example of an ordered sum, which corresponds to concatenations of words.

34

4 Operations for monadic second-order logic

Definition 4.9. Let C be a set of colours. (a) A C-coloured order is a structure of the form A = ⟨A, ≤, (Pc )c∈C ⟩ where ≤ is a linear ordering on A and the Pc are unary predicates. (b) Let I = ⟨I, ≤⟩ be a linear order and let A i ∶= ⟨A i , ≤ i , P¯i ⟩, i ∈ I, be a family or C-coloured linear orders indexed by I. The ordered sum ∑ Ai i∈I

is the linear order with universe L ∶= { ⟨i, a⟩ ∣ i ∈ I , a ∈ A i } and order ⟨i, a⟩ ≤ ⟨ j, b⟩

: iff

i< j

or

(i = j and a ≤ i b) .

The colour predicates are Pc ∶= ⋃ (Pi )c . i∈I

If I = [2], we simply write A0 + A1 for the ordered sum.

⌟

Proposition 4.10. Let A0 , A1 , B0 , B1 be C-coloured linear orders and let L be FO, MSO, WMSO, or CMSO. Then A0 ≡ m L B0

and

A1 ≡ m L B1

implies

A0 + A1 ≡ m L B0 + B1 .

Proof. We have A0 + A1 ≅ τ(A0 ⊕ A1 ) , where τ is a quantifier-free L-interpretation that corrects the order relation. It has the definition scheme δ(x) ∶= true , φ≤ (x, y) ∶= x ≤ y ∨ (Left(x) ∧ Right(y)) , φ Pc (x) ∶= Pc x .

35

I. Logics and Their Expressive Powers

As a final application let us show that first-order logic cannot compute the length of a linear order. Proposition 4.11. Let A be a C-coloured linear order and m < ω a constant. Then m k × A ≡FO l ×A,

for all k, l ≥ 2m − 1 ,

where k × A ∶= ∑ i 0. We check the forth property. (As usual the back property follows by symmetry.) Hence, let a be an element of k × A and suppose that a belongs to the i-th copy of A. In l × A, we choose the same element in the j-th copy of A where

⎧ ⎪ ⎪i j ∶= ⎨ ⎪ l − (k − i) ⎪ ⎩

if i ≤ 2m−1 , otherwise .

Let us denote this element by b. By inductive hypothesis it follows that k × A, a ≅ (i − 1) × A + A, a + (k − i) × A m−1 ≡FO ( j − 1) × A + A, b + (l − j) × A ≅ l × A, b .

For MSO and CMSO, we obtain the following result. Proposition 4.12. For every m < ω, there exist numbers k, k, l ′ , l ′ < ω such that m ⟨A, ≤⟩ ≡MSO ⟨B, ≤⟩

iff

∣A∣ = ∣B∣ < k ,

or

∣A∣, ∣B∣ ≥ k and ∣A∣ ≡ ∣B∣ (mod l) , m

⟨A, ≤⟩ ≡CMSO ⟨B, ≤⟩

iff

∣A∣ = ∣B∣ < k ′ ,

or

∣A∣, ∣B∣ ≥ k ′ and ∣A∣ ≡ ∣B∣ (mod l ′ ) , for all finite linear orders ⟨A, ≤⟩ and ⟨B, ≤⟩.

36

4 Operations for monadic second-order logic

Proof. Let L be one of MSO or CMSO. Let Θ be the set of all L m -theories of finite linear orders. It follows by Proposition 4.10 that we can define a binary operation + on Θ such that m m Thm L (A) + Th L (B) = Th L (A + B) ,

for all finite linear orders A and B. This turns Θ into a finite semigroup. Let σ be the theory of the 1-element linear order. Since Θ is finite, there is some number n > 1 such that σ n = σ k , for some k < n. We choose n minimal. Let l ∶= n − k. Then σ k+i l + j = σ k+ j ,

for all i, j ,

and it follows that σi = σ j

iff

i= j

or

i, j ≥ k and i − k ≡ j − k (mod l) .

∣A∣ Since Thm the claim follows. L (A) = σ

Corollary 4.13. For each CMSO-formula φ there exists an CMSO-formula φ∗ such that ⟨A, ≤⟩ ⊧ φ

iff

⟨A⟩ ⊧ φ∗ .

Proof. Let m be the quantifier-rank of φ and k ′ , l ′ the constants from the preceding lemma. By the lemma, there exist sets K ⊆ [k ′ ] and L ⊆ [l ′ ] such that ⟨A, ≤⟩ ⊧ φ

iff

∣A∣ ∈ K ,

or

∣A∣ ≥ k ′ and ∣A∣ mod l ′ ∈ L .

