Languages, logics and automata

Transcription

1 Languages, logics and automata Anca Muscholl LaBRI, Bordeaux, France EWM summer school, Leiden / 89

2 Before all that.. Sonia Kowalewskaya Emmy Noether Julia Robinson All this attention has been gratifying but also embarrassing. What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered, as a mathematician should, simply for the theorems I have proved and the problems I have solved. (J. Robinson) 2 / 89

3 Plan 1 Automata 2 Logic 3 Schützenberger s theorem 4 Infinite words and trees 3 / 89

4 Plan 1 Automata 2 Logic 3 Schützenberger s theorem 4 Infinite words and trees 4 / 89

5 Automata Automata in Computer Science Fundamental notion for Theoretical Computer Science, rooted in computability theory and logic (Turing, Church, Kleene, 1930) and formal language theory (Chomsky, 1960). Mathematical model for computers. Applications verification of digital circuits, programs, systems,... modelization and dynamics of biological systems speech and image processing... 5 / 89

6 Languages Languages Words (strings) over Σ = {a, b,...}: Concatenation: ɛ, a, b, aa, ab,... ab ba = abba Free monoid (Σ,, ɛ): set of all words over Σ, together with concatenation (associative) Language: set of words, L Σ Beyond finite words Infinite words, ordinal words Binary trees (finite and infinite) Pictures Finite structures: partial orders, graphs Infinite structures: automatic structures, structures defined by logics 6 / 89

7 Automata on finite words 7 / 89

8 Automata Finite automata Finite automata (or finite-state machines): computers with limited memory. Example: switch press off on release states: on, off transitions: (off, press, on), (on, release, off ) initial state: off 8 / 89

9 Finite automata A finite automaton is a tuple A = S, Σ, δ, s 0, F with: S finite set of states Σ alphabet (finite set of letters) δ : S Σ S partially defined transition function s 0 S initial state, F S set of accept states Runs, accepted language Run: sequence of transitions Accepting run: s n F Accepted language L(A) Σ : a s 0 a 0 1 a n 1 s1 s n a L(A) = {a 0 a n 1 s 0 a 0 1 a n 1 s1 s n is accepting run} 9 / 89

10 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 10 / 89

11 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 11 / 89

12 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 12 / 89

13 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 13 / 89

14 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 14 / 89

15 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 15 / 89

16 Non-determinism Multiple transitions with same label from a state, multiple initial states: non-deterministic automaton A = S, Σ, δ, I, F. Transition relation: δ S Σ S L(A) is the set of words labeling some accepting run of A. Example a, b a, b a a a 3 S = {0, 1, 2, 3} Σ = {a, b} I = {0, 3}, F = {2, 3} aaa L(A) 16 / 89

17 Expressions Regular expressions Expressions built from, ɛ and a (a Σ), using the operations union, product and Kleene iteration ( ). Example E 1 + E 2 L(E 1) L(E 2) E 1E 2 L(E 1) L(E 2) = {uv u L(E 1), v L(E 2)} E L(E ) = S n 0 L(E) L(E) {z } n L(A) = (a + b) a + ɛ + (a + b) aa(a + b) 17 / 89

18 Expressions Regular expressions Expressions built from, ɛ and a (a Σ), using the operations union, product and Kleene iteration ( ). Example E 1 + E 2 L(E 1) L(E 2) E 1E 2 L(E 1) L(E 2) = {uv u L(E 1), v L(E 2)} E L(E ) = S n 0 L(E) L(E) {z } n L(A) = (a + b) a + ɛ + (a + b) aa(a + b) Star-free expressions Expressions built from, ɛ and a (a Σ), using the operations union, product and complementation ( co ). Example (a + b) = co 18 / 89

19 Regular languages (Non-deterministic) finite automata and regular expressions define the same class of word languages, called regular languages. Remark Regular languages over a given alphabet form a Boolean algebra. 19 / 89

