Boundedness Games. Séminaire du LIGM, April 16th, Nathanaël Fijalkow. Institute of Informatics, Warsaw University Poland

Transcription

1 Boundedness Games Séminaire du LIGM, April 16th, 2013 Nathanaël Fijalkow Institute of Informatics, Warsaw University Poland LIAFA, Université Paris 7 Denis Diderot France (based on joint works with Krishnendu Chatterjee, Thomas Colcombet, Florian Horn and Martin Zimmermann)

2 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

3 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

4 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

5 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

6 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

7 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs controlled by Eve controlled by Adam

8 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ωb winning condition: parity and all counters are bounded

9 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs 2 0 parity condition: the minimal priority seen infinitely often is even

10 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 0 c 2 = 0 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

11 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 0 c 2 = 0 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

12 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 0 c 2 = 1 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

13 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 0 c 2 = 1 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

14 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 1 c 2 = 0 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

15 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i i, i i,ε i, i c 1 = 1 c 2 = 0 ε, i i, r i,ε ε,ε ε, i r, i ε : nothing i : increment r : reset ε, r

16 Definition of ωb-games 1 Two-player turn-based games over finite or infinite graphs ε, i 3 2 i, i ε, i i, r 4 i,ε i, i 0 ε,ε 2 3 ε, i i,ε 1 r, i ωb winning condition: parity and all counters are bounded ε, r

17 Strategy (for Eve) 2 General form σ : V + V

18 Strategy (for Eve) 2 General form σ : V + V Positional or memoryless σ : V V

19 Strategy (for Eve) 2 General form σ : V + V Positional or memoryless σ : V V Theorem (Müller and Schupp) In parity games, both players have memoryless winning strategies.

20 Strategy (for Eve) 2 General form σ : V + V Positional or memoryless σ : V V Theorem (Müller and Schupp) In parity games, both players have memoryless winning strategies. What about ωb games?

21 Strategy (for Eve) 2 General form σ : V + V Positional or memoryless σ : V V Theorem (Müller and Schupp) In parity games, both players have memoryless winning strategies. What about ωb games? Finite-memory { σ : V M V µ : M E M

22 Why proving the existence of finite-memory strategies? 3 Thomas Colcombet s habilitation: Existence of finite-memory strategies in (some) boundedness games = Decidability of cost MSO over infinite trees = Decidability of the index of the non-deterministic Mostowski s hierarchy!

23 Quantification 4 Eve wins means: σ (strategy for Eve), π (paths), N N, σ (strategy for Eve), N N, π (paths), π satisfies parity and each counter is bounded by N.

24 Quantification 4 Eve wins means: σ (strategy for Eve), π (paths), N N, σ (strategy for Eve), N N, π (paths), π satisfies parity and each counter is bounded by N. Is this: 1 Equivalent? 2 Decidable?

25 Quantification 4 Eve wins means: σ (strategy for Eve), π (paths), N N, σ (strategy for Eve), N N, π (paths), π satisfies parity and each counter is bounded by N. Is this: 1 Equivalent? Sometimes... 2 Decidable? Not always...

26 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies.

27 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies. Cost-parity games [F. and Zimmermann, 2012].

28 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies. Cost-parity games [F. and Zimmermann, 2012]. 2 Over pushdown graphs: decide the winner and construct finite-memory strategies.

29 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies. Cost-parity games [F. and Zimmermann, 2012]. 2 Over pushdown graphs: decide the winner and construct finite-memory strategies. Pushdown finitary games [Chatterjee and F., 2013].

30 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies. Cost-parity games [F. and Zimmermann, 2012]. 2 Over pushdown graphs: decide the winner and construct finite-memory strategies. Pushdown finitary games [Chatterjee and F., 2013]. 3 Over infinite graphs: prove the existence of finite-memory winning strategies.

