Constructive modal logics I

We often have to draw conclusions about states of machines in computer science and about states of knowledge and belief in artificial intelligence (AI) based on partial information. Nerode (1990) suggested using constructive (equivalently, intuitionistic) logic as the language to express such deductions and also suggested designing appropriate intuitionistic Kripke frames to express the partial information. Following this program, Nerode and Wijesekera (1990) developed syntax, semantics and completeness for a system of intuitionistic dynamic logic for proving properties of concurrent programs. Like all dynamics logics, this was a logic of many modalities, each expressing a program, but in intuitionistic rather than in classical logic. In that logic, both box and diamond are needed, but these two are not intuitionistically interdefinable and, worse, diamond does not distribute over ‘or’, except for sequential programs. This also happens in other contemplated computer science and AI applications, and leads outside the class of constructive logics investigated in the literature. The present paper fills this gap. We provide intuitionistic logics with independent box and diamond without assuming distribution of diamond over ‘or’. The completeness theorem is based on intuitionistic Kripke frames (partially ordered sets of increasing worlds), but equipped with an additional, quite separate accessibility relation between worlds. In the interpretation of Nerode and Wijesekera (1990), worlds under the partial order represent states of partial knowledge, the accessibility represents change in state of partial knowledge resulting from executing a specific program. But there are many other computer science interpretations. This formalism covers all computer science applications of which we are aware. We also give a cut elimination theorem and algebraic and topological formulations, since these present some new difficulties. Finally, these results were obtained prior to those in Nerode and Wijesekera (1990).