20 Plan 1 Automata 2 Logic 3 Schützenberger s theorem 4 Infinite words and trees 20 / 89

21 Logic Logic on words: syntax First-order variables x, y,... and second-order variables X, Y,.... Atomic propositions a(x) (a Σ), x < y, y = x + 1, x X. Boolean connectors,,,..., quantifiers,. MSOL: monadic second-order logic, FOL: first-order logic. Semantics Relational structure associated to w = a 1 a n Σ + with dom(w) = {1,..., n}: dom(w), succ, <, {a a Σ} succ = {(k, k + 1) 1 k < n} < linear order on dom(w) a = {k {1,..., n} a k = a} First-order (second-order, resp.) variables = positions (sets of positions, resp.). 21 / 89

22 Language of ϕ w ϕ L(ϕ) = {w Σ + w ϕ} w models ϕ language of ϕ 22 / 89

23 Language of ϕ w ϕ L(ϕ) = {w Σ + w ϕ} w models ϕ language of ϕ Examples X 0 X 1 x `(x X 0 x X 1) (x X 0 x / X 1) 1 X 1 x `(x X 0 x + 1 X 1) (x X 1 x + 1 X 0) x (x X 1 a(x)) =: ODD a Every odd position carries an a. ϕ := ODD a EVEN b L(ϕ) = (ab) + 23 / 89

24 Language of ϕ w ϕ L(ϕ) = {w Σ + w ϕ} w models ϕ language of ϕ Examples X 0 X 1 x `(x X 0 x X 1) (x X 0 x / X 1) 1 X 1 x `(x X 0 x + 1 X 1) (x X 1 x + 1 X 0) x (x X 1 a(x)) =: ODD a Every odd position carries an a. ϕ := ODD a EVEN b L(ϕ) = (ab) + ϕ := x a(x) b(x + 1) b(x) a(x + 1) a(1) b(n) 24 / 89

25 Logic and automata Büchi (1960) and Elgot (1961) show that finite automata and monadic second-order logic have the same expressive power. Büchi (1962), McNaughton (1966), Rabin (1969) extend the equivalence to infinite words and trees. Motivation: decision problems for logical systems and synthesis problems. Also: descriptive complexity theory relates logics with complexity classes Ptime, NP, Pspace,... (Fagin 1974). Today automata are a central algorithmic tool for logic-related problems: language theory, model-checking, synthesis/control, navigation in XML documents, / 89

26 Büchi Büchi s theorem, 1960 A language is regular if and only if it is definable in MSOL. 26 / 89

27 Büchi Büchi s theorem, 1960 A language is regular if and only if it is definable in MSOL. Proof ( ) Describe accepting runs in MSOL: Partition dom(w) into sets X s, one for each state s. First position belongs to X s0, last one to S f F X f. Consistency of automaton transitions: for each 1 k < n, s S, a Σ, (k X s a(x)) [ (k + 1) X s s :(s,a,a ) δ 27 / 89

28 Büchi s theorem A language is regular if and only if it is definable in MSOL. Proof ( ) Formula ϕ(x 1,..., x k, X k+1,..., X l ) with free variables x i, X j is interpreted over Σ {0, 1} l. Automata for atomic predicates: succ(x 1, x 2) Σ {0} 2 Σ {0} 2 (, 1, 0) (, 0, 1) Boolean connectors, : closure under union, complementation. Quantifier corresponds to a projection from Σ {0, 1} l to Σ {0, 1} l / 89

29 Application: Presburger arithmetic Peano arithmetic FOL(N, +, ) is undecidable (even the existential fragment, i.e. diophantine equations, Matyasevich 1970). Presburger arithmetic FOL(N, +) is decidable (Presburger 1929). Decision procedure via automata: encode natural numbers in binary (lest significant bit left). Automaton for addition: (0, 0, 0), (0, 1, 1) (0, 1, 0), (1, 0, 0) (1,1,0) 0 1 (1, 0, 1) (0, 0, 1) (1, 1, 1) The automata-based procedure applied to formulas in prenex normal form yields automata of triple exponential size (Klaedtke 2004). Quantifier elimination yields a double alternating time procedure. 29 / 89

30 Definability Definability problem Given a restricted fragment of MSOL, say FOL, how can we decide whether a language is definable in that fragment? 30 / 89

