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)

This is the first paper on constructive concurrent dynamic logic . For the first time, either for concurrent or sequential dynamic logic, we give a satisfactory treatment of what statements are forced to be true by partial information about the underlying computer. Dynamic logic was developed by Pratt [V. Pratt, Semantical considerations on Floyd–Hoare logic, in: 17th Annual IEEE Symp. on Found. Comp. Sci., New York, 1976, pp. 109–121, V. Pratt, Applications of modal logic to programming, Studia Logica 39 257–274] (...) for nondeterministic sequential programs, and by Peleg [D. Peleg, Concurrent dynamic logic, Journal of the Association for Computing Machinery 34 , D. Peleg, Communication in concurrent dynamic logic, Journal of Computer and System Sciences 35 ] for concurrent programs, for the purpose of proving properties of programs such as correctness. Here we define what it means for a dynamic logic formula to be forced to be true knowing only partial information about the results of assignments and tests. This informal CCDL semantics is formalized by intuitionistic Kripke frames modeling this partial information, and each such frame is interpreted as an idealized concurrent machine . In CCDL, proofs and deductions are ω-height, ω-branching, well-founded labeled subtrees of ωω. These are a generalization of the signed tableaux of Nerode [A. Nerode, Some lectures in modal logic, Technical Report, M.S.I. Cornell University, 1989, CIME Logic and Computer Science Montecatini Volume, Springer-Verlag Lecture Notes, 1990, A. Nerode, Some lectures in intuitionistic logic, Technical Report, M.S.I. Cornell University, 1988, Marktoberdorf Logic and Computation NATO Summer School Volume, NATO Science Series, 1990 ] stemming from the prefix tableaux of Fitting [M.C. Fitting, Proof Methods for Modal and Intuitionistic Logic, Reidel, 1983]. We demonstrate the correctness of our tableau proofs, define consistency properties, prove that consistency properties yield models, construct systematic tableaux, prove that systematic tableaux yield a consistency property, and conclude that CCDL is complete. This infinitary semantics and proof procedure will be the primary guide for defining, in a sequel, the correct finitary CCDL based on induction principles. FCCDL is suitable for implementation in constructive logic software systems such as Constable’s NUPRL or