Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

Transcription

1 Weighted Automata on Infinite Words in the Context of Attacker-Defender Games V. Halava 1,2 T. Harju 1 R. Niskanen 1 I. Potapov 2 1 Department of Mathematics and Statistics University of Turku, Finland 2 Department of Computer Science University of Liverpool, UK Workshop on Automatic Sequences, 2015 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

2 Attacker-Defender Games Two players: Attacker, Defender. Players play in turns using available moves. Initial and target configurations. Configuration is a sequence of alternating moves. Play is an infinite sequence of configurations. Attacker wins if the target configuration is reachable in a play starting from the initial configuration. Otherwise Defender wins. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

3 Games we consider 1 Weighted Word Games 2 Word Game on pairs of group words 3 Matrix Games 4 Braid Games Theorem It is undecidable whether Attacker has a winning strategy in these games. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

4 Weighted Word Games Players are given sets of words over free group alphabet. They play words in turns. Attacker s goal to reach a certain word. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

5 Word Games on pairs of group words Similar to Weighted Word Games but now the weight is encoded as a unary word. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

6 Matrix Games Players are given sets of matrices from SL(n, Z). We encode words from Word Games on pair of words into 4 4 matrices. M2 M3 M1 x0 x M4 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

7 Braid Games Two variants - played on 3 (B 3 ) or 5 strands (B 5 ). Players are given sets of braid words. = Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

8 Universality Problem for Finite Automata For given Finite Automaton A, over alphabet A, is its language L(A) = A? For given Büchi Automaton B, over alphabet A, is its language L(B) = A ω? Both are known to be decidable. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

9 Universality Problem for Weighted Automata Extend automaton by adding weight function γ to transitions. For given Weighted Automaton A γ, over alphabet A, is its language L(A γ ) = A? Shown to be undecidable by Halava and Harju in Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

10 Weighted automaton on Infinite Words Let A = (Q, A, σ, q 0, F, Z) be a finite automaton, where Q is the set of states, A is the alphabet, σ is the set of transitions, q 0 is the initial state, F is the set of final states, and Z is the additive group of integers. Transitions In the form t = q, a, q, z. Weight of a path Let π = t i0 t i1 be an infinite path of A. Let p π be a finite prefix. Weight of p is γ(p) = z z n Z. Acceptance condition Let w A ω. It is accepted by A if there exists a computation π such that infinite number of its prefixes p have zero weight. (a, 1) (a, 1) (a, 0), (b, 0) (b, 2) (a, 0), (b, 0) Words beginning with aab, aba or baa are accepted. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

11 Infinite Post Correspondence Problem (ωpcp) In ωpcp we are given two morphisms h, g : A B. Does there exist an infinite word w such that for all prefixes p either h(p) < g(p) or g(p) < h(p)? Shown to be undecidable by Halava and Harju for domain alphabets A 9 and improved to A 8 by Dong and Liu. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

12 Idea of construction The goal is to construct an automaton A such that L(A) = A ω if and only if the instance of ωpcp has no solution. An infinite word w A ω is accepted by A if and only if for some finite prefix p of w, g(p) h(p) and h(p) g(p). Such a prefix p does exist for all infinite words except for the solutions of the instance (h, g). We call the verification of such a prefix p error checking. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

13 One possible path in the automaton h : g : abaab ab aab aa babaa k aba a babaab a aba b abaa A B C l D j k i l Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

14 Example Relevant part of the automaton ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) q2 (1, 0) (2, 0) q4 (1, 5) (2, 4) We ll show that non-solutions are accepted and why the solution is not accepted. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

15 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Clearly w = is the solution. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

16 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Clearly w = is the solution. Computation path (1, 1) The counter is 1, i.e. this is not an accepting path. q0 (1, 0) (2, 0) q4 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

17 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Clearly w = is the solution. Computation path q0 (1, 5) (1, 3), (2, 3) q2 (1, 5) (2, 4) (1, 0) (2, 0) q4 The counter is 5 3 2k 4 (when (2, 4) is the final transition) or 5 3 (2k + 1) 5 (when (1, 5) is the final transition) for some k N, i.e. this is not an accepting path. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

18 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Clearly w = is the solution. Computation path q0 (1, 3) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) q2 (1, 0) (2, 0) q4 (1, 5) (2, 4) Let s consider one of the possible paths. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

19 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Clearly w = is the solution. Computation path q0 (1, 3) q1 (1, 1) (1, 3), (2, 3) q2 (1, 0) (2, 0) q4 (2, 4) Now the counter is 3 2k k 4 for some k, k N, i.e. this is not an accepting path. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

20 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Consider w = 2. Computation path (2, 0) The counter is 0, i.e. this is an accepting path. q0 (1, 0) (2, 0) q4 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

