Translating a Modal Language with Embedded Implication into Horn Clause Logic

In this paper we present a method for translating Horn clauses extended with modalities and embedded implication (which provide reasoning capabilities in a multiagent situation and hypothetical reasoning) into Horn clauses, therefore suitable for SLD resolution. The translation takes two steps: the rst one eliminates embedded implications by introducing new modalities; the second eliminates modalities by adding an argument which represents the worlds of the Kripke semantics to all predicates. 1 Introduction Modal logics are widely used in computer science and artiicial intelligence to deal with knowledge and beliefs, time, actions, and several researchers have proposed modal extensions of logic programming languages 7, 1, 13, 19, 17, 6]. In particular, in 5] we have deened a modal logic programming language which allows both multiple modalities and embedded implications. The language has been shown to be well suited for structuring knowledge, and, in particular, for deening module constructs within programs, for representing agent beliefs and also for hypothetical reasoning. This language has a goal directed operational semantics which has been proved to be sound and complete with respect to the Kripke semantics. The operational derivability of a goal is deened with respect to a notion of modal context, which consists of a sequence of modal operators. The modal context keeps track of the new clauses which are added to the program when evaluating implication goals. The modal operators of this language are rather speciic, since the language was deened for providing constructs for modularizing programs, and for allowing simple reasoning capabilities in a multiple agent environment. However, the goal directed procedure can be extended to similar languages whose modal operators have diierent properties. In the next section we introduce the modal language, and recall its operational semantics. The goal directed procedure gives a precise deenition of the operation behaviour of a program, and provides a means for executing a program. However the actual implementation of the procedure can raise several problems. The simplest solution of building an interpreter (say in Prolog), may turn out to be extremely ineecient, since the interpreter will have to deal with the modal context, and, in particular, with the dynamically added clauses. In this paper we present a diierent approach, based on translating our language into Horn clause logic, so that the translated program can be executed by any Prolog interpreter or compiler, with the advantage that many features, such as uniication or variable renaming, are …