31 Definability Definability problem Given a restricted fragment of MSOL, say FOL, how can we decide whether a language is definable in that fragment? For word languages, the answer is provided using algebra. 31 / 89

32 Definability Definability problem Given a restricted fragment of MSOL, say FOL, how can we decide whether a language is definable in that fragment? For word languages, the answer is provided using algebra. Recognizable languages Finite monoid M,, 1 : x (y z) = (x y) z, 1 x = x 1 = x Language L Σ, homomorphism h : Σ M. h recognizes L if L = h 1 (h(l)). L is recognized by M if there exists a homomorphism h : Σ M recognizing L. Languages recognized by finite monoids are precisely the regular languages. 32 / 89

33 Example (a + b) aa(a + b) is recognized by M = {1, a, b, 0, ab, ba}, h(a) = a, h(b) = b with a a = 0, b b = b, x 0 = 0 y = 0, ab b = ab, ab a = a,... h 1 (1) = ɛ h 1 (0) = (a + b) aa(a + b) h 1 (a) = (ab + ) a h 1 (b) = b(ab + ) h 1 (ab) = (ab + ) h 1 (ba) = b(ab + ) a 33 / 89

34 Plan 1 Automata 2 Logic 3 Schützenberger s theorem 4 Infinite words and trees 34 / 89

35 Aperiodicity e x n+1 x x 2 x n x n+p 1 x n = x n+p A monoid is aperiodic if it satisfies x n = x n+1 for some n. Example with M = {1, a, b, 0, ab, ba}, h(a) = a, h(b) = b a a = 0, b b = b, x 0 = 0 y = 0, ab b = ab, ab a = a, ab ab = ab,... is aperiodic: a 2 = a 3 and x 2 = x for every x a 35 / 89

36 Aperiodic language A language is aperiodic if it is recognized by some finite aperiodic monoid. Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. McNaughton & Papert A regular language is star-free if and only if it is definable in FOL. 36 / 89

37 Aperiodic language A language is aperiodic if it is recognized by some finite aperiodic monoid. Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. McNaughton & Papert A regular language is star-free if and only if it is definable in FOL. Example is defined by (a + b) aa(a + b) x y. a(x) a(y) succ(x, y) 37 / 89

38 McNaughton & Papert A regular language is star-free if and only if it is definable in FOL. 38 / 89

39 McNaughton & Papert A regular language is star-free if and only if it is definable in FOL. Proof ( ) Translate union, complementation and concatenation into FOL. Let L i = L(ϕ i), i = 1, 2. Formula for L 1 L 2: x. `ϕ 1 x ϕ 2 >x ϕ x : ϕ relative to dom(w) {i i x}. 39 / 89

40 McNaughton & Papert A regular language is star-free if and only if it is definable in FOL. Proof ( ) Translate union, complementation and concatenation into FOL. Let L i = L(ϕ i), i = 1, 2. Formula for L 1 L 2: x. `ϕ 1 x ϕ 2 >x ϕ x : ϕ relative to dom(w) {i i x}. Proof ( ) x y. a(x) a(y) {z} {z} 1 2 succ(x, y) {z } 3 {z } 4 E 1 = (a + b) a x(a + b), E 2 = (a + b) a y(a + b) E 3 = (a + b) (a x + b x)(a y + b y)(a + b) E 4 = E 1 E 2 E 3 = (a + b) a xa y(a + b) E = (a + b) aa(a + b) 40 / 89

41 Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. Proof( ) Syntactic congruence on Σ Σ : u L v if xuy L xvy L, x, y Σ Show that for every star-free L Σ there exists n > 0 such that w n L w n+1, for all w Σ. 41 / 89

42 Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. Proof( ) Syntactic congruence on Σ Σ : u L v if xuy L xvy L, x, y Σ Show that for every star-free L Σ there exists n > 0 such that w n L w n+1, for all w Σ. w w {a} w w w {a} 42 / 89

43 Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. Proof( ) Syntactic congruence on Σ Σ : u L v if xuy L xvy L, x, y Σ Show that for every star-free L Σ there exists n > 0 such that w n L w n+1, for all w Σ. w w w w L 1 L 2 n 1 n 2 43 / 89

