Formale Systeme 2. Prof. Dr. Peter H. Schmitt KIT I NSTITUT F U R T HEORETISCHE I NFORMATIK

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1 Formale Systeme 2 Prof. Dr. Peter H. Schmitt KIT I NSTITUT F U R T HEORETISCHE I NFORMATIK KIT Universita t des Landes Baden-Wu rttemberg und nationales Forschungszentrum in der Helmholtz-Gemeinschaft

2 Repetition Linear Temporal Logic Prof. Dr. Peter H. Schmitt FM2 2/36

3 LTL Formulas Definition Let PVar be a set of propositional atoms. The set of LTL formulas, LTLFor is defined by 1. PVar LTLFor 2. 1, 0 LTLFor 3. If A, B in LTLFor, then also A, A B, A B, A B 4. if A, B LTLFor then also 4.1 GA and FA, (sometimes also A and A) 4.2 A U B LTLFor 4.3 X A Prof. Dr. Peter H. Schmitt FM2 3/36

4 Temporal Structures A temporal structure is a triple T = (S, R, I) with S a nonempty set, called the set of abstract time points. R a strict transitive relation, the temporal before relation, I a valuation function I : (PVar S) {W, F} An omega structure is the special case with (S, R) = (N, <). We will present omega structures in the equivalent form T = (N, <, ξ) with ξ : N 2 PVar and the intuition p ξ(n) in T atom p is true at time point n I(p, n) = W For ξ : N 2 PVar and n N we will use ξ n to stand for the final segment of ξ starting at n. In symbol ξ n (m) = ξ(n + m). In particular ξ 0 = ξ. Prof. Dr. Peter H. Schmitt FM2 4/36

5 LTL Semantics Let T = (N, <, ξ) be an omega structure of the propositional signature PVar. ξ = p iff p ξ(0) (p a propositional atom) ξ = op(a, B) for a propositional combination op(a, B) of A and B as usual ξ = GA iff for all n N it is true ξ n = A ξ = FA iff there exists an n N with ξ n = A ξ = A U B iff there is n N with ξ n = B and for all m with 0 m < n we have ξ m = A ξ = X A iff ξ 1 = A Prof. Dr. Peter H. Schmitt FM2 5/36

6 Simple LTL Tautologies The following formulas are true for all omega structures: 1. FA 1 U A 2. GA F A (1 U A) Prof. Dr. Peter H. Schmitt FM2 6/36

7 Repetition Büchi Automata Prof. Dr. Peter H. Schmitt FM2 7/36

8 Omega Words Let V be a finite alphabet. V ω denotes the set of infinite words made up of letters from V. V denotes the set of all finite words over V. w(n) refers to the n-th letter in w for n 0. w (n) stands for the finite initial segment w(0)... w(n) of w. Omega words are sometime also referenced as ω words using the last letter of the Greek alphabet. Prof. Dr. Peter H. Schmitt FM2 8/36

9 Operations on Omega Words Let K V and J V ω. 1. K ω denotes the set of omega words w 1... w i... such that w i K for all i 2. The concatenation of K with J is defined by 3. KJ = {w 1 w 2 w 1 K, w 2 J} Concatenation of two sets of infinite words would not make sense. K = {w V ω w (n) K for infinitely many n} Since we may think of K as a kind of limit of K (in the mathematical sense of the word) some authors denote it by lim(k ). Prof. Dr. Peter H. Schmitt FM2 9/36

10 Büchi Automata A Büchi automaton A = (S, V, s 0, δ, F) is a non-deterministic finite automaton with S V s 0 S δ : S V P(S) F S a finite set of states an alphabet the initial state the transition function the set of accepting (or final) states Prof. Dr. Peter H. Schmitt FM2 10/36

11 Büchi Acceptance A run for A is an infinite sequence s 0,..., s n,... of states, starting with s 0 such that for all 0 n there is an a V with s n+1 δ(s n, a). A run (s n ) n 0 is an accepting run if s n F is true for infinitely many n. Given w V ω we call a run s 0,..., s n,... a run for w, if for all 0 n s n+1 δ(s n, w(n)) is true. The language accepted by A is defined by L ω (A) = {w V ω there is an accepting run for w} The sets L of omega-words accepted by a Büchi automaton A are also called omega-regular sets. Prof. Dr. Peter H. Schmitt FM2 11/36

