We often have to draw conclusions about states of machines in computer science and about states of knowledge and belief in artificial intelligence based on partial information. Nerode suggested using constructive 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 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 , but equipped with an additional, quite separate accessibility relation between worlds. In the interpretation of Nerode and Wijesekera , 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. (shrink)

Sets out the basic theory of normal modal and temporal propositional logics; applies this theory to logics of discrete (integer), dense (rational), and continuous (real) time, to the temporal logic of henceforth, next, and until, and to the propositional dynamic logic of regular programs.

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...) an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic. (shrink)

This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of (...) formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated. (shrink)

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.