44 Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. Proof( ) Syntactic congruence on Σ Σ : u L v if xuy L xvy L, x, y Σ Show that for every star-free L Σ there exists n > 0 such that w n L w n+1, for all w Σ. w w w w L 1 L 2 n 1+1 n 2 44 / 89

45 Schützenberger s theorem A regular language is star-free if and only if it is aperiodic. Proof( ) Original proof by Schützenberger uses algebra. Simpler proof due to Wilke (1998) by induction over the size of the alphabet Σ and the size of an (aperiodic) monoid recognizing L Σ. Even more simple proof due to Diekert/Kufleitner, / 89

46 Proof by Diekert/Kufleitner 1/3 A tiny bit of algebra M,, 1 monoid, m M Define operation on mm Mm: xm my = xmy mm Mm,, m is a monoid mm Mm,, m aperiodic if M aperiodic mm Mm < M if M aperiodic and m 1 46 / 89

47 Proof by Diekert/Kufleitner 2/3 Preliminary L Σ recognized by homomorphism h : Σ M means L = [ h 1 (m) m h(l) Sufficient to know that each h 1 (m) is star-free. 47 / 89

48 Proof by Diekert/Kufleitner 2/3 Preliminary L Σ recognized by homomorphism h : Σ M means L = [ h 1 (m) m h(l) Sufficient to know that each h 1 (m) is star-free. h 1 (1) M aperiodic and x 1 x n = 1 implies x i = 1 for each i. h 1 (1) = {a Σ h(a) = 1} is star-free 48 / 89

49 Proof by Diekert/Kufleitner 2/3 Preliminary L Σ recognized by homomorphism h : Σ M means L = [ h 1 (m) m h(l) Sufficient to know that each h 1 (m) is star-free. h 1 (1) M aperiodic and x 1 x n = 1 implies x i = 1 for each i. h 1 (m), m 1 h 1 (1) = {a Σ h(a) = 1} is star-free Every word w with h(w) = m can be decomposed as c c c c h 1 (m 1) h 1 (m 2) h 1 (m 3) for some m i with m = m 1 m 2 m 3 and c Σ with h(c) 1. h 1 (m 1), h 1 (m 3): induction over Σ. 49 / 89

50 Proof by Diekert/Kufleitner 3/3 h 1 (m) (cσ Σ c), h(c) 1 T = h((σ \ {c}) ) v 1 v 2 v k w Σ c c c c c c c c T h h h h(v 1) h(v 2) h(v k ) g g h(c)m Mh(c) h(cv 1c) h(cv 2c) h(cv k c) = h(w) Recall: h(c)m Mh(c) is aperiodic and smaller than M. g 1 (m) T is star-free (induction on M) h 1 (t) (Σ \ {c}) star-free (induction on Σ) Star-free languages are closed under substitution. 50 / 89

51 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a 51 / 89

52 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a 52 / 89

53 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a 53 / 89

54 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a 54 / 89

55 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a 55 / 89

56 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). 56 / 89

57 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. 57 / 89

58 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 58 / 89

59 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 59 / 89

60 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 60 / 89

61 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 61 / 89

62 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 62 / 89

63 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a 63 / 89

64 Ehrenfeucht-Fraïssé games FOL definability Ehrenfeucht-Fraïssé games: general method for determining FOL definability (words, graphs etc.) EF-game: Played on two structures S, s, T, t. Spoiler picks an element from either S or T. Duplicator answers with an element in the other structure. Duplicator wins if she can always extend the partial isomorphism. Example FOL on words Structures S = aaba, T = aba: Duplicator wins the 2-round game for FOL with =, < a a b a a b a Duplicator looses the 3-round game (Spoiler chooses 3 a s in aaba). Structures S = a k, T = a l : Duplicator wins the m-round EF-game, iff either max(k, l) > 2 m or k = l. a a a a a a a a Thus, Even = {a n n is even} is not FOL-definable. Syntactic monoid of Even is Z 2 (group, period 2). 64 / 89