This is a condition that can be expressed in CMSO. Remark. We can rephrase this statement by saying that the reduct operation ⌟ ⟨A, ≤⟩ ↦ ⟨A⟩ is CMSO-compatible.

37

I. Logics and Their Expressive Powers

Example. (a) There does not exist an FO-formula φ that holds in an undirected graph if, and only if, the graph is connected. For a contradiction, suppose that such a formula φ exists. We will construct a new formula ψ that holds in a finite linear order if, and only if, this order has an even number of elements. Let m be the quantifier rank of ψ and let A and B be linear orders of size 2m and 2m + 1, respectively. Then A⊧ψ

and

B⊭ψ,

in contradiction to the statement in the above exercise. To construct the desired formula ψ, we define an FO-interpretation τ = ⟨δ, φ E ⟩ mapping linear orders to undirected graphs as follows. The formula δ is true while φ E (x, y) states that ◆ in the order ≤ there is exactly one element between x and y, or ◆ x is the first element and y is the last one, or ◆ y is the first element and x is the last one. Then τ maps finite linear orders of even size to paths and finite linear orders of odd size (at least 3) to the disjoint union of a path and a cycle. Orders of size 1 are mapped to a loop.

Hence, τ(A) ⊧ φ

iff

A has either exactly one, or an even number of elements.

Consequently, the formula ψ ∶= φ τ ∧ ∃x y(x ≠ y) has the desired properties. (b) There does not exist an FO-formula φ(x, y) such that G ⊧ φ(u, v)

38

iff

the graph G contains a path from u to v .

4 Operations for monadic second-order logic

Otherwise, the formula ∀x∀yφ(x, y) would express that the graph is connected.

⌟

Example. We consider undirected graphs as structures over the signature {E}. (a) There does not exist an MSO-formula φ that holds in a finite complete bipartite graph K m,n if, and only if, m = n. The proof is similar to that of Proposition 4.10. Suppose that such a formula φ exists and let k be its quantifier rank. Let A and B be graphs without any edges that have, respectively, m ∶= 2 k and n ∶= 2 k + 1 vertices. Then K m,m ∶= τ(A ⊕ A) and K m,n ∶= τ(A ⊕ B) , where τ is a quantifier-free interpretation that adds all edges between a vertex k in Left and a vertex in Right. Since A ≡MSO B it follows that k K m,m = τ(A ⊕ A) ≡MSO τ(A ⊕ B) = K m,n .

A contradiction, since φ distinguishes between these two graphs. (b) There does not exist an MSO-formula φ that holds in a finite graph if, and only if, all vertices have the same number of neighbours. For a contradiction, suppose that such a formula φ exists. For a complete bipartite graph K m,n it follows that K m,n ⊧ φ

iff

m = n.

This contradicts (a). (c) There does not exist an MSO-formula φ that holds in a finite undirected graph if, and only if, the graph has a Hamiltonian cycle. For a contradiction, suppose that such a formula φ exists. Since a complete bipartite graph Km,n contains an Hamiltonian cycle if, and only if, m = n, it follows that Km,n ⊧ φ

iff

This contradicts (a).

m = n. ⌟

39

I. Logics and Their Expressive Powers

Quotients In some contexts it is usual to combine interpretations with a quotient operation. To simplify the presentation we present these operations separately. Definition 4.14. Let A be a (Σ + ≈)-structure where ≈A is an equivalence relation on A. The quotient of A by ≈ is the Σ-structure A/≈ with universe A/≈ ∶= { [a]≈ ∣ a ∈ A } and relations RA/≈ ∶= { ⟨[a 0 ]≈ , . . . , [a n−1 ]≈ ⟩ ∣ a¯ ∈ RA } .

⌟

Proposition 4.15. Let L be one of FO, MSO, WMSO, or GSO, let Σ be a signature with relations of arity at most r, and let A, B be a (Σ + ≈)-structures such that ≈A and ≈B are equivalence relations. Then A ≡m+r B L

implies

A/≈ ≡m L B/≈ .

Proof. Given φ ∈ L m , we construct a formula φ′ ∈ L m+r such that iff

A/≈ ⊧ φ

A ⊧ φ′ .

We obtain φ′ by ◆ replacing each atomic subformula of the form x = y by x ≈ y, ◆ replacing each atomic subformula of the form R x¯ (where R is either a relation symbol or a guarded second-order variable) by ∃ y¯[⋀ y i ≈ x i ∧ R y¯] . i

To reach the desired quantifier-rank for L = GSO, we have to make sure in this translation that the arity of guarded variables is bounded by r. But note that, since every guarded tuple has at most r distinct components, we can replace each guarded variable Z of arity n > r by one of arity r (or rather a tuple (Z σ )σ of such variables indexed by all surjective functions σ ∶ [n] → [r]).

