Alternating-Time Temporal Logic

Transcription

1 Alternating-Time Temporal Logic R.Alur, T.Henzinger, O.Kupferman Rafael H. Bordini School of Informatics PUCRS Logic Club 5th of September, 2013

2 ATL All the material in this presentation is taken almost as-is from [Alur et al., 1997], though much summarised It s all about trends... But also really cool!

3 Syntax The syntax of ATL is defined with respect to a finite set Π of propositions and a finite set Σ = {1,..., k} of players (i.e., agents) The well-formed ATL formulæ ϕ are defined by the following grammar: ϕ ::= p ϕ ϕ 1 ϕ 2 A ϕ A ϕ A ϕ 1 U ϕ 2 where p Π, and A Σ

4 Intuition The operator is a path quantifier All other operators are as usual in temporal logics Note also that other operators such as are introduced as the usual abbreviations The ATL logic is similar to CTL except that path quantifiers are parameterised by sets of players Intuitively, A ψ means that players in A can cooperate to make ψ true (i.e., they can enforce ψ)

5 Semantics (I) The semantics of ATL is given in terms of a concurrent game structure S = k, Q, Π, π, d, δ where: k 1 is a natural number stating the number of players (players are identified with the numbers 1,..., k); Q is a finite set of states; Π is as above (propositions are also called observables); π is a labelling (or observation) function such that for each state q Q, it defines the set of propositions π(q) Π which are true at q; and d and δ are defined as follows.

6 Semantics (II) Move and Transition Functions For each player a {1,..., k} and each state q Q, the natural number d a (q) 1 is the number of different moves available at state q to player a; For each state q Q, a move vector at q is a tuple j1,..., j k such that 1 j a d a (q) for each player a; Given a state q Q, we write D(q) for the set {1,..., d 1 (q)}... {1,..., d k (q)} of move vectors; Function D is called the move function; For each state q Q and each move vector j 1,..., j k D(q), the state δ(q, j 1,..., j k ) Q is the one that results from state q if every player a {1,..., k} chooses move j a ; Function δ is called the transition function.

7 Semantics (III) Computations For two states q and q, we say that q is a successor of q if there is a move vector j 1,..., j k D(q) such that q = δ(q, j 1,..., j k ) Thus, q is a successor of q iff whenever the game is in state q, the players can possibly choose moves so that q is the next state A computation of S is an infinite sequence λ = q0, q 1, q 2,... of states such that for all positions i 0, state q i+1 is a successor of the state q i A computation starting at state q is called a q-computation For a computation λ and a position i 0, notation: λ[i] denotes the ith state of λ; λ[0, i] denotes the finite prefix q 0, q 1,..., q i of λ; and λ[i, ] denotes the infinite suffix q i, q i+1,... of λ.

8 Semantics (IV) Formulæ in ATL are interpreted over states of a concurrent game structure S that has the same propositions and players In order to define the semantics of ATL formally, the notion of strategy needs to be introduced first Strategies Considering a game structure S, a strategy for player a Σ is a function f a that maps every nonempty finite state sequence λ Q + to a natural number such that if the last state of λ is q, then f a (λ) d a (q) (i.e., it determines, for a prefix of a computation, the player s next move) Each strategy fa for a player a induces a set of computations that player a can enforce

9 Semantics (V) Outcomes Given a state q Q, a set A {1,..., k} of players, and a set F A = {f a a A} of strategies, one for each player in A, we define the outcomes of F A from q to be the set out(q, F A ) of q-computations that the players in A enforce when they follow the strategies in F A That is, a computation λ = q0, q 1, q 2,... is in out(q, F A ) if q 0 = q and for all positions i 0, there is a move vector j 1,..., j k D(q i ) such that: (i) j a = f a(λ[0, i]) for all players a A, and (ii) δ(q i, j 1,..., j k ) = q i+1 Formal semantics is then given to ATL by the definition of a satisfaction relation; S, q = ϕ indicates that state q in structure S satisfies the formula ϕ; when S is clear from the context it can be omitted and we write q = ϕ instead