65 Plan 1 Automata 2 Logic 3 Schützenberger s theorem 4 Infinite words and trees 65 / 89

66 Words Infinite words Word w : N Σ. Denote by Σ ω the set of infinite words over Σ. Büchi automata A Büchi automaton is a tuple A = S, Σ, δ, s 0, F with: S finite set of states Σ alphabet δ S Σ S transition relation s 0 S initial state, F S set of accept states Runs, languages Run: infinite sequence of transitions a s 0 a 0 1 s1 Accepting run: {n s n F } is infinite Accepted language L(A) Σ ω : a L(A) = {a 0a 1 s i : s 0 a 0 1 s1 accepting run} 66 / 89

67 Büchi automata and determinism Deterministic Büchi automata are strictly weaker than deterministic ones. L = {w {a, b} ω w i = a for finitely many i} a, b b b 0 1 Assume L = L(A) for A deterministic, A = S, Σ, s 0, δ, F. b ω L run s 0 b k 1 s 1 F for some k 1. b k 1 ab ω L run s 0 b k 1 s 1 b k 2 s 2 F for some k 2. Thus one constructs b k 1 ab k 2 a L(A) \ L. 67 / 89

68 Regular languages Büchi automata and determinism Büchi automata can be determinized into automata with more powerful acceptance conditions (Muller, Rabin,... ). Cf. Safra s construction. Regular omega-word languages Characterizations of regular word languages extend to omega-word languages: Acceptance by Büchi automata Definability in MSOL Definability by ω-regular expressions Recognition by finite monoids 68 / 89

69 Logic First-order omega-word languages Characterizations of star-free word languages extend to omega-word languages (Ladner/Thomas, Perrin,... ): Definability in FOL Definability by star-free ω-regular expressions Recognition by finite aperiodic monoids 69 / 89

70 Logic First-order omega-word languages Characterizations of star-free word languages extend to omega-word languages (Ladner/Thomas, Perrin,... ): Definability in FOL Definability by star-free ω-regular expressions Recognition by finite aperiodic monoids Linear temporal logics (LTL), Kamp 1968 ϕ ::== tt a Σ ϕ ϕ ϕ Xϕ ϕ U ϕ Interpreted over w = w 1w 2... Σ ω : a holds if the current position is labeled by a, Next: Xϕ holds at position k if ϕ holds at position k + 1, Until: ϕ 1 U ϕ 2 holds at position i if ϕ 2 holds at some (later) position k i and ϕ 1 holds in-between (for all i j < k). 70 / 89

71 Logic First-order omega-word languages Characterizations of star-free word languages extend to omega-word languages (Ladner/Thomas, Perrin,... ): Definability in FOL Definability by star-free ω-regular expressions Recognition by finite aperiodic monoids Linear temporal logics (LTL), Kamp 1968 ϕ ::== tt a Σ ϕ ϕ ϕ Xϕ ϕ U ϕ Interpreted over w = w 1w 2... Σ ω : a holds if the current position is labeled by a, Next: Xϕ holds at position k if ϕ holds at position k + 1, Until: ϕ 1 U ϕ 2 holds at position i if ϕ 2 holds at some (later) position k i and ϕ 1 holds in-between (for all i j < k). LTL operators Future Fϕ ::== tt U ϕ Always Gϕ ::== (F( ϕ)) 71 / 89

72 From LTL to FOL... Translate LTL formula ϕ into FOL formula ϕ(x): ϕ = a ϕ = Xψ ϕ = ψ 1 U ψ 2 ϕ(x) = a(x) ϕ(x) = y. `succ(x, y) ψ(y) ϕ(x) = z. `x z ψ 2(z) y. (x y < z ψ 1(y)) 72 / 89

73 From LTL to FOL... Translate LTL formula ϕ into FOL formula ϕ(x): ϕ = a ϕ = Xψ ϕ = ψ 1 U ψ 2 ϕ(x) = a(x) ϕ(x) = y. `succ(x, y) ψ(y) ϕ(x) = z. `x z ψ 2(z) y. (x y < z ψ 1(y))... and from FOL to LTL Original proof (by Kamp) very technical. Alternative proof (by Wilke 98) over aperiodic monoids. 73 / 89

