Games with Costs and Delays
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1 Games with Costs and Delays Martin Zimmermann Saarland University June 20th, 2017 LICS 2017, Reykjavik, Iceland Martin Zimmermann Saarland University Games with Costs and Delays 1/14
2 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. Martin Zimmermann Saarland University Games with Costs and Delays 2/14
3 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) Martin Zimmermann Saarland University Games with Costs and Delays 2/14
4 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : O: b Martin Zimmermann Saarland University Games with Costs and Delays 2/14
5 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b O: a Martin Zimmermann Saarland University Games with Costs and Delays 2/14
6 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a O: a Martin Zimmermann Saarland University Games with Costs and Delays 2/14
7 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a O: a a Martin Zimmermann Saarland University Games with Costs and Delays 2/14
8 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b O: a a Martin Zimmermann Saarland University Games with Costs and Delays 2/14
9 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b O: a a I wins! Martin Zimmermann Saarland University Games with Costs and Delays 2/14
10 Gale-Stewart Games Büchi-Landweber: The winner of a zero-sum two-player game of infinite duration with ω-regular winning condition can be determined effectively. ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b O: a a I wins! Many possible extensions... we consider two: Interaction: one player may delay her moves. Winning condition: quantitative instead of qualitative. Martin Zimmermann Saarland University Games with Costs and Delays 2/14
11 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) Martin Zimmermann Saarland University Games with Costs and Delays 3/14
12 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : O: b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
13 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a O: Martin Zimmermann Saarland University Games with Costs and Delays 3/14
14 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b O: Martin Zimmermann Saarland University Games with Costs and Delays 3/14
15 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b O: b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
16 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b O: b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
17 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b O: b b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
18 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a O: b b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
19 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a O: b b a Martin Zimmermann Saarland University Games with Costs and Delays 3/14
20 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a O: b b a Martin Zimmermann Saarland University Games with Costs and Delays 3/14
21 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a O: b b a a Martin Zimmermann Saarland University Games with Costs and Delays 3/14
22 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b O: b b a a Martin Zimmermann Saarland University Games with Costs and Delays 3/14
23 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b O: b b a a b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
24 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b b O: b b a a b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
25 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b b O: b b a a b b Martin Zimmermann Saarland University Games with Costs and Delays 3/14
26 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b b O: b b a a b b O wins! Martin Zimmermann Saarland University Games with Costs and Delays 3/14
27 Delay Games Allow Player O to delay her moves: ( )( ) α(0) α(1) L, if β(i) = α(i + 2) for every i β(0) β(1) I : b a b b a a b b O: b b a a b b O wins! Typical questions: How often does Player O have to delay to win? How hard is determining the winner of a delay game? Does the ability to delay allow Player O to improve the quality of her strategies? Martin Zimmermann Saarland University Games with Costs and Delays 3/14
28 Previous Work If winning conditions given by deterministic parity automata: Theorem (Klein, Z. 15) If Player O wins delay game induced by A, then also by delaying at most 2 A 2 times. Martin Zimmermann Saarland University Games with Costs and Delays 4/14
29 Previous Work If winning conditions given by deterministic parity automata: Theorem (Klein, Z. 15) If Player O wins delay game induced by A, then also by delaying at most 2 A 2 times. Lower bound 2 A (already for safety automata). Martin Zimmermann Saarland University Games with Costs and Delays 4/14