10 Semantics (VI) Definition (Semantics of ATL) The satisfaction relation = is defined inductively, for all states q of S, as follows: q = p, for propositions p Π, iff p π(q); q = ϕ iff q = ϕ; q = ϕ 1 ϕ 2 iff q = ϕ 1 or q = ϕ 2 ; q = A ϕ iff there exists a set F A of strategies, one for each player in A, such that for all computations λ out(q, F A ), we have λ[1] = ϕ; q = A ϕ iff there exists a set F A of strategies, one for each player in A, such that for all computations λ out(q, F A ) and all positions i 0, we have λ[i] = ϕ; q = A ϕ 1 U ϕ 2 iff there exists a set F A of strategies, one for each player in A, such that for all computations λ out(q, F A ), there is a position i 0 such that λ[i] = ϕ 2 and for all positions 0 j < i, we have λ[j] = ϕ 1.

11 Model Checking Definition. (Model-Checking Problem for ATL) The model-checking problem for ATL asks, given a game structure S =< k, Q, Π, π, d, δ > and an ATL formula ϕ, for the states in Q that satisfy ϕ. That set of states is denoted by [ϕ] S or simply [ϕ] when the game structure S is implicitly understood. Next we will look at the symbolic model checking algorithm for ATL, which manipulates state sets of S, as given in [Alur et al., 2002]

12 Primitive Operations Used in the Algorithm Function Sub, when given an ATL formula ϕ, returns a queue of syntactic subformulæ of ϕ such that if ϕ 1 is a subformula of ϕ and ϕ 2 is a subformula of ϕ 1, then ϕ 2 precedes ϕ 1 in the queue Sub(ϕ) Function Reg, when given a proposition p Π, returns the set of states in Q that satisfy p Function Pre, when given a set A Σ of players and a set ρ Q of states, returns the set of states q such that from q, the players in A can cooperate and enforce the next state to lie in ρ. Formally, Pre(A, ρ) contains state q Q if for every play6er a A, there exists a move j a {1,..., d a (q)} such that for all players b Σ \ A and moves j b 1,..., d b (q), we have δ(q, j 1,..., j k ) ρ Union, intersection, difference, and inclusion test for state sets. Note also that we write [true] for the set Q of all states, and [false] for the empty set of states.

13 Symbolic Model Checking Algorithm for ATL foreach ϕ in Sub(ϕ) do case ϕ = p : [ϕ ] := Reg(p) case ϕ = θ : [ϕ ] := [true] \ [θ] case ϕ = θ 1 θ 2 : [ϕ ] := [θ 1] [θ 2] case ϕ = A θ : [ϕ ] := Pre(A, [θ]) case ϕ = A θ : ρ := [true] τ := [θ] while ρ τ do ρ := τ τ := Pre(A, ρ) [θ] [ϕ ] := ρ case ϕ = A θ 1 U θ 2 : ρ := [false] τ := [θ 2] while τ ρ do ρ := ρ τ τ := Pre(A, ρ) [θ 1] [ϕ ] := ρ end return [ϕ]

14 Some Results Theorem (Correctness) The symbolic model checking algorithm for ATL is correct. Proof (Sketch). Partial correctness of the algorithm can be proved by induction on the structure of the input formula ϕ. Termination is guaranteed because the state space Q is finite. Theorem (Complexity) The model checking problem for ATL is PTIME-Complete and can be solved in time O(m l) for a game structure with m transitions and an ATL formula of length l. The problem is PTIME-hard even for a fixed formula, and even in the special case of turn-based synchronous game structures. Proof (Sketch). graphs. By reduction to reachability in AND-OR

15 References Alur, R., Henzinger, T. A., and Kupferman, O. (1997). Alternating-time temporal logic. In 38th Annual Symposium on Foundations of Computer Science (FOCS 97), Miami Beach, Florida, USA, October 19-22, pages IEEE Computer Society. Alur, R., Henzinger, T. A., and Kupferman, O. (2002). Alternating-time temporal logic. Journal of the ACM, 49(5):