Game-Theoretic Semantics for Alternating- Time Temporal Logic Valentin Goranko Stockholm University Joint work with Antti Kuusisto and Raine Rönnholm Highlights of Logic, Games and Automata 2016 Brussels, September 8, 2016 V Goranko
Logic and games Two approaches for relating logic and games: 1. Analyzing games with logic ATL 2. Analyzing logic with games GTS We bring these two approaches together. V Goranko
Introduction: ATL and GTS Alternating-time temporal logic (ATL): a logical formalism for reasoning about strategic abilities of agents and coalitions in multi-agent systems. The formula � � A � � ϕ of ATL says, intuitively, that the coalition of agents A have a strategy to co-operatively ensure that ϕ holds, regardless of the behaviour of the other agents. In game-theoretic semantics (GTS), truth of a formula ϕ is determined in a formal dispute, called evaluation game , between two players: Eloise, who is trying to verify ϕ , and Abelard, who is trying to falsify it. GTS equates truth of ϕ with the existence of a winning strategy for Eloise in the evaluation game for ϕ . We develop GTS for ATL. A 2-way interaction of logic and games. V Goranko
Preliminaries: concurrent game models and ATL A concurrent game model (CGM) consists of a set of agents A gt, a set of states St, together mappings act , v , and o , defining for every s ∈ St: ◮ a set v ( s ) of atomic propositions p i that are true at s , ◮ a set act ( s , a ) of actions available at s , for each agent a ∈ A gt, ◮ a unique outcome / successor state o ( s , � α ) of s for every admissible tuple of actions � α chosen by all the agents in A gt. For simplicity, I will only consider the following subset of ATL-formulae: ϕ ::= p | ¬ ϕ | ( ϕ ∨ ϕ ) | � � A � � F ϕ | � � A � � G ϕ The semantics of ATL is based on truth at a state in a CGM, via the clauses: ◮ � � A � � X ϕ : ‘ The coalition A has a collective action that ensures that every possible outcome (state) satisfies ϕ ’, ◮ � � A � � F ϕ : ‘ The coalition A has a collective strategy to ensure that every possible outcome play eventually satisfies ϕ ’, ◮ � � A � � G ϕ : ‘ The coalition A has a collective strategy to ensure that every possible outcome play forever satisfies ϕ ’, V Goranko
Example: 2-round ‘Rock-paper-scissors’ s 1 s 3 M : RS,PR,SP WIN 1 RS,PR,SP RR,PP,SS s 0 RS,PR,SP SR,RP,PS RR,PP,SS SR,RP,PS RR,PP,SS SR,RP,PS A gt = { a 1 , a 2 } WIN 2 s 2 s 4 act ( s i , a i ) = { R , P , S } V Goranko
The unbounded evaluation game for ATL Consider a CGM M , a state s 0 in M and an ATL-formula ϕ . We introduce the unbounded evaluation game G = G ( M , s 0 , ϕ ). A position in G is a tuple ( P , s , ψ ), where P ∈ { Eloise , Abelard } , s is a state in M and ψ a subformula of ϕ . The opposing player of P is denoted by P . The game G begins from the initial position (Eloise , s 0 , ϕ ) and proceeds according to the following rules. V Goranko
The unbounded evaluation game for ATL: rules ◮ In a position ( P , s , p ) for an atomic proposition p , the game ends. P wins the game if p ∈ v ( s ). Otherwise P wins the game. ◮ In a position ( P , s , ψ ∨ θ ), the player P chooses a disjunct α ∈ { ψ, θ } , and then the game continues from ( P , s , α ). ◮ In a position ( P , s , ¬ ψ ), the game moves to the position ( P , s , ψ ). ◮ In a position ( P , s , � � A � � X ψ ), the following one-step game g X is played with starting position ( P , A , s ), as follows: 1. P chooses actions for the agents in A . 2. P chooses actions for the agents in A gt \ A . Thus, an action profile � α is selected. 3. The one-step game g X ends in position ( P , s ′ , ψ ), where s ′ = o ( s , � α ) is the resulting state. V Goranko
The unbounded evaluation game for ATL: rules (cont.) ◮ In a position ( P , s , � � A � � F ψ ), the game enters an embedded subgame g F . It begins from the state s and proceeds by playing repeatedly the one-step game g X , starting from ( P , A , s ) and each next round starting at the ending position of the previous one. In the embedded game g F , the player P is the controlling player who may decide to end g F at any state s ′ that is reached. When (if) the embedded game g F ends, in a position ( P , s ′ , ψ ), the evaluation game then continues from that position. If the subgame g F goes on forever, the controlling player P loses the entire evaluation game. ◮ In a position ( P , s , � � A � � G ψ ), the players enter a dual embedded subgame g G , just like g F , but now the controlling player is P . V Goranko
Unbounded game-theoretic semantics for ATL Unbounded game-theoretic semantics for ATL : M , s | = GTS ϕ iff Eloise has a winning strategy in G ( M , s , ϕ ) . Theorem The unbounded GTS for ATL is equivalent to the standard (compositional) semantics of ATL . The unbounded evaluation games are determined, but possibly infinite. A drawback. V Goranko
Finitely bounded evaluation games for ATL We now modify the evaluation game G by associating with every embedded subgame ( g F or g G ) a (finite) time limit n ∈ N . The time limit is selected and announced in the beginning of the subgame by the controlling player. The controlling player must end the subgame before time runs out. Evaluation games with time limit define finitely bounded GTS for ATL. ⊲ NB: an analogy with FOR-loops and WHILE-loops. Theorem The finitely bounded and unbounded GTS are generally non-equivalent. However, they are equivalent on image finite models. V Goranko
Example M : 0 A gt = { a } p ¬ p act ( s 0 , a ) = N + 1 0 0 p p ¬ p 2 s 0 3 0 0 0 p p p p ¬ p 4 0 0 0 0 p p p p ¬ p Here Eloise can win G ( M , s 0 , � � a � � G p ) with time limits, but not with unbounded GTS. = fin Therefore, M , s 0 | GTS ϕ but M , s 0 �| = GTS ϕ . V Goranko
Ordinal bounded evaluation games for ATL We now generalize the notion of time limit, by allowing time limits for the embedded subgames to be any ordinals . The initial time limit of the subgame is chosen by the controlling player. Every round, the time limit decreases: – if a successor ordinal, by 1; – if a limit ordinal, the controlling player must choose a strictly smaller one. Ordinal bounded GTS: using evaluation games with ordinal time limits. NB. Since ordinals are well-founded, the ordinal bounded semantics guarantees that evaluation games end in finite number of rounds. Theorem The ordinal bounded and the unbounded GTS are equivalent. V Goranko
Example M : 0 A gt = { a } p ¬ p act ( s 0 , a ) = N + 1 0 0 p p ¬ p 2 s 0 3 0 0 0 p p p p ¬ p 4 0 0 0 0 p p p p ¬ p Here Abelard can win G ( M , s 0 , � � a � � G p ) with ordinal bounded GTS by choosing the ordinal ω as the initial time limit. After the 1st round, where Eloise chooses an action, Abelard can always reduce the time limit appropriately to win. V Goranko
Concluding remarks We argue that: ◮ the game-theoretic semantics is a natural framework for reasoning about games by using games. ◮ limitations on time resources in evaluation games lead to interesting semantic variants for ATL. Ongoing work: ◮ develop game-theoretic semantics for ATL ∗ and other extensions of ATL. ◮ consider other evaluation games with limiting time resources. V Goranko
Thank you for your attention! V Goranko
Recommend
More recommend
