Reasoning About Epistemic States of Agents by Modal Logic Programming [chapter]

Linh Anh Nguyen
2006 Lecture Notes in Computer Science  
Modal logic programming is one of appropriate approaches to deal with reasoning about epistemic states of agents. We specify here the least model semantics, the fixpoint semantics, and an SLD-resolution calculus for modal logic programs in the multimodal logic KD4Ig5a, which is intended for reasoning about belief and common belief of agents. We prove that the presented SLD-resolution calculus is sound and complete. We also present a formalization of the wise men puzzle using a modal logic
more » ... m in KD4Ig5a. This shows that it is worth to study modal logic programming for multi-agent systems. A Kripke frame is a tuple W, τ, R 1 , . . . , R m , where W is a nonempty set of possible worlds, τ ∈ W is the actual world, and R i is a binary relation on W , called the accessibility relation for the modal operators 2 i , 3 i . If R i (w, u) holds then we say that the world u is accessible from the world w via R i . A fixed-domain Kripke model with rigid terms, hereafter simply called a (Kripke) model, is a tuple M = D, W, τ, R 1 , . . . , R m , π , where D is a set called the domain, W, τ, R 1 , . . . , R m is a Kripke frame, and π is an interpretation of symbols. For a constant symbol a, π(a) is an element of D, denoted by a M . For an n-ary function symbol f , π(f ) is a function from D n to D, denoted by f M . For an n-ary predicate symbol p and a world w ∈ W , π(w)(p) is an n-ary relation on D, denoted by p M,w . (We adopt here the version with fixed-domain and rigid terms, as it is most popular. This work can be extended for other versions of Kripke semantics, e.g. with varying domain and flexible terms; see a discussion in [26] .) A model graph is a tuple W, τ, R 1 , . . . , R m , H , where W, τ, R 1 , . . . , R m is a Kripke frame and H is a function that maps each world of W to a set of formulas. Every model graph W, τ, R 1 , . . . , R m , H corresponds to a Herbrand model M = U, W, τ, R 1 , . . . , R m , π specified by: U is the Herbrand universe (i.e. the set of all ground terms), c M = c, f M (t 1 , . . . , t n ) = f (t 1 , . . . , t n ), and ((t 1 , . . . , t n ) ∈ p M,w ) ≡ (p(t 1 , . . . , t n ) ∈ H(w)), where t 1 , . . . , t n are ground terms. We will sometimes treat a model graph as its corresponding model. A variable assignment V w.r.t. a Kripke model M is a function that maps each variable to an element of the domain of M . The value of t M [V ] for a term t is defined as usual. Given some Kripke model M = D, W, τ, R 1 , . . . , R m , π , some variable assignment V , and some world w ∈ W , the satisfaction relation M, V, w |= ψ for a formula ψ is defined as follows: M, V, w |= p(t 1 , . . . , t n ) iff (t M 1 [V ], . . . , t M n [V ]) ∈ p M,w ; M, V, w |= 2 i ϕ iff for all v ∈ W such that R i (w, v), M, V, v |= ϕ; M, V, w |= ∀x.ϕ iff for all a ∈ D, (M, V , w |= ϕ), where V (x) = a and V (y) = V (y) for y = x;
doi:10.1007/11750734_3 fatcat:fnzle2g6orhkjjszrqritxn3se