74 Temporal logics and program verification Verification Two basic problems: model-checking and synthesis. Model-checking program specification model formula 74 / 89

75 Temporal logics and program verification Verification Two basic problems: model-checking and synthesis. Model-checking program specification model Kripke structure formula Temporal logics 75 / 89

76 Model-checking Given a Büchi automaton A and an LTL formula ϕ, check whether each run in A satisfies ϕ. 76 / 89

77 Model-checking Given a Büchi automaton A and an LTL formula ϕ, check whether each run in A satisfies ϕ. Translate ϕ into Büchi automaton B ϕ and check empty intersection L(A) L(B f )? = 77 / 89

78 Synthesis C Church s problem (1950): given an input/output relation R, construct a circuit C implementing R. Example: if the current input is 1, then the corresponding output is 1; if the input is 0 then the output is the parity of the number of 0 s seen so far. A finite memory circuit C does the job. 78 / 89

79 Synthesis C Church s problem (1950): given an input/output relation R, construct a circuit C implementing R. Example: if the current input is 1, then the corresponding output is 1; if the input is 0 then the output is the parity of the number of 0 s seen so far. A finite memory circuit C does the job. Program synthesis Program synthesis amounts to write a program that satisfies a specification, no matter how the environment acts. 79 / 89

80 Synthesis C Church s problem (1950): given an input/output relation R, construct a circuit C implementing R. Example: if the current input is 1, then the corresponding output is 1; if the input is 0 then the output is the parity of the number of 0 s seen so far. A finite memory circuit C does the job. Program synthesis Program synthesis amounts to write a program that satisfies a specification, no matter how the environment acts. Trees, games Reasoning about the synthesis problem: Branching behaviors needed. Consider trees instead of words. Program/environment interaction can be seen as a two-player game. Constructing a circuit/program/... amounts to find a winning strategy. 80 / 89

81 Tree automata Automata on infinite, binary trees A tree automaton is is a tuple A = S, Σ, δ, s 0 with: δ S Σ S 2 transition relation Acceptance condition: parity (Mostowski 1984, Emerson/Jutla 1991). Infinite binary trees t : {0, 1} Σ. Children of node i {0, 1} : i0, i1. Run ρ of A on t: labeling by states ρ : {0, 1} S compatible with δ: (ρ(i), t(i), ρ(i0), ρ(i1)) δ. Coloring χ : S {0,... n} of states with priorities. Accepting run: For each path, the minimal priority seen infinitely often, is even. 81 / 89

82 Automata and logic Büchi tree automata Büchi tree automata are strictly weaker than parity tree automata. Example: each path of the tree has finitely many a-nodes. Parity tree automaton: S = {s a, s b }, Σ = {a, b}, δ = {s a} {a} S 2 {s b } {b} S 2. χ(s b ) = 0, χ(s a) = 1. Complementation (Rabin 1972) Parity tree automata can be effectively complemented. Complementation via games (Gurevich/Harrington 1982) Two players, Automaton and Pathfinder. Player Automaton chooses a transition, player Pathfinder chooses a direction (left/right) to proceed. The outcome is an infinite sequence from (Σ S) ω. Automaton wins the game if each outcome satisfies the parity condition. 82 / 89

83 Example a b a 83 / 89

84 Example a s a b a 84 / 89

85 Example a s a b a 85 / 89

86 Example a s a b s b a 86 / 89

87 Example a s a b s b Outcome (a, s a)(b, s b )(a, s a) a s a Winning strategy for Automaton = accepting run for the parity tree automaton. The above game is a parity game. In particular, it is determined (Martin s theorem), i.e. either Automaton or Pathfinder has a (memoryless) winning strategy. Turn Pathfinder s winning strategy into a parity tree automaton (accepting the complementary tree language). 87 / 89

88 Rabin Rabin s theorem, 1972 MSOL is decidable over infinite binary trees. Applications Various applications: decidability of synthesis problems decidability of MSOL over trees with countable branching decidability of modal logics over Kripke structures / 89

89 Thank you! 89 / 89