30 Previous Work If winning conditions given by deterministic parity automata: Theorem (Klein, Z. 15) If Player O wins delay game induced by A, then also by delaying at most 2 A 2 times. Lower bound 2 A (already for safety automata). Determining the winner is EXPTIME-complete (hardness already for safety automata). Martin Zimmermann Saarland University Games with Costs and Delays 4/14
31 Previous Work If winning conditions given by deterministic parity automata: Theorem (Klein, Z. 15) If Player O wins delay game induced by A, then also by delaying at most 2 A 2 times. Lower bound 2 A (already for safety automata). Determining the winner is EXPTIME-complete (hardness already for safety automata). Note: This improved similar results by Holtmann, Kaiser, and Thomas with doubly-exponential upper bounds and no lower bounds. Martin Zimmermann Saarland University Games with Costs and Delays 4/14
32 Previous Work If winning conditions given by formula in (quantitative) linear temporal logics: Theorem (Klein, Z. 16) If Player O wins delay game induced by ϕ, then also by delaying at most 2 22 ϕ times. There is a matching lower bound. Determining the winner is 3EXPTIME-complete. Note: Quantitative conditions not harder than qualitative ones. Martin Zimmermann Saarland University Games with Costs and Delays 5/14
33 Uniformization of Relations A strategy σ for O in a game induces a mapping f σ : Σ ω I Σ ω O σ is winning { ( α f σ(α)) α Σ ω I } L (f σ uniformizes L) Martin Zimmermann Saarland University Games with Costs and Delays 6/14
34 Uniformization of Relations A strategy σ for O in a game induces a mapping f σ : Σ ω I Σ ω O σ is winning { ( α f σ(α)) α Σ ω I } L (f σ uniformizes L) Continuity in terms of strategies (in Cantor metric): Strategy without lookahead: i-th letter of f σ (α) only depends on first i letters of α (very strong notion of continuity). Martin Zimmermann Saarland University Games with Costs and Delays 6/14
35 Uniformization of Relations A strategy σ for O in a game induces a mapping f σ : Σ ω I Σ ω O σ is winning { ( α f σ(α)) α Σ ω I } L (f σ uniformizes L) Continuity in terms of strategies (in Cantor metric): Strategy without lookahead: i-th letter of f σ (α) only depends on first i letters of α (very strong notion of continuity). Strategy with bounded delay: f σ Lipschitz-continuous. Martin Zimmermann Saarland University Games with Costs and Delays 6/14
36 Uniformization of Relations A strategy σ for O in a game induces a mapping f σ : Σ ω I Σ ω O σ is winning { ( α f σ(α)) α Σ ω I } L (f σ uniformizes L) Continuity in terms of strategies (in Cantor metric): Strategy without lookahead: i-th letter of f σ (α) only depends on first i letters of α (very strong notion of continuity). Strategy with bounded delay: f σ Lipschitz-continuous. Strategy with arbitrary (finite) delay: f σ (uniformly) continuous. Martin Zimmermann Saarland University Games with Costs and Delays 6/14
37 Uniformization of Relations A strategy σ for O in a game induces a mapping f σ : Σ ω I Σ ω O σ is winning { ( α f σ(α)) α Σ ω I } L (f σ uniformizes L) Continuity in terms of strategies (in Cantor metric): Strategy without lookahead: i-th letter of f σ (α) only depends on first i letters of α (very strong notion of continuity). Strategy with bounded delay: f σ Lipschitz-continuous. Strategy with arbitrary (finite) delay: f σ (uniformly) continuous. Holtmann, Kaiser, Thomas: for ω-regular L L uniformizable by continuous function L uniformizable by Lipschitz-continuous function Martin Zimmermann Saarland University Games with Costs and Delays 6/14
38 Finitary Parity Automata a 1 a 0 a 2 b Martin Zimmermann Saarland University Games with Costs and Delays 7/14
39 Finitary Parity Automata a 1 a 0 a 2 Parity acceptance: Almost every odd priority is followed by a larger even one. b L(A) = a(b aaa) b ω + a(b aaa) ω Martin Zimmermann Saarland University Games with Costs and Delays 7/14
40 Finitary Parity Automata a 1 a 0 a 2 Parity acceptance: Almost every odd priority is followed by a larger even one. b L(A) = a(b aaa) b ω + a(b aaa) ω Finitary parity acceptance: There is a bound n such that almost every odd priority is followed by a larger even one within n steps. L(A) = a(b aaa) b ω + n N a(b n aaa) ω Martin Zimmermann Saarland University Games with Costs and Delays 7/14
41 Finitary Parity Automata Remark Safety automata can be transformed into finitary parity automata of the same size. Martin Zimmermann Saarland University Games with Costs and Delays 8/14
42 Finitary Parity Automata Remark Safety automata can be transformed into finitary parity automata of the same size. Proof: Turn all unsafe states into sinks with an odd color, all safe states get even color. Martin Zimmermann Saarland University Games with Costs and Delays 8/14
43 Finitary Parity Automata Remark Safety automata can be transformed into finitary parity automata of the same size. Proof: Turn all unsafe states into sinks with an odd color, all safe states get even color. Thus: exponential lower bounds on complexity and necessary lookahead for delay games with finitary parity conditions. Martin Zimmermann Saarland University Games with Costs and Delays 8/14
