Games where you can play optimally with fi nite memory Patricia Bouyer 1 ephane Le Roux 1 Youssouf Oualhadj 2 St´ Mickael Randour 3 Pierre Vandenhove 3 1 LSV – CNRS & ENS Paris-Saclay 2 LACL – UPEC 3 F.R.S.-FNRS & UMONS – Universit´ e de Mons October 10, 2019 GT ALGA annual meeting 2019
Games where you can play optimally with fi nite memory Patricia Bouyer 1 ephane Le Roux 1 Youssouf Oualhadj 2 St´ Mickael Randour 3 Pierre Vandenhove 3 1 LSV – CNRS & ENS Paris-Saclay 2 LACL – UPEC 3 F.R.S.-FNRS & UMONS – Universit´ e de Mons October 10, 2019 GT ALGA annual meeting 2019
Games where you can play optimally with fi nite memory A sequel to the critically acclaimed blockbuster by Gimbert & Zielonka Patricia Bouyer 1 ephane Le Roux 1 Youssouf Oualhadj 2 St´ Mickael Randour 3 Pierre Vandenhove 3 1 LSV – CNRS & ENS Paris-Saclay 2 LACL – UPEC 3 F.R.S.-FNRS & UMONS – Universit´ e de Mons October 10, 2019 GT ALGA annual meeting 2019
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. � Good? Performance evaluated through objectives / payo ff s . Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. � Good? Performance evaluated through objectives / payo ff s . Question When are simple strategies su ffi cient to play optimally? Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. � Good? Performance evaluated through objectives / payo ff s . Question When are simple strategies su ffi cient to play optimally? Two directions for fi nite-memory determinacy : Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. � Good? Performance evaluated through objectives / payo ff s . Question When are simple strategies su ffi cient to play optimally? Two directions for fi nite-memory determinacy : 1 lifting under objective combination (with S. Le Roux and A. Pauly, in FSTTCS’18 [LPR18]), Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion The talk in one slide Strategy synthesis for two-player turn-based games Finding good controllers for systems interacting with an antagonistic environment. � Good? Performance evaluated through objectives / payo ff s . Question When are simple strategies su ffi cient to play optimally? Two directions for fi nite-memory determinacy : 1 lifting under objective combination (with S. Le Roux and A. Pauly, in FSTTCS’18 [LPR18]), 2 complete characterization and lifting from one-player games (ongoing work). Games where you can play optimally with fi nite memory Mickael Randour 1 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion 1 Memoryless determinacy 2 Finite-memory determinacy and Boolean combinations 3 Characterization and lifting corollary 4 Conclusion Games where you can play optimally with fi nite memory Mickael Randour 2 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion 1 Memoryless determinacy 2 Finite-memory determinacy and Boolean combinations 3 Characterization and lifting corollary 4 Conclusion Games where you can play optimally with fi nite memory Mickael Randour 3 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Two-player turn-based zero-sum games on graphs Games where you can play optimally with fi nite memory Mickael Randour 4 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Two-player turn-based zero-sum games on graphs We consider fi nite arenas with vertex colors in C . Two players: circle ( P 1 ) and square ( P 2 ). Strategies C ∗ × V i → V . � A winning condition is a set W ⊆ C ω . v 1 v 2 v 3 v 4 v 5 v 6 Games where you can play optimally with fi nite memory Mickael Randour 4 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Two-player turn-based zero-sum games on graphs We consider fi nite arenas with vertex colors in C . Two players: circle ( P 1 ) and square ( P 2 ). Strategies C ∗ × V i → V . � A winning condition is a set W ⊆ C ω . v 1 v 2 v 3 v 4 v 5 v 6 From where can P 1 ensure to reach v 6 ? How complex is his strategy? Games where you can play optimally with fi nite memory Mickael Randour 4 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Two-player turn-based zero-sum games on graphs We consider fi nite arenas with vertex colors in C . Two players: circle ( P 1 ) and square ( P 2 ). Strategies C ∗ × V i → V . � A winning condition is a set W ⊆ C ω . v 1 v 2 v 3 v 4 v 5 v 6 From where can P 1 ensure to reach v 6 ? How complex is his strategy? Memoryless strategies ( V i → V ) always su ffi ce for reachability (for both players). Games where you can play optimally with fi nite memory Mickael Randour 4 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion When are memoryless strategies su ffi cient to play optimally? Games where you can play optimally with fi nite memory Mickael Randour 5 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion When are memoryless strategies su ffi cient to play optimally? Virtually always for simple winning conditions! Examples: reachability, safety, B¨ uchi, parity, mean-payo ff , energy, total-payo ff , average-energy, etc. Games where you can play optimally with fi nite memory Mickael Randour 5 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion When are memoryless strategies su ffi cient to play optimally? Virtually always for simple winning conditions! Examples: reachability, safety, B¨ uchi, parity, mean-payo ff , energy, total-payo ff , average-energy, etc. Can we characterize when they are? Games where you can play optimally with fi nite memory Mickael Randour 5 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion When are memoryless strategies su ffi cient to play optimally? Virtually always for simple winning conditions! Examples: reachability, safety, B¨ uchi, parity, mean-payo ff , energy, total-payo ff , average-energy, etc. Can we characterize when they are? Yes, thanks to Gimbert and Zielonka [GZ05]. Games where you can play optimally with fi nite memory Mickael Randour 5 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Gimbert and Zielonka’s characterization Memoryless strategies su ffi ce for a preference relation � (and the induced winning conditions) if and only if 1 it is monotone , 2 it is selective . Games where you can play optimally with fi nite memory Mickael Randour 6 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Gimbert and Zielonka’s characterization Memoryless strategies su ffi ce for a preference relation � (and the induced winning conditions) if and only if 1 it is monotone , � Intuitively, stable under pre fi x addition. 2 it is selective . Games where you can play optimally with fi nite memory Mickael Randour 6 / 19
Memoryless determinacy FM determinacy and Boolean combinations Characterization and lifting corollary Conclusion Gimbert and Zielonka’s characterization Memoryless strategies su ffi ce for a preference relation � (and the induced winning conditions) if and only if 1 it is monotone , � Intuitively, stable under pre fi x addition. 2 it is selective . � Intuitively, stable under cycle mixing. Games where you can play optimally with fi nite memory Mickael Randour 6 / 19
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