40

4 Operations for monadic second-order logic

In the above lemma the quantifier rank increases when going from a structure to its quotient. Sometimes this can be avoided by using the following simple version of a quotient. Definition 4.16. Let A be a (Σ+{P})-structure where P is a unary predicate. The fusion fuseP (A) of A is the Σ-reduct of the quotient A/≈ where a≈b

: iff

a = b or a, b ∈ P .

⌟

Since a fusion is a quotient by an equivalence relation with just one nontrivial class, we can avoid increasing the quantifier rank by annotating the structure by information about the elements in this class. Proposition 4.17. Let L be one of FO, MSO, WMSO, CMSO, or GSO, and let A and B be (Σ + {P})-structures. Then ¯ ≡m ⟨B∣B∖P , V⟩ ¯ ⟨A∣A∖P , U⟩ implies L

fuseP (A) ≡m L fuse P (B) ,

¯ = (U R,w )R,w and V¯ = (VR,w )R,w contain, for every where the parameters U R ∈ Σ of arity n and every set w ⊆ [n], the predicate U R,w ∶= { a¯∣w ∣ a¯ ∈ RA , a i ∈ P ⇔ i ∉ w } ¯ (and similarly for V). Proof. Let C be the Σ-structure with one element and empty relations. There exists a quantifier-free interpretation τ such that ¯ ⊕ C) . fuseP (A) = τ(⟨A∣A∖P , U⟩ Hence, the result follows from Propositions 4.2 and 4.5. It turns out that when constructing a structure A from smaller parts, we ¯ instead, which can often construct the annotated substructure ⟨A∣A∖P , U⟩ then allows us to compute fuseP (A) by the above proposition. Exercise 4.2. Prove that, for every L-interpretation τ, there is some Linterpretation σ such that τ(A/≈) = σ(A)/≈ .

⌟

41

I. Logics and Their Expressive Powers

Exercise 4.3. Prove that, for every L-interpretation τ, there is some Linterpretation σ such that ℘(τ(A)) = σ(℘(A)) .

⌟

The Copying Operation Next, let us introduce a variant of the disjoint union that will be used extensively in Chapter X. Definition 4.18. The k-copy operation is of the form copy k (A) ∶= ⟨A ⊕ ⋅ ⋅ ⋅ ⊕ A, H 0 , . . . , H k−1 , I⟩ . That is copy k (A) consists of k disjoint copies of A with unary predicates H i ∶= { ⟨i, a⟩ ∣ a ∈ A } containing the i-th copy, and a binary relation I ∶= { ⟨⟨i, a⟩, ⟨ j, a⟩⟩ ∣ a ∈ A , i, j < k } that relates all copies of the same element.

⌟

Proposition 4.19. Let L be one of FO, MSO, WMSO, CMSO, or GSO, and ¯ = s and ¯ with ∣x¯∣ = r and ∣Y∣ let m, k < ω. For every L m -formula φ(x¯ , Y) s ′ ′ ¯ every tuple u¯ ∈ [k] , there exists an L mk -formula φ u¯ (x¯ , Y ) with ∣Y ′ ∣ = sk such that

iff

copy k (A) ⊧ φ(a 0 , . . . , a r−1 , P0 , . . . , Ps−1 ) A ⊧ φ′ (a′ , . . . , a ′ , P¯ ′ , . . . P¯ ′ ) , u¯

0

r−1

0

s−1

¯ P¯ ′ that are related via for all structures A, elements a¯, a¯′ , and sets P, a i = ⟨u i , a′i ⟩ and

42

(Pi′ )v = { b ∈ A ∣ ⟨v, b⟩ ∈ Pi } .

4 Operations for monadic second-order logic

Proof. We can construct φ′u¯ by induction on φ. We replace each variable Yi by a k-tuple Y¯ i′ = ⟨Yi′,0 , . . . , Yi′,k−1 ⟩. ⎧ ⎪ ⎪x i = x j if u i = u j , (x i = x j )′u¯ ∶= ⎨ ⎪ false otherwise , ⎪ ⎩ ⎧ ⎪ ⎪Rx i . . . x i n−1 if u i 0 = ⋅ ⋅ ⋅ = u i n−1 , (Rx i 0 . . . x i n−1 )′u¯ ∶= ⎨ 0 ⎪ false otherwise , ⎪ ⎩ (Yi x j )′u¯ ∶= Yi′,u j x j , (φ

∨ ψ)′u¯

∶=

φ′u¯

∨ ψ ′u¯

(∃zφ)′u¯ ∶= ⋁ ∃z¬φ′u¯ v , v 0 such that a n = a n+k . We choose them minimal. If k = 1, we are done. Hence, suppose that k > 1. Let m be the number such that n ≤ m < n + k and m ≡ 1 modulo k. It follows that (a m ) i = a m i = a m+i−1 . Hence, the element a m generates the subsemigroup {a m−1 , a m , . . . , a m+k−1 } which is isomorphic to Z/kZ, a group with k > 1 elements. We can already prove the following part of Theorem 5.1.