12 Example Büchi Automaton a s 0 s 1 {a, b} a This automaton, N afin, accepts all words that eventually contain only the letter a. Alternatively: L ω (N afin ) = {a, b} a ω Prof. Dr. Peter H. Schmitt FM2 12/36

13 Theorem Review For every LTL formula C there is a Büchi automaton A C such that L ω (A C ) = {ξ V ω ξ = C}. If PVar are the propositional atoms in C, the label vocabulary V of the Büchi automaton consists of all subsets of PVar, i.e., V = P(PVar). Prof. Dr. Peter H. Schmitt FM2 13/36

14 A 1 A 2 S = S 1 S 2 {1, 2} s 0 = (s1 0, s0 2, 1) F = F 1 S 2 {1} for all s 1 S 1, s 2 S 2, i {1, 2} (t 1, t 2, 1) δ((s 1, s 2, 1), a) t 1 δ 1 (s 1, a) and t 2 δ 2 (s 2, a) and s 1 F 1 (t 1, t 2, 2) δ((s 1, s 2, 1), a) t 1 δ 1 (s 1, a) and t 2 δ 2 (s 2, a) and s 1 F 1 (t 1, t 2, 2) δ((s 1, s 2, 2), a) t 1 δ 1 (s 1, a) and t 2 δ 2 (s 2, a) (t 1, t 2, 1) δ((s 1, s 2, 2), a) t 1 δ 1 (s 1, a) and t 2 δ 2 (s 2, a) and s 2 F 2 Prof. Dr. Peter H. Schmitt FM2 14/36

15 Second Order Definability of Omega Regular Sets Prof. Dr. Peter H. Schmitt FM2 15/36

16 Representation of Omega Words Vocabulary Let Σ V be the following vocabulary a constant symbol 0 the first element a unary function symbol s the successor function a unary relation symbol a(x) for any letter a in V a binary relation symbol < the order relation For a Σ V structure W we think of the elements in the universe W as the positions of a word. 0 W will be the first position s W (p) will yield the next position after p. W = a(p) for a position p W is supposed to mean that at p the letter a V occurs. Of course not every Σ V structure makes sense. Prof. Dr. Peter H. Schmitt FM2 16/36

17 Representation of Omega Words The axioms FOW 1. 0 is interpreted as the smallest element. x(x = 0 0 < x) 2. For every p exactly one of the atomic formulas a(p) for a V is true. x a V (a(x)) a,b V,a b (a(x) b(x)) 3. For all elements p s(p) is the successor of p. x(x < s(x)) z(x < z z < s(x)) 4. The successor s is injective. x y(s(x) = s(y) x = y) 5. < is a transitive and irreflexive relation. x y z(x < y y < z x < z) x (x < x) 6. All elements are of the form s n (0) for some n N. X(X(0) x(x(x) X(s(x))) y(x(y))) Prof. Dr. Peter H. Schmitt FM2 17/36

18 Omega Words as FOW Structures There is a 1-1 correspondence between omega words over the alphabet V and Σ V structures satisfying FOW. For w V ω we will denote the correspondig structure by W w. Without loss of generality we will assume that the universe of W w is the set N of natural numbers. Prof. Dr. Peter H. Schmitt FM2 18/36

19 Definability of the order relation The formula less(x, y) = X(X(x) z(x(z) X(s(z))) X(y)) x y defines the order relation in all structures satisfying FOW. Prof. Dr. Peter H. Schmitt FM2 19/36

20 Definable Sets of Omega Words Let F be a formula in monadic second order logic with signature Σ v. Example L ω (F) = {w V ω W w = F } F = n m(n < m a(m)) defines the set L ω (N afin ). Prof. Dr. Peter H. Schmitt FM2 20/36

21 Definability Theorem Let L V ω be a set of infinite words. There is a Büchi automaton accepting L iff there is a second order formula in signature Σ V defining L. Second Order Definability of Büchi acceptance Let L V ω be a set of infinite words. If there is a Büchi automaton N accepting L, i.e., L ω (N ) = L, then there is a second order formula Φ defining L. i.e., L ω (Φ) = L. Omega Regularity of Second Order Definable Sets For any second order formula Φ in the signature Σ V the set L ω (Φ) = L is omega regular. Prof. Dr. Peter H. Schmitt FM2 21/36

