Boundedness Games. Séminaire de l équipe MoVe, LIF, Marseille, May 2nd, Nathanaël Fijalkow. Institute of Informatics, Warsaw University Poland
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1 Boundedness Games Séminaire de l équipe MoVe, LIF, Marseille, May 2nd, 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 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 (open for 40 years)!
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 ωb 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 ωb 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 ωb 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 Counting conditions 2 Equivalence for pushdown ωb 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 Counting conditions 2 Equivalence for pushdown ωb 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 Counting conditions 2 Equivalence for pushdown ωb 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 Conjecture s status 10 Conjecture (Thomas Colcombet s conjecture) Over arbitrary graphs / chronological graphs / tree-like graphs, if Eve has a strategy ensuring Parity Bound(N), then she has a finite-memory strategy (of size independent of N) ensuring Parity Bound(α(N)).
42 Conjecture s status 10 Conjecture (Thomas Colcombet s conjecture) Over arbitrary graphs / chronological graphs / tree-like graphs, if Eve has a strategy ensuring Parity Bound(N), then she has a finite-memory strategy (of size independent of N) ensuring Parity Bound(α(N)). Theorem The conjecture does not hold over arbitrary graphs.
43 Conjecture s status 10 Conjecture (Thomas Colcombet s conjecture) Over arbitrary graphs / chronological graphs / tree-like graphs, if Eve has a strategy ensuring Parity Bound(N), then she has a finite-memory strategy (of size independent of N) ensuring Parity Bound(α(N)). Theorem The conjecture does not hold over arbitrary graphs. With some more effort: Theorem The conjecture does not hold over chronological graphs.
44 Potato-trees 11
45 Outline 11 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb games The case of finite graphs The case of pushdown graphs Application: ωb-games with max
46 Counting conditions 12 A subclass of ωb conditions where the counters and parity are not independent.
47 Counting conditions 12 A subclass of ωb conditions where the counters and parity are not independent. Theorem Over general graphs: Eve has positional winning strategies in counting Büchi games. Eve has finite-memory winning strategies in finitary parity games.
48 Counting conditions 12 A subclass of ωb conditions where the counters and parity are not independent. Theorem Over general graphs: Eve has positional winning strategies in counting Büchi games. Eve has finite-memory winning strategies in finitary parity games. In some sense, the class of counting conditions is the maximal subclass of ωb conditions for such results.
49 Outline 12 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb games The case of finite graphs The case of pushdown graphs Application: ωb-games with max
50 Some more examples (1) 13 pop(a) push(a) r pop(a) i Eve should maintain a low stack.
51 Objective 14 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 14 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 15 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb 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 16 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 16 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 16 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 16 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 16 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb games The case of finite graphs The case of pushdown graphs Application: ωb-games with max
60 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
61 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
62 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
63 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
64 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
65 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
66 Some more examples (2) 17 push(a) pop(a) push(b) pop(b) i Adam may use the stack as credit.
67 Proof sketch 18 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 18 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 19 Theorem (derived from Serre) For all N, W E (N) is a regular set of configurations, recognized by an alternating automaton of size Q (independent of N).
70 Decidability 20 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 20 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 20 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 20 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb games The case of finite graphs The case of pushdown graphs Application: ωb-games with max
74 The max operator 21 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 22 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 22 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 23 We reduce ωb-games with max to pushdown ωb-games (without max).
78 Reduction 23 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 23 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. 23 Thank you!
81 Outline 23 1 Finite-memory strategies Some examples Counting conditions 2 Equivalence for pushdown ωb games The case of finite graphs The case of pushdown graphs Application: ωb-games with max
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