94

5 First-order logic

Proposition 5.4. A language L ⊆ Σ∗ is FO-definable if, and only if, it is recognised by a homomorphism into a finite aperiodic semigroup. Proof. (⇒) Suppose that L is defined by an FO-formula of quantifierrank m. It follows by Proposition I.4.10 that we can define a binary operation on the set Θ m of all FOm -theories that turns Θ m into a semigroup and the m theory map ThFO ∶ Σ+ → Θ m into a semigroup homomorphism. As this homomorphism recognises every FOm -definable language, it is therefore sufficient to show that Θ m is aperiodic. By Proposition I.4.11, we have m m ThFO (w n+1 ) = ThFO (w n ) ,

for every w ∈ Σ+ and every n ≥ 2m − 1. Consequently, aperiodicity follows by Lemma 5.3. (⇐) Let η ∶ Σ+ → S be a homomorphism recognising L where S is finite and aperiodic. We will construct FO-formulae φ a (x, y), for a ∈ S, such that w ⊧ φ a (i, k)

iff

η(w[i, k)) = a .

We proceed by induction on the J-class J of a. By inductive hypothesis, we have already constructed formulae φ c for all c >J a. First, we construct a formula ϑ J such that w ⊧ ϑJ

iff

η(w) ∈ J .

Let us call a factor w[i, k) of a word w an J-factor if ◆ η(w[i, k)) >J a, ◆ either k = ∣w∣ or η(w[i, k + 1)) ≤J a, ◆ either i = 0 or η(w[i − 1, k)) ≤J a. We can define a formula ψ Jc (x, y) stating that x[x, y − 1) is a J-factor and η(w[x, y)) = c by expressing that ◆ φ b (x, y − 1) holds for some b >J a, ◆ Pd y holds for some d with b ⋅ η(d) = c ≤J a, and

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II. Finite Words

◆ x is either the least element, or we have Pd ′ (x − 1) for some d ′ with η(d ′ ) ⋅ b ≤J a. Let ϑ J be the formula saying that ◆ every J-factor u of w satisfies η(u) ≥J a, ◆ if u and v are consecutive J-factors of w, then η(uv) ≥J a. Then it follows by Corollary 2.11 (b) that w ⊧ ϑ J implies η(w) ≥J a, as desired. To conclude the proof note that, S being aperiodic, we have η(w) = a

iff

η(w) ≡H a

iff

η(w) ≡L a and η(w) ≡R a .

Furthermore, η(w) ≡L η(v)

and

η(w) ≡R η(u) ,

where u is the first J-factor of w and v the last one. Consequently, we obtain the desired formula φ a (x, y) by stating the following three conditions: ◆ η(w) ∈ J . ◆ η(u) ≡R a, where u is the first J-factor of w, ◆ η(v) ≡L a, where v is the first J-factor of w. By the above remarks, each of them can be expressed in first-order logic. Remark. This result can be used to decide whether a given regular language L is first-order definable. Given an automaton for L, we start by computing a semigroup recognising it using the construction from the proof above. Unfortunately, simply checking this semigroup for aperiodicity is not enough since we need to know whether some semigroup recognising L is aperiodic. One can show that amoung all semigroups recognising a given language L there is always a minimal one, the so-called syntactic semigroup of L. This semigroup can be computed from any other semigroup recognising L by taking a suitable quotient. As aperiodicity is presvered under quotients it follows that, if any semigroup recognising L is aperiodic, so is its syntactic semigroup. Hence, from the semigroup we computed above we can construct ⌟ the syntactic semigroup and check it for aperiodicity.

