Time-optimal Winning Strategies in Infinite Games Martin Zimmermann RWTH Aachen University zimmermann@automata.rwth-aachen.de Gasics meeting Brussels, March 5-6, 2009 Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 1
Introduction Two-player games of infinite duration on graphs � Solution to the synthesis problem for reactive systems. � Well-developed theory with nice results. � Classical quality measure: memory size of a winning strategy . Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 2
Introduction Two-player games of infinite duration on graphs � Solution to the synthesis problem for reactive systems. � Well-developed theory with nice results. � Classical quality measure: memory size of a winning strategy . But: many winning conditions allow other quality measures. � “From qualitative to quantitative games.” � “Optimal controller synthesis.” Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 2
Outline Outline � Definitions & Related Work � Poset Games � Time-optimal Winning Strategies for Poset Games Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 3
1. Definitions & Related Work Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 4
Arenas, Plays and Strategies An (initialized) Arena G = ( V, V 0 , V 1 , E, s 0 ) consists of � a finite directed graph ( V, E ) , � a partition { V 0 , V 1 } of V denoting the positions of Player 0 and 1 , � an initial vertex s 0 ∈ V . A play ρ 0 ρ 1 ρ 2 . . . in G is an infinite path starting in s 0 . A strategy for Player i is a (partial) mapping σ : V ∗ V i → V such that ( s, σ ( ws )) ∈ E for all w ∈ V ∗ and all s ∈ V i . ρ 0 ρ 1 ρ 2 . . . is consistent with σ if ρ n +1 = σ ( ρ 0 . . . ρ n ) for all ρ n ∈ V i . Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 5
Outcome of a Play The outcome of a play can be � qualitative : win or lose � one player wins a play, the other loses it. � B¨ uchi, Co-B¨ uchi, Rabin, Streett, Parity, Muller,... � σ winning strategy for Player i : every play that is consistent with σ is won by Player i . Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 6
Outcome of a Play The outcome of a play can be � qualitative : win or lose � one player wins a play, the other loses it. � B¨ uchi, Co-B¨ uchi, Rabin, Streett, Parity, Muller,... � σ winning strategy for Player i : every play that is consistent with σ is won by Player i . � quantitative : a payoff for each player � each player tries to maximize her payoff. � Mean-Payoff, Discounted Payoff,... � Value of σ : payoff of the worst play consistent with σ . Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 6
Optimal Strategies Idea: � The outcome of a play is still binary: win or lose. � But the quality of the (winning) plays and strategies is measured: � determine optimal (w.r.t. given quality measure) winning strategies for Player 0 . Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 7
An Example Request-Response Game G = ( G, ( Q j , P j ) j =1 ,...,k ) where Q j , P j ⊆ V . � Player 0 wins a play if every visit to a Q j vertex is responded by a later visit to P j . � Waiting times: start a clock for every request that is stopped as soon as it is responded (and ignore subsequent requests). � Accumulated waiting time: sum up the clock values up to that position (quadratic influence). � Value of a play: limit superior of the average accumulated waiting time; corresponding notion of optimal strategies. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 8
An Example Theorem: (Horn, Thomas, Wallmeier) If Player 0 has a winning strategy for an RR Game, then she also has an optimal winning strategy, which is finite-state and effectively computable. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 9
Other Winning Conditions Many other winning conditions have a natural notion of waiting times. � Reachability Games: the number of steps to the target vertices. � B¨ uchi Games: the periods between visits of the target vertices. � Co-B¨ uchi Games: the number of steps until the target vertices are reached for good. � Parity Games: the periods between visits of vertices colored with a maximal even color (which can be optimized as well). Some classical algorithms compute optimal winning strategies. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 10
2. Poset Games Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 11
Motivation Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 12
Motivation green 0 green 1 raise 0 raise 1 crossing free train go lower 0 lower 1 red 0 red 1 Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 12
Motivation green 0 green 1 raise 0 raise 1 crossing free train go lower 0 lower 1 Request: still a singular event. red 0 red 1 Response: partially ordered set of events. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 12
Definition Poset Game G = ( G, ( q j , P j ) j =1 ,...,k ) , P set of atomic propositions � G arena (labeled with l G : V → 2 P ) � q j ∈ P request � P j = ( D j , � j ) labeled poset where D j ⊆ P Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 13
Definition cont’d Embedding of P j in ρ 0 ρ 1 ρ 2 . . . : function f : D j → N such that � d ∈ l G ( ρ f ( d ) ) for all d ∈ D j � d � j d ′ implies f ( d ) ≤ f ( d ′ ) for all d, d ′ ∈ D j Player 0 wins ρ 0 ρ 1 ρ 2 . . . if ∀ j ∀ n ( q j ∈ l G ( ρ n ) → ρ n ρ n +1 . . . allows embedding of P j ) “Every request q j is responded by a later embedding of P j in ρ .“ Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 14
An Example train go lower 0 lower 1 red 0 red 1 { req } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
An Example train go lower 0 lower 1 red 0 red 1 { req } { red 0 } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
An Example train go lower 0 lower 1 red 0 red 1 { req } { red 0 } { lower 0 } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
An Example train go lower 0 lower 1 red 0 red 1 { req } { red 0 } { lower 0 } { red 1 } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
An Example train go lower 0 lower 1 red 0 red 1 { req } { red 0 } { lower 0 } { red 1 } { lower 1 } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
An Example train go lower 0 lower 1 red 0 red 1 { req } { red 0 } { lower 0 } { red 1 } { lower 1 } { train go } Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 15
Overlapping Embeddings train go lower 0 red 0 { red 0 } { red 0 } { lower 0 } { train go } req req Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 16
Overlapping Embeddings train go lower 0 red 0 { red 0 } { red 0 } { lower 0 } { train go } req req Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 16
Overlapping Embeddings train go lower 0 red 0 { red 0 } { red 0 } { lower 0 } { train go } req req Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 16
Solving Poset Games Theorem: Poset Games are reducible to B¨ uchi Games. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 17
Solving Poset Games Theorem: Poset Games are reducible to B¨ uchi Games. Proof: Use memory to � store elements of the posets that still have to be embedded, � deal with overlapping embeddings, and � implement a cyclic counter to ensure that every request is responded by an embedding. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 17
3. Time-optimal Winning Strategies for Poset Games Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 18
Waiting Times As desired, there is a natural definition of waiting times � Start a clock if a request is encountered... � ... that is stopped as soon as the embedding is completed. Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 19
Waiting Times As desired, there is a natural definition of waiting times � Start a clock if a request is encountered... � ... that is stopped as soon as the embedding is completed. � Need a clock for every revisit of a request (while the request is already active). Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite Games 19
Recommend
More recommend
