Towards Abstract Interpretation of Epistemic Logic

The model-checking problem is to decide, given a formula φ and an interpretation M , whether M satisfies φ, written M |= φ. Model-checking algorithms for temporal logics were initially developed with finite models (such as models of hardware) in mind so that M |= φ is decidable. As interest grew in model-checking infinite systems, other approaches were developed based on approximating the model-checking algorithm so that it still terminates with some useful output. In this work we present a model-checking algorithm for a multiagent epistemic logic containing operators for common and distributed knowledge. The model-checker is developed as a function directly from the semantics of the logic, in a style that could be applied straightforwardly to derive model-checkers for other logics. Secondly, we consider how to abstract the model-checker using abstract interpretation, yielding a procedure applicable to infinite models. The abstract model-checker allows model-checking with infinite-state models. When applied to the problem of whether M |= φ, it terminates and returns the set of states in M at which φ might hold. If the set is empty, then M definitely does not satisfy φ, while if the set is non-empty then M possibly satisfies φ. 1 Syntax and semantics of the logic CMAEL(CD) We consider the logic CMAEL(CD) [1, 7] whose formulas φ ∈ Φ are defined by the following grammar. φ ::= p | ¬φ | (φ1 ∧ φ2) | DAφ | CAφ. The variable p ranges over the set AP of atomic propositions, typically denoted by p, q, r, . . .; the variable A ranges over the set of coalitions P(Σ), which is the set of of non-empty subsets of Σ, where Σ is a finite, non-empty set of (names for) agents, typically denoted by a, b, . . .. The epistemic operators DA and CA are read as it is distributed knowledge among A that . . . and it is common knowledge among A that . . . respectively. When A is a singleton {a} we often write it as a subscript a instead of {a}, for example Da instead of D{a}. The semantics of CMAEL(CD) is given in terms of coalitional multiagent epistemic models (CMAEMs). A CMAEM is a tuple (Σ,S, {RA}A∈P+(Σ), {RA}A∈P+(Σ), L), 1. Σ is a finite, non-empty set of agents; 2. S 6= ∅ is a set of states; 3. for every A ∈ P(Σ), RA is an equivalence relation on S, satisfying the condition RA = ⋂ a∈ARa ; 4. for every A ∈ P(Σ), RA is the transitive closure of ⋃ a∈ARa ; 5. L : S 7→ P(AP) is a labelling function, assigning to every state s the set L(s) of atomic propositions true at s. Let S be a set. We define functions pre : ((S×S)×P(S))→ P(S) and p̃re : ((S×S)×P(S))→ P(S). – pre(R)(X) = {s | ∃s ′ ∈ X : (s, s ′ ) ∈ R} returns the set of states having at least one of their successors (in relation R) in the set X ⊆ S;