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5 First-order logic

Star-Free Expressions We can also characterise the first-order definable languages via a certain kind of regular expressions. Definition 5.5. (a) A star-free regular expression α over an alphabet Σ is a term built up from binary operations ⋅ , ∩, ∪, a unary operation ∼, and constant symbols ∅ and a, for each letter a ∈ Σ. (b) The language L(α) ⊆ Σ∗ of such an expression α is defined inductively as follows. L(∅) ∶= ∅ , L(a) ∶= {a} ,

for a ∈ Σ ,

L(α ∩ β) ∶= L(α) ∩ L(β) , L(α ∪ β) ∶= L(α) ∪ L(β) , L(∼α) ∶= Σ∗ ∖ L(α) , L(α ⋅ β) ∶= L(α) ⋅ L(β) .

⌟

Examples. (a) ∼∅ ⋅ a ⋅ ∼∅ ⋅ a ⋅ ∼∅ describes the language of all words containing at least two occurrences of the letter a. (b) ∼(∼∅ ⋅ (aa ∪ bb) ⋅ ∼∅) ∩ (a ⋅ ∼∅ ⋅ b) defines (ab)+ .

⌟

To show the equivalence of star-free expressions and first-order logic, we use the following variant of the back-and-forth property for FO. Lemma 5.6. For words u, v ∈ Σ∗ and a number m < ω, we have m+1 u ≡FO v

iff

(u ∈ EaF ⇔ v ∈ EaF) m m for all a ∈ Σ and all ≡FO -classes E, F ∈ Σ∗ /≡FO .

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II. Finite Words

Proof. By Proposition I.3.6, we have m+1 u ≡FO v

iff

m for every i < ∣u∣ there is some k < ∣v∣ with u, i ≡FO v, k , and m for every k < ∣v∣ there is some i < ∣u∣ with u, i ≡FO v, k

iff

for every factorisation u = u 0 au 1 (with a ∈ Σ) there is some m m factorisation v = v 0 av 1 with u 0 ≡FO v 0 and u 1 ≡FO v 1 , and

for every factorisation v = v 0 av 1 (with a ∈ Σ) there is some m m factorisation u = u 0 au 1 with u 0 ≡FO v 0 and u 1 ≡FO v1

iff iff

u ∈ EaF ⇒ v ∈ EaF ,

m for all ≡FO -classes E, F and a ∈ Σ

v ∈ EaF ⇒ u ∈ EaF ,

m for all ≡FO -classes E, F and a ∈ Σ

u ∈ EaF ⇔ v ∈ EaF ,

m for all ≡FO -classes E, F .

Exercise 5.1. Show that, over the class of all finite words, every first-order formula is equivalent to a formula that uses only three variables (which can ⌟ be quantified several times). Proposition 5.7. A language L ⊆ Σ∗ is FO-definable if, and only if, it can be expressed by a star-free regular expression. Proof. (⇐) Given a star-free expression α we construct an FO-formula φ α (x, y) such that w ⊧ φ α (i, j)

iff

w[i, j] ∈ L(α) .

As usual the definition proceeds by induction on α. φ∅ (x, y) ∶= false , φ a (x, y) ∶= x = y ∧ Pa x , φ α∩β (x, y) ∶= φ α (x, y) ∧ φ β (x, y) , φ α∪β (x, y) ∶= φ α (x, y) ∨ φ β (x, y) , φ∼α (x, y) ∶= x ≤ y ∧ ¬φ α (x, y) ,

98

5 First-order logic

φ α⋅β (x, y) ∶= ∃u∃v[x ≤ u ∧ u + 1 = v ∧ v ≤ y ∧ φ α (x, u) ∧ φ β (v, y)] ∨ ψ α ,β (x, y) ∨ ψ β,α (x, y) , where in the last definition we have used the formula ⎧ ⎪ ⎪φ β (x, y) ψ α,β (x, y) ∶= ⎨ ⎪ false ⎪ ⎩

if ⟨⟩ ∈ L(α) , otherwise .

m m (⇒) It is sufficient to construct, for every ≡FO -class K ∈ Σ+ /≡FO ,a star-free expression defining K. We do so by induction on m. m If m = 0, all words are ≡FO -equivalent. Hence, K = Σ+ and we can use the m+1 star-free expression ∼∅. For the inductive step, let K be an ≡FO -class. By Lemma 5.6, it follows that K can be written as a finite boolean combination m of languages of the form EaF where a ∈ Σ and E, F are ≡FO -classes. We can use the inductive hypothesis to obtain expressions α and β for, respectively, E and F. Hence, α ⋅ a ⋅ β defines EaF. As star-free expressions are closed under boolean operations, we can combine these expressions to get one for K.

