Time-optimal Strategies for Infinite Games Martin Zimmermann RWTH Aachen University March 10th, 2010 DIMAP Seminar Warwick University, United Kingdom Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 1/32
Introduction Model Checking: program P , specification ϕ , does P | = ϕ ? Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 2/32
Introduction Model Checking: program P , specification ϕ , does P | = ϕ ? Synthesis: environment E , specification ϕ . Generate program P such that E × P | = ϕ . Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 2/32
Introduction Model Checking: program P , specification ϕ , does P | = ϕ ? Synthesis: environment E , specification ϕ . Generate program P such that E × P | = ϕ . Synthesis as a game: no matter what the environment does, the program has to guarantee ϕ . Beautiful and rich theory based on infinite graph games. typically: a player either wins or loses (zero-sum). here: adding quantitative aspects to infinite games. Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 2/32
Outline 1. Infinite Games 2. Poset Games 3. Parametric LTL Games 4. Finite-time Muller Games 5. Conclusion Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 3/32
Definitions An arena A = ( V , V 0 , V 1 , E , v 0 , l ) consists of a finite directed graph ( V , E ) without dead-ends, a partition { V 0 , V 1 } of V denoting the positions of Player 0 (circles) and Player 1 (squares), an initial vertex v 0 ∈ V , a labeling function l : V → 2 P for some set P of atomic propositions. p q , r v 1 v 3 p , q v 0 v 2 v 4 ∅ r Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 4/32
Definitions cont’d Play in A : infinite path ρ 0 ρ 1 ρ 2 . . . starting in v 0 . Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 5/32
Definitions cont’d Play in A : infinite path ρ 0 ρ 1 ρ 2 . . . starting in v 0 . Strategy for Player i ∈ { 0 , 1 } : mapping σ : V ∗ V i → V such that ( s , σ ( ws )) ∈ E . σ is finite-state: σ computable by finite automaton with output. ρ 0 ρ 1 ρ 2 . . . is consistent with σ : ρ n +1 = σ ( ρ 0 . . . ρ n ) for all n such that ρ n ∈ V i . Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 5/32
Definitions cont’d Play in A : infinite path ρ 0 ρ 1 ρ 2 . . . starting in v 0 . Strategy for Player i ∈ { 0 , 1 } : mapping σ : V ∗ V i → V such that ( s , σ ( ws )) ∈ E . σ is finite-state: σ computable by finite automaton with output. ρ 0 ρ 1 ρ 2 . . . is consistent with σ : ρ n +1 = σ ( ρ 0 . . . ρ n ) for all n such that ρ n ∈ V i . Game: G = ( A , Win ) with Win ⊆ V ω . ρ winning for Player 0: ρ ∈ Win . ρ winning for Player 1: ρ ∈ V ω \ Win . Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 5/32
Definitions cont’d Play in A : infinite path ρ 0 ρ 1 ρ 2 . . . starting in v 0 . Strategy for Player i ∈ { 0 , 1 } : mapping σ : V ∗ V i → V such that ( s , σ ( ws )) ∈ E . σ is finite-state: σ computable by finite automaton with output. ρ 0 ρ 1 ρ 2 . . . is consistent with σ : ρ n +1 = σ ( ρ 0 . . . ρ n ) for all n such that ρ n ∈ V i . Game: G = ( A , Win ) with Win ⊆ V ω . ρ winning for Player 0: ρ ∈ Win . ρ winning for Player 1: ρ ∈ V ω \ Win . σ winning strategy for Player i : all plays ρ consistent with σ are winning for Player i . G determined: one player has a winning strategy. Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 5/32
Outline 1. Infinite Games 2. Poset Games 3. Parametric LTL Games 4. Finite-time Muller Games 5. Conclusion Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 6/32
Motivation Request-Reponse conditions are a typical requirement on reactive systems. There is a natural definition of waiting times and they allow time-optimal strategies. Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 7/32
Motivation Request-Reponse conditions are a typical requirement on reactive systems. There is a natural definition of waiting times and they allow time-optimal strategies. Goal: Extend the Request-Response condition to partially ordered objectives.. .. while retaining the notion of waiting times and the existence of time-optimal strategies. Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 7/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 t 2 : 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 t 2 : 0 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 t 2 : 0 0 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 t 2 : 0 0 0 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 t 2 : 0 0 0 0 1 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 t 2 : 0 0 0 0 1 2 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 t 2 : 0 0 0 0 1 2 3 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 3 t 2 : 0 0 0 0 1 2 3 4 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 3 4 t 2 : 0 0 0 0 1 2 3 4 5 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 3 4 5 t 2 : 0 0 0 0 1 2 3 4 5 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 3 4 5 0 t 2 : 0 0 0 0 1 2 3 4 5 0 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
Request-Response games Request-response game: ( A , ( Q j , P j ) j =1 ,..., k ) where Q j , P j ⊆ V . Player 0 wins a play if every visit to Q j (request) is responded by a later visit to P j . P 1 Q 2 P 2 Q 1 t 1 : 0 1 2 0 0 1 2 3 4 5 0 t 2 : 0 0 0 0 1 2 3 4 5 0 0 p i = t 1 + t 2 : 0 1 2 0 1 3 5 7 9 5 0 Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 8/32
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