21 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Consider w = 11. Computation path q0 (1, 5) q2 (1, 0) (2, 0) q4 (1, 5) The counter is 0, i.e. this is an accepting path. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

22 Example ω PCP instance Let g(1) = 12, g(2) = 12, h(1) = 1, h(2) = 2. Relevant part of the automaton q0 (1, 1), (2, 0) (1, 5), (2, 5) q1 (1, 1), (1, 5) (2, 1), (2, 5) (1, 3), (2, 3) (1, 0) (2, 0) q2 q4 (1, 5) (2, 4) Consider w = 122. Computation path q0 (1, 3) q1 (2, 1) q2 (1, 0) (2, 0) q4 (2, 4) The counter is 0, i.e. this is an accepting path. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

23 Additional paths are needed for different forms of PCP instances: Image under h is longer and error is far away (part C is non-empty). Image under g is longer and error is far away (part C is non-empty). Image under h is longer and error is close (part C is empty, parts B and D are done simultaneously). Image under g is longer and error is close (part C is empty, parts B and D are done simultaneously). Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

24 Theorem It is undecidable whether or not L(A) = A ω holds for 5-state integer weighted automata A on infinite words over alphabet A. Theorem It is undecidable whether or not L R (B) = A ω holds for 5-state integer weighted automata B on infinite words over alphabet A using reverse acceptance condition. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

25 Attacker-Defender Games Two players: Attacker, Defender. Players play in turns using available moves. Initial and target configurations. Configuration is a sequence of alternating moves. Play is an infinite sequence of configurations. Attacker wins if target configuration is in reachable in a play starting from initial configuration. Otherwise Defender wins. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

26 Games we consider 1 Weighted Word Games 2 Word Game on pairs of group words 3 Matrix Games 4 Braid Games Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

27 Weighted Word Games Players are given sets of words over free group alphabet. Defender plays a word of the automaton letter by letter. Attacker plays words corresponding to the letter played by Defender and respecting the structure of the automaton. Attacker tries to reach the word corresponding to the accepting configuration starting from the word corresponding to the initial configuration. Attacker has a winning strategy if and only if every word that Defender plays is accepted, that is the automaton is universal. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

28 Word Games on pairs of group words Similar to Weighted Word Games but now the weight is encoded as a unary word. Using an additional trick, initial and final words are (ε, ε). Attacker has a winning strategy if and only if every word that Defender plays is accepted, that is the automaton is universal. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

29 Idea of construction for Word Game Initial word q 0. Defender plays letter a. Current word: q 0 a Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

30 Idea of construction for Word Game Initial word q 0. Attacker plays word a q 0 q 1 corresponding to transition q 0, a, q 1. Current word: q 0 a(a q 0 q 1 ) = q 1 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

31 Idea of construction for Word Game Initial word q 0. Defender plays letter b. Current word: q 1 b Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

32 Idea of construction for Word Game Initial word q 0. Attacker plays word bq 1 q 2 corresponding to transition q 1, b, q 2. Current word: q 1 b(bq 1 q 2 ) = q 2 If Attacker does not match the letter or plays incorrect transition, uncancellable elements will remain. In actual Word Game, the construction is slightly more complicated. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

33 Counter Example (In previous construction, Attacker commits to a path, while Defender does not commit to a word.) Universal automaton q0 (a, 1) q1 (b, 0) (a, 1) (b, 0) (a, 1) (a, 0) (b, 1) q2 (a, 0) (b, 0) Now from q 0, Defender plays a. If Attacker plays a q 0 q 1, then Defender will play only b and there is no path with 0 weight. If Attacker plays a q 0 q 2, then Defender will play only a and there is no path with 0 weight. Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

34 Matrix Games Players are given sets of matrices from SL(n, Z). Starting from initial vector x 0, players apply their matrices in turns. Attacker s goal is to reach x 0. We encode words from Word Games on pair of words into 4 4 matrices. Since matrices are from SL(4, Z), this is only possible when matrix played by the players is the identity matrix. Identity matrix is reachable if and only if the empty word is reachable in the Word Game. M3 M2 M1 x0 x M4 Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

35 Braid Games Two variants - played on 3 (B 3 ) or 5 strands (B 5 ). Players are given sets of braid words. In B 3, starting from a braid word corresponding to the initial word of Weighted Word Game, Attacker s aim is to reach the trivial braid. B 5 contains direct product of two free groups of rank 2 as a subgroup. We encode words of Word Game on pair of words into braids. In both variants, the trivial braid is reachable if and only if the empty word is reachable in the corresponding Word Game. = Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27

36 THANK YOU! Halava, Harju, Niskanen, Potapov Weighted Automata on Infinite Words in... Automatic / 27