Linear Temporal Logic Finally, we can also use a certain form of modal logic. Definition 5.8. Let Σ be an alphabet. The formulae of linear temporal logic LTL are built up from atomic formulae of the form Pa with a ∈ Σ using (i) boolean operations and (ii) a binary modal operator U. We read φ U ψ as ‘φ until ψ’. The semantics is defined as follows. Given a word w ∈ Σ+ of length n > 0, we set w ⊧ Pa

: iff

w(0) = a ,

w ⊧ φUψ

: iff

there is some 0 < k < n such that w[k, n) ⊧ ψ and w[i, n) ⊧ φ for all 0 < i < k .

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II. Finite Words

The boolean operations are interpreted in the usual way. In addition we use the following abbreviations: Xφ ∶= false U φ

(‘next φ’) ,

Fφ ∶= true U φ

(‘finally φ’) ,

Gφ ∶= ¬F¬φ

(‘generally φ’) .

We also introduce reflexive versions of U, F, G: φ U∗ ψ ∶= ψ ∨ (φ U ψ) , F∗ φ ∶= φ ∨ Fφ , G∗ φ ∶= φ ∧ Gφ .

⌟

Examples. (a) F∗ (Pa ∧ F∗ Pa ) defines the language of all words containing at least two occurrences of the letter a. (b) G∗ (Pa → F∗ Pb ) says that every letter a is followed (not necessarily immediately) by a b. (c) ¬Xtrue states that the word consists of a single letter. ⌟ (d) Pa ∧ Pa U GPb defines the language a ∗ b+ . Clearly, the logic LTL can be embedded into first-order logic. Lemma 5.9. For every LTL-formula φ, there exists an FO-formula ψ such that w⊧φ

iff

w ⊧ψ,

for all words w .

Proof. Given φ we construct an FO-formula φ∗ (x) such that w⊧φ

iff

uw ⊧ φ∗ (∣u∣) ,

for all w, u ∈ Σ∗ .

The definition proceeds by induction on φ. Pa∗ (x) ∶= Pa x ,

(φ ∧ ψ)∗ (x) ∶= φ∗ (x) ∧ ψ ∗ (x) , (¬φ)∗ (x) ∶= ¬φ∗ (x) , (φ U ψ)∗ (x) ∶= ∃y[x < y ∧ ψ ∗ (y) ∧ ∀z[x < z < y → φ∗ (z)]] .

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5 First-order logic

To conclude the proof of Theorem 5.1 it is now sufficient to show that first-order definable languages can also be defined in LTL. This is the hardest part of the theorem and requirese a bit of preparation. Lemma 5.10. For every LTL-formula φ and every set ∆ ⊆ Σ, there exists an LTL-formula φ(∆) such that w ⊧ φ(∆)

iff

u⊧φ

for the maximal prefix u of w with u ∈ ∆+ .

Proof. We start by transforming the given formula φ into negation normal form where negations are only allowed in front of the atomic predicates Pa . This can be done using the laws of de Morgan and the equivalences ¬Fψ ≡ G¬ψ , ¬Gψ ≡ F¬ψ , ¬(ψ U ϑ) ≡ G(ψ ∧ ¬ϑ) ∨ (ψ ∧ ¬ϑ) U (¬ψ ∧ ¬ϑ) . After this simplification, we can construct φ(∆) by induction on φ as follows. Setting P∆ ∶= ⋁c∈∆ Pc , we define ⎧ ⎪ if a ∈ ∆ , ⎪Pa Pa(∆) ∶= ⎨ ⎪ false otherwise , ⎪ ⎩ ⎧ ⎪ ⎪¬Pa ∧ P∆ if a ∈ ∆ , ¬Pa(∆) ∶= ⎨ ⎪ true otherwise , ⎪ ⎩ (∆) (∆) (∆) (φ ∧ ψ) ∶= φ ∧ ψ , (φ ∨ ψ)(∆) ∶= φ(∆) ∨ ψ (∆) , (φ U ψ)(∆) ∶= P∆ ∧ [φ(∆) ∧ P∆ ] U ψ (∆) . The second construction we need is the following analogue of an interpretation for LTL. Definition 5.11. Let Σ and Γ be alphabets, ◻ ∉ Γ a new letter, and let (ψ c )c∈Γ∪{◻} be a family of LTL-formulae such that, for every w ∈ Σ+ , there exists exactly one c ∈ Γ ∪ {◻} with w ⊧ ψ c .

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II. Finite Words

The LTL-transduction τ ∶ Σ+ → Γ + defined by (ψ c )c is the following function. Given a word w ∈ Σ+ of length n ∶= ∣w∣, let c i ∈ Γ ∪ {◻}, for i < n, be the letters such that w, i ⊧ ψ c i . Then τ(w) is the word obtained from ⟨c 0 , . . . , c n−1 ⟩ by deleting all letters ⌟ that are equal to ◻. Lemma 5.12. Let τ ∶ Σ+ → Γ + be an LTL-transduction. For every LTLformula φ, there exists an LTL-formula φ τ such that iff

τ(w) ⊧ φ

w ⊧ φτ ,

for all w ∈ Σ+ .

