A Fixpoint Semantics and an SLD-Resolution Calculus for Modal Logic Programs

We propose a modal logic programming language called MProlog, which is as expressive as the general modal Horn fragment. We give a fixpoint semantics and an SLD-resolution calculus for MProlog in all of the basic serial modal logics KD , T , KDB , B , KD4, S4, KD5, KD45, and S5. For an MProlog program P and for L being one of the mentioned logics, we define an operator TL,P , which has the least fixpoint IL,P . This fixpoint is a set of formulae, which may contain labeled forms of the modal operator 3, and is called the least L-model generator of P . The standard model of IL,P is shown to be a least L-model of P . The SLD-resolution calculus for MProlog is designed with a similar style as for classical logic programming. It is sound and complete. We also extend the calculus for MProlog in the almost serial modal logics KB , K5, K45, and KB5.