22 Second Order Definability of Büchi acceptance Proof Given: N = (S, V, s 0, δ, F) with S = {q 1,..., q n } and F = {q 1,..., q f }. w L ω (N ) iff there is a run s = (s i ) 0 i of N for w with infinitely many final states. iff there are subsets X k N, 1 k n such that 1. 1 k n X k = N 2. the X k are mutually disjoint 3. the sequence of states (s i ) 0 i definied by s i = q k i X k is a run of N for w with infinitely many final states. Prof. Dr. Peter H. Schmitt FM2 22/36

23 Second Order Definability of Büchi acceptance Proof w L ω (N ) iff there are subsets X k N, 1 k n such that 1. 1 k n X k = N 2. the X k are mutually disjoint 3. (s i ) 0 i definied by s i = q k i X k is a run of N for w with many final states. iff W w = Φ = X 1... X n (U D R F) with 1. U = x( 1 k n X k(x)) 2. D = 1 k<r n x(x k(x) X r (x)) 3. R = X 0 (0) x( 1 k,r n (X k(x) X r (s(x)) a V k,r a(x))) with V k,r = {a V q r δ(q k, a)} 4. F = 1 k f x y(x < y X k(y)) Prof. Dr. Peter H. Schmitt FM2 23/36

24 Omega Regularity of Definable Sets Proof Claim For any second order formula Φ in the signature Σ V the set L ω (Φ) = L is omega regular. Naturally, the proof will proceed by induction on the complexity of Φ. This necessitates that we also consider formulas Φ(x 1,..., x p, X 1,..., X q ) with free first-order and second-order variables. We will for any numbers p of first-order and q of second-order free variables define new vocabularies V p,q such that V 0,0 = V and Φ(x 1,..., x p, X 1,..., X q ) defines an omega-regular subset of V ω p,q. Prof. Dr. Peter H. Schmitt FM2 24/36

25 A Short Note on Notation Instead of we will write val M,β,γ (Φ(x 1,..., x p, X 1,..., X q )) = 1, M = Φ(x 1,..., x p, X 1,..., X q )[s 1,..., s p, S 1,..., S q ] with s i = β(x i ) and S j = γ(x j ). Postfixing the assignments in square brackets to the formula is more convenient than manipulating the βs and γs Prof. Dr. Peter H. Schmitt FM2 25/36

26 The Vocabularies V p,q V p,q = V {0, 1} p {0, 1} q For p = q = 0 we get V p,q = V, as promissed. As an example consider p = 3, q = 2 and a V. Then b 1 = a, 0, 0, 0, 0, 0 b 2 = a, 0, 0, 1, 0, 0 b 3 = a, 0, 1, 0, 0, 1 are examples of letters in V 3,2. Let a[i] for 0 i < p + q + 1 denote the i-th position of the letter a. Thus, e.g. b 1 [0] = a b 2 [3] = 1 b 3 [5] = 1 Prof. Dr. Peter H. Schmitt FM2 26/36

27 The Subset K p,q V ω p,q The subset K p,q V ω p,q will play a crucial role. K p,q = {w Vp,q ω for all 1 i p there is exactly one k s.t. the i-th position in the letter w(k) equals 1} Or using the a[i] notation K p,q = {w Vp,q ω for all 1 i p there is exactly one k with w(k)[i] = 1} Observe that K 0,0 = V ω. Exercise Show that all K p,q are omega regular set in the vocabulary V p,q. Prof. Dr. Peter H. Schmitt FM2 27/36

28 The Hidden Meaning of K p,q A word w K p,q codes a word 1. w 0 V ω, 2. p elements s i N and 3. q subsets S j N w 0 (n) = w(n)[0] n N s i = the unique n with w(n)[i] = 1 1 i p S j = {n N w(n)[p + j] = 1} 1 j q Prof. Dr. Peter H. Schmitt FM2 28/36

29 The Inductive Claim For any formula Φ(x 1,..., x p, X 1,..., X q ) the set L ω (Φ( x, X)) = {w K p,q W w 0 = Φ[s 1,..., s p, S 1,..., S q ]} is omega regular W w 0 is the structure already used above. Note that the special case of p = q = 0 is the claim of the theorem we want to prove. The proof proceeds by structural induction on Φ Prof. Dr. Peter H. Schmitt FM2 29/36