Proof. Given φ, we will define a formula φ∗ such that iff

τ(w) ⊧ φ

w ⊧ φ∗ ,

for all w ∈ Σ+ with w ⊧ ¬ψ◻ .

Then we can set φ τ ∶= ψ◻ U∗ (¬ψ◻ ∧ φ∗ ). To define φ∗ we proceed by induction on φ. Pa∗ ∶= ψ a ,

(φ ∧ ϑ)∗ ∶= φ∗ ∧ ϑ ∗ , (¬φ)∗ ∶= ¬(φ∗ ) , (φ U ϑ)∗ ∶= (¬ψ◻ → φ∗ ) U (¬ψ◻ ∧ ϑ ∗ ) . As an application, let us show how to compute products in aperiodic semigroups using LTL. Proposition 5.13. Let S be a finite, aperiodic semigroup. For every element d ∈ S, there exists an LTL-formula φ d such that w ⊧ φd

102

iff

π(w) = d ,

for all w ∈ S + .

5 First-order logic

Proof. We will prove the following more general claim. Given a finite, aperiodic semigroup S, a non-empty subset C ⊆ S, and an element d ∈ S, there exists an LTL-formula φ d such that iff

w ⊧ φd

π(u) = d

where u is the maximal prefix of w with u ∈ C + .

The proof proceeds by induction on ∣S∣ and ∣C∣. If S = {c}, we can set φ c ∶= true. If C = {c}, we have to check whether w = c n v where c n = d and v does not start with c. As S is aperiodic, there exists some number k such that c n = c k , for all n ≥ k. Setting ψ 1 ∶= Pc

and ψ n+1 ∶= Pc ∧ Xψ n ,

we obtain formulae such that w = cnv ,

iff

w ⊧ ψn

for some v ∈ S ∗ .

Since there exists at most one number n ≤ k with c n = d, we can now set ⎧ ψ n ∧ ¬ψ n+1 ⎪ ⎪ ⎪ ⎪ φ d ∶= ⎨ψ k ⎪ ⎪ ⎪ ⎪ ⎩false

if d = c n with n < k , if d = c k , if c n ≠ d for all n ≤ k .

For the inductive step, suppose that we have already proved the claim for all semigroups S′ and all subsets C ′ ⊆ S ′ such that either ∣S ′ ∣ < ∣S∣, or ∣S ′ ∣ = ∣S∣ and ∣C ′ ∣ < ∣C∣. We first consider the case where, for every element c ∈ C, left-multiplication σc ∶= a ↦ ca by c is bijective. Since S is aperiodic, there exists some number k such that c k+1 = c k . Consequently, σck+1 = σck . As σc is bijective, we can divide this equvation by σck and obtain σc = id. Hence, we have ca = a, for all c ∈ C and a ∈ S, and it follows that π(w) = d

iff

the last element of w is equal to d ,

for w ∈ C + .

Thus, we can set φ d ∶= [ ⋁ Pc ] U [Pd ∧ ¬X ⋁ Pc ] . c∈C

c∈C

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II. Finite Words

It remains to consider the case where there is some c ∈ C such that the function a ↦ ca is not bijective. Set T ∶= cS and D ∶= C ∖ {c}. By assumption, T ⊂ S. Furthermore, T induces a subsemigroup of S since ca ⋅ cb = c(acb) ∈ T. Let us define a block of w ∈ C + to be a maximal factor of the form c n u with n < ω and u ∈ D∗ . To compute π(w) we will proceed in two steps: first we multiply every block of w and then we multiply the results. To accomplish the former we define formulae ψ d , for d ∈ S, such that w ⊧ ψd

iff

w = ac n uv

and

π(c n u) = d

where c n u is a block of w , a ∈ D , v ∈ C ∗ . Note that this formula is supposed to be evaluated at the position preceding the block in question. This is because we need to verify that we are at the beginning of a block and we cannot look backwards in LTL. By inductive hypothesis, we can construct formulae (φ ca ) a and (φ D a ) a evaluating products of sequences in, respectively, {c}+ and D + . We set ψ d ∶= ¬Pc ∧ X(Pc ∧ ψ ′d ) where ψ ′d ∶= [φ dc ∧ GPc ] ∨ ⋁ [φ ca ∧ [Pc U φ bD ]] . a,b∈S ab=d