31 The questions I am interested in 5 Boundedness games: 1 Over finite graphs: decide the winner efficiently and construct small finite-memory strategies. Cost-parity games [F. and Zimmermann, 2012]. 2 Over pushdown graphs: decide the winner and construct finite-memory strategies. Pushdown finitary games [Chatterjee and F., 2013]. 3 Over infinite graphs: prove the existence of finite-memory winning strategies. Ongoing work with Thomas Colcombet and Florian Horn.

32 Outline 6 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

33 Outline 6 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

34 Outline 6 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

35 Eve needs some memory 7 i 1, r 2 v 1,2 r 1, i 2 to v 0 i 1, r 3 v 0 v 1,3 r 1, i 3 to v 0 i 2, r 3 v 2,3 r 2, i 3 to v 0

36 Adam needs infinite memory 8... r i i...

37 Adam needs infinite memory 8... push(a) ε pop(a) i r i i... r

38 Eve needs infinite memory 9 i r r r i i i

39 Eve needs infinite memory 9 i r r r i i i

40 Eve needs infinite memory 9 i 0 0 r 1 i 1 0 r 1 i 1 0 r 1 i

41 Outline 9 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

42 Playing for the worst, hoping for the best 10 Condition: the only counter is bounded by N. Observation For every vertex v, there is a maximal N(v) N such that Eve wins. Idea: play from v as you would with counter value N(v).

43 Playing for the worst, hoping for the best 10 Condition: the only counter is bounded by N. Observation For every vertex v, there is a maximal N(v) N such that Eve wins. Idea: play from v as you would with counter value N(v). Bottom-line: possible scenarios are linearly ordered.

44 Playing for the worst, hoping for the best 10 Condition: the only counter is bounded by N. Observation For every vertex v, there is a maximal N(v) N such that Eve wins. Idea: play from v as you would with counter value N(v). Bottom-line: possible scenarios are linearly ordered. Lemma Eve has positional winning strategies for the condition the counter is bounded by N.

45 Outline 10 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

46 Finitary Büchi 11 Condition: there exists N such that Büchi vertices are visited every N steps.

47 Finitary Büchi 11 Condition: there exists N such that Büchi vertices are visited every N steps. (roughly speaking: actions i and r, noε)

48 Finitary Büchi 11 Condition: there exists N such that Büchi vertices are visited every N steps. (roughly speaking: actions i and r, noε) Theorem Eve has positional winning strategies in finitary Büchi games. Eve has finite-memory winning strategies in finitary parity games.

49 Outline 11 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

50 Some more examples (1) 12 pop(a) push(a) r pop(a) i Eve should maintain a low stack.

51 Objective 13 Theorem For all pushdown games, the following are equivalent: σ (strategy for Eve), π (paths), N N, π satisfies parity and each counter is bounded by N. σ (strategy for Eve), N N, π (paths), π satisfies parity and eventually each counter is bounded by N.

52 Objective 13 Theorem For all pushdown games, the following are equivalent: σ (strategy for Eve), π (paths), N N, π satisfies parity and each counter is bounded by N. σ (strategy for Eve), N N, π (paths), π satisfies parity and eventually each counter is bounded by N.

53 Counter-example for the general case r i r i i... Eve wins but she does not know the bound! n

54 Outline 14 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

55 A simple proof for the case of finite graphs 15 Condition: parity and all counters are bounded. Define: W E (N) the set of vertices where Eve wins for the bound N. W E the set of vertices where Eve wins for some (non-uniform) bound.

56 A simple proof for the case of finite graphs 15 Condition: parity and all counters are bounded. Define: Lemma W E (N) the set of vertices where Eve wins for the bound N. W E the set of vertices where Eve wins for some (non-uniform) bound. 1 W E (0) W E (1) W E (N) W E (N + 1) W E.

57 A simple proof for the case of finite graphs 15 Condition: parity and all counters are bounded. Define: Lemma W E (N) the set of vertices where Eve wins for the bound N. W E the set of vertices where Eve wins for some (non-uniform) bound. 1 W E (0) W E (1) W E (N) W E (N + 1) W E. 2 There exists N such that W E (N) = W E (N + 1) =.