30 Inductive Proof Initial Case: Φ x i = x k Let A i,k V p,q be the set of all letters a with a[i] = a[k] = 0 and B i,k V p,q be the set of all letters b with b[i] = b[k] = 1. Then L ω (x i = x k ) = {w K p,q W w 0 = x i = x k [s i, s k ]} = A i,k B i,ka ω i,k K p,q Thus L ω (x i = x k ) is omega regular. Prof. Dr. Peter H. Schmitt FM2 30/36

31 Inductive Proof Initial Case: Φ a(x i ) Let D a,i be the set of letters d V p,q with d[0] = a and d[i] = 1. Then L ω (a(x i )) = {w K p,q W w 0 = a(x i )[s i ]} = for the unique n with w(n)[i] = 1 we must have w(n)[0] = a = K p,q Vp,qD a,i Vp,q ω Thus L ω (a(x i )) is omega regular. The remaining initial cases x i = s(x k ) and X j (x i ) are easy variations. Prof. Dr. Peter H. Schmitt FM2 31/36

32 Induction Step Case: Φ( x, X) = x i Φ 0 ( x, X) The remaining inductions steps Φ = Φ 1 Φ = Φ 1 Φ 2 Φ = Φ 1 Φ 2 and Φ = X j Φ 0 are either simple or similar. Notational Simplification: i = p. Thus Φ(x 1,..., x p 1, X) = x p Φ 0 (x 1,..., x p 1, x p, X). By induction hypothesis L 1 = {w K p,q W w 0 = Φ 0 ( x, x p, X)[ s, s p, S]} is an omega regular subset of V ω p,q. We want to show that L = {w K p 1,q W w 0 = x p Φ 0 ( x, X)[ s, S]} is an omega regular subset of V ω p 1,q. Prof. Dr. Peter H. Schmitt FM2 32/36

33 Homomorphisms The still open proof obligation requires us to compare omega regular sets over different vocabularies. This is new! Let V 1, V 2 be two vocabularies, µ : V 1 V 2 an arbitrary mapping. The extension of µ to µ : V1 ω V 2 ω is called an homomorphism from V1 ω to V 2 ω. Lemma L an omega regular subset of V ω 1 and µ : V ω 1 V ω 2 an homorphism. Then µ(l) is an omega regular subset of V ω 2. Prof. Dr. Peter H. Schmitt FM2 33/36

34 Induction Step Case: Φ( x, X) = x i Φ 0 ( x, X) Assume L 1 = {w K p,q W w 0 = Φ 0 ( x, x p, X)[ s, s p, S]} is ω regular in V ω p,q. Show L = {w K p 1,q W w 0 = x p Φ 0 ( x, X)[ s, S]} is ω regular in Vp 1,q ω. Define µ : V p,q V p 1,q by µ( a, c 1,..., c p 1, c p, d 1,..., d q ) = a, c 1,..., c p 1, d 1,..., d q Claim : L = {µ(w) w L 1 } = µ(l 1 ). From this claim we obtain by the homomorphism lemma that L is omega regular, as desired. Prof. Dr. Peter H. Schmitt FM2 34/36

35 Proof of Claim To increase readability we supress all variables except x p. w L 1 iff W w 0 = Φ 1 (x p )[s p ] W w 0 = x p Φ 1 iff W µ(w) 0 = x p Φ 1 µ(w) 0 = w 0 and µ(w)(z) = w(z) for z not shown since µ only touches position p iff µ(w) L w L iff W (w ) 0 = x pφ 1 iff W (w ) 0 = Φ 1(x p )[n p ] for some n p N iff W w 0 = Φ 1 (x p )[s p ] for an appropriate w Vp,q ω iff w L 1 w is { obtained by replacing w (n) = (a, c 1,..., c p 1, d 1,..., d q ) a, c1,..., c by p 1, 1, d 1,..., d q if n = n p a, c 1,..., c p 1, 0, d 1,..., d q if n n p Prof. Dr. Peter H. Schmitt FM2 35/36

36 Example The formula X(X(0) x(x(x) X(x + 2)) x(x(x) a(x))) describes the set of all omega words with the letter a occuring at all even positions. At uneven positions a may occur or not. By the theorem we know that there is a Büchi automaton for this set of words. By contrast we know, that this set if not LTL-definable. Prof. Dr. Peter H. Schmitt FM2 36/36