(The first part deals with the special case where we are in the last block and this block is of the form c n without elements from D. The second part is for the more common case where the current block does contain elements from D.) Together with the formula ψ◻ ∶= Pc ∨ ¬XPc we obtain a family (ψ d )d∈S∪{◻} that defines an LTL-transduction τ that maps w to the sequence of products of the blocks (excluding the first block which we treat separately). This sequence belongs to T + . Since T is a proper

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5 First-order logic

subsemigroup of S we can use the inductive hypothesis to obtain formulae φ dT for evaluating the resulting product. This leads to the following definition. φ d ∶= [ψ ′d ∧ Gψ◻ ] ∨ ⋁ [ψ ′a ∧ [ψ◻ U (φ bT )τ ]] . a,b∈S ab=d

(The first clause is for the case where there is only one block, the second one if there are more.) As we have already established the equivalence between FO-definability and recognisability in an aperiodic semigroup, we now immediately obtain the last missing piece for the proof of Theorem 5.1. Corollary 5.14. Every FO-definable language is LTL-definable. Proof. Let L ⊆ Σ+ be FO-definable. By Proposition 5.4, we can find a homomorphism η ∶ Σ+ → S to a finite aperiodic semigroup S such that L = η−1 [P] for some P ⊆ S. We construct an LTL-formula ψ defining L as follows. Let φ d , d ∈ S, be the LTL-formulae from Proposition 5.13. Given a word w = a 0 . . . a n−1 ∈ Σ+ , let w η = η(a 0 ) . . . η(a n−1 ) ∈ S + be the word obtained from w by replacing each letter by its image under η. It follows that w∈L

iff

η(w) ∈ P

iff

w η ⊧ ⋁ φa

iff

w ⊧ψ,

a∈P

where ψ is the formula obtained from ⋁ a∈P φ a by replacing every predicate Pc with c ∈ S by the formula ϑ c ∶=

⋁

b∈η −1 (c)∩Σ

Pb .

Notes Ramseyan splits were introduced by Colcombet, extending earlier results by Simon [137] on factorisation trees. Their existence for arbitrary linear orders

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II. Finite Words

is due to [31]. Our presentation follows expositions by Bojańczyk [18] and Colcombet [32]. The equivalence between monadic second-order logic and automata was independently discovered by Büchi [23], Elgot [48], and Trakhtenbrot [143]. The equivalence between star-free regular expressions and aperiodic monoids is due to Schützenberge [133], the one between star-free regular expressions and first-order logic due to McNaughton and Papert [94], and the equivalence to LTL is due to Kamp [70].

106

III Infinite Words 1 Ramsey Theory ur next aim is to do what we just did in Section II.4 for languages of infinite words. Unfortunately this entails a bit of technical overhead. In particular, we need a few results from a branch of combinatorics called Ramsey Theory. We have already seen one result of this kind in Section II.3: the Lemma of Simon. In this section we will derive several more. The simplest example of such a result is the statement that every infinite undirected graph contains an infinite clique or an infinite independent set.

O

Definition 1.1. Let A be a linear order. (a) We denote by [A]2 the set of all pairs ⟨i, k⟩ ∈ A2 with i < k. (b) A finite colouring of A is a function λ ∶ [A]2 → C where C is a finite set of colours. (c) Let S be a finite semigroup. A finite colouring λ ∶ [A]2 → S is additive if λ(x, y) ⋅ λ(y, z) = λ(x, z) ,

for all x < y < z .

⌟

Theorem 1.2 (Ramsey). Let λ ∶ [ω]2 → C be a finite colouring of ω. There exists an infinite subset I ⊆ ω such that λ(i, k) = λ( j, l) ,

for all i < k and j < l in I .

Proof. We construct an increasing sequence n 0 < n 1 < ⋯ of indices, a sequence c 0 , c 1 , . . . ∈ C of colours, and a decreasing sequence J 0 ⊇ J 1 ⊇ ⋯ of infinite sets such that, for every i < ω, ni ∈ Ji

and

λ(n i , k) = c i ,

monadic second -order model theory 2023-12-19

for all k ∈ J i+1 .

—

©achim blumensath

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III. Infinite Words

We start with n 0 ∶= 0 and J 0 ∶= ω. By induction, suppose that we have already defined n i and J i . For c ∈ C, set L c ∶= { k ∈ J i ∣ k > n i and λ(n i , k) = c } . Then J i ∖ [n i + 1] = ⋃c∈C L c . As J i is infinite and C is finite, there is some element c i ∈ C such that L c i is infinite. We set J i+1 ∶= L c i

and

n i+1 ∶= min J i+1 .

Having defined (n i ) i