58 A simple proof for the case of finite graphs 15 Condition: parity and all counters are bounded. Define: Lemma W E (N) the set of vertices where Eve wins for the bound N. W E the set of vertices where Eve wins for some (non-uniform) bound. 1 W E (0) W E (1) W E (N) W E (N + 1) W E. 2 There exists N such that W E (N) = W E (N + 1) =. 3 For such N, Adam wins from V \W E (N), hence W E = W E (N).

59 Outline 15 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

60 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

61 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

62 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

63 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

64 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

65 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

66 Some more examples (2) 16 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.

67 Proof sketch 17 Condition: parity and all counters are bounded. Define: W E (N) the set of vertices where Eve wins for the bound N in the limit. W E the set of vertices where Eve wins for some (non-uniform) bound.

68 Proof sketch 17 Condition: parity and all counters are bounded. Define: W E (N) the set of vertices where Eve wins for the bound N in the limit. W E the set of vertices where Eve wins for some (non-uniform) bound. Proposition 1 W E (0) W E (1) W E (N) W E (N + 1) W E. 2 There exists N such that W E (N) = W E (N + 1) =. 3 For such N, Adam wins from V \W E (N), hence W E = W E (N). Why is 2. true?

69 Regularity of the winning regions 18 Theorem (derived from Serre) For all N, W E (N) is a regular set of configurations, recognized by an alternating automaton of size independent of N.

70 Decidability 19 Theorem For all pushdown games, the following are equivalent: σ (strategy for Eve), π (paths), N N, π satisfies parity and each counter is bounded by N. σ (strategy for Eve), N N, π (paths), π satisfies parity and eventually each counter is bounded by N.

71 Decidability 19 Theorem For all pushdown games, the following are equivalent: σ (strategy for Eve), π (paths), N N, π satisfies parity and each counter is bounded by N. σ (strategy for Eve), N N, π (paths), π satisfies parity and eventually each counter is bounded by N. Corollary Determining the winner in a pushdown ωb-game is decidable.

72 Decidability 19 Theorem For all pushdown games, the following are equivalent: σ (strategy for Eve), π (paths), N N, π satisfies parity and each counter is bounded by N. σ (strategy for Eve), N N, π (paths), π satisfies parity and eventually each counter is bounded by N. Corollary Determining the winner in a pushdown ωb-game is decidable. Remark: one can show that the collapse bound is doubly-exponential!

73 Outline 19 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max

74 The max operator 20 We add a new feature for counters: γ max(γ 1,γ 2 ). Theorem (derived from Bojańczyk and Toruńczyk) Deterministic max-automata are equivalent to Weak MSO +U. We consider ωb-games with max.

75 Co-determinisation 21 i r max(γ 1,γ 2 ) i i max(γ 2,γ 3 ) i r i i i r max(γ 1,γ 2 )

76 Co-determinisation 21 i r max(γ 1,γ 2 ) i i max(γ 2,γ 3 ) i r i i i r max(γ 1,γ 2 ) To prove that a counter value is high, one can count backwards!

77 Reduction 22 We reduce ωb-games with max to pushdown ωb-games (without max).

78 Reduction 22 We reduce ωb-games with max to pushdown ωb-games (without max). Idea: simulate the game and store the play in the stack. Whenever he wants, Adam can declare this counter has a very large value : from there, play backwards using the stack until a reset is met.

79 Reduction 22 We reduce ωb-games with max to pushdown ωb-games (without max). Idea: simulate the game and store the play in the stack. Whenever he wants, Adam can declare this counter has a very large value : from there, play backwards using the stack until a reset is met. Theorem Determining the winner in an ωb-game with max is decidable.

80 The end. 22 Thank you!

81 Outline 22 1 Finite-memory strategies Some examples Worst-case strategies Finitary conditions 2 Equivalence for pushdown games The case of finite graphs The case of pushdown graphs Application: ωb-games with max