44 Results If winning conditions given by deterministic finitary parity automata: Theorem If Player O wins delay game induced by A, then also by delaying at most 2 A 6 times. Martin Zimmermann Saarland University Games with Costs and Delays 9/14
45 Results If winning conditions given by deterministic finitary parity automata: Theorem If Player O wins delay game induced by A, then also by delaying at most 2 A 6 times. Lower bound 2 A. Martin Zimmermann Saarland University Games with Costs and Delays 9/14
46 Results If winning conditions given by deterministic finitary parity automata: Theorem If Player O wins delay game induced by A, then also by delaying at most 2 A 6 times. Lower bound 2 A. Determining the winner is EXPTIME-complete. Note: Again, quantitative conditions not harder than qualitative ones. Martin Zimmermann Saarland University Games with Costs and Delays 9/14
47 Tradeoff Lookahead vs. Quality ( ( 1 0 ( 0)... 1) ( 1 ( 0 Martin Zimmermann Saarland University Games with Costs and Delays 10/14
48 Tradeoff Lookahead vs. Quality ( ( 1 0 ( 0)... 1) ( 1 ( 0 Theorem For every n > 0, there is a language L n recognized by a finitary Büchi automaton with n + 2 states such that an optimal strategy without delay has cost n, but an optimal strategy delaying once has cost 1. Martin Zimmermann Saarland University Games with Costs and Delays 10/14
49 Tradeoff Lookahead vs. Quality Theorem For every n > 0, there is a language L n recognized by a finitary Büchi automaton with O(n) states such that an optimal strategy delaying 2 n times has cost 0, and an optimal strategy delaying less than 2 n times has cost n. 0) 1) delay 2 n cost 0 Martin Zimmermann Saarland University Games with Costs and Delays 11/14
50 Tradeoff Lookahead vs. Quality f ( 1 ( 0 ( 0 e ( 1 d ( 0 c ( 1 b ( 0 ( a ( 1 (3,0) ) (3,1) ) ) (2,0) (2,1) (1,0) ) ) ) (1,1) Martin Zimmermann Saarland University Games with Costs and Delays 12/14
51 Tradeoff Lookahead vs. Quality ( 0 a ( 1 ) (3,0) ) (3,1) Martin Zimmermann Saarland University Games with Costs and Delays 12/14
52 Tradeoff Lookahead vs. Quality ( 0 ) (2,0) c b a ( 1 ) (2,1) Martin Zimmermann Saarland University Games with Costs and Delays 12/14
53 Tradeoff Lookahead vs. Quality ( 0 e ( 1 d c b a ) (1,0) ) (1,1) Martin Zimmermann Saarland University Games with Costs and Delays 12/14
54 Tradeoff Lookahead vs. Quality ( 1 f e d c b a ) (1,0) ) (1,1) ( 0 Martin Zimmermann Saarland University Games with Costs and Delays 12/14
55 Tradeoff Lookahead vs. Quality f ( 1 ( 0 ( 0 e ( 1 d ( 0 c ( 1 b ( 0 ( a ( 1 (3,0) ) (3,1) ) ) (2,0) (2,1) (1,0) ) ) ) (1,1) Theorem For every n > 0, there is a language L n recognized by a finitary Büchi automaton with O(n 2 ) states such that for every 0 j n: an optimal strategy delaying j times has cost 2(n + 1) j. Martin Zimmermann Saarland University Games with Costs and Delays 12/14
56 More Results acceptance lookahead complexity Martin Zimmermann Saarland University Games with Costs and Delays 13/14
57 More Results acceptance lookahead complexity parity exp. EXPTIME-complete finitary parity exp. EXPTIME-complete Martin Zimmermann Saarland University Games with Costs and Delays 13/14
58 More Results acceptance lookahead complexity parity exp. EXPTIME-complete finitary parity exp. EXPTIME-complete parity w. costs exp. EXPTIME-complete Martin Zimmermann Saarland University Games with Costs and Delays 13/14
59 More Results acceptance lookahead complexity parity exp. EXPTIME-complete finitary parity exp. EXPTIME-complete parity w. costs exp. EXPTIME-complete finitary Streett exp./doubly-exp. EXPTIME/2EXPTIME Streett w. costs exp./doubly-exp. EXPTIME/2EXPTIME Martin Zimmermann Saarland University Games with Costs and Delays 13/14
60 More Results acceptance lookahead complexity parity exp. EXPTIME-complete finitary parity exp. EXPTIME-complete parity w. costs exp. EXPTIME-complete finitary Streett exp./doubly-exp. EXPTIME/2EXPTIME Streett w. costs exp./doubly-exp. EXPTIME/2EXPTIME Theorem Optimal strategies in delay games with Streett conditions with costs may require doubly-exponential lookahead. Martin Zimmermann Saarland University Games with Costs and Delays 13/14
61 Conclusion Quantitative delay games with parity conditions are not harder than qualitative ones. Lookahead allows to improve the quality of strategies. Martin Zimmermann Saarland University Games with Costs and Delays 14/14
62 Conclusion Quantitative delay games with parity conditions are not harder than qualitative ones. Lookahead allows to improve the quality of strategies. Open Problems Close the gaps for Streett conditions (qualitative and quantitative). Study other tradeoffs, e.g., lookahead vs. memory size. Determine the complexity of finding optimal strategies (smallest cost or smallest lookahead). Martin Zimmermann Saarland University Games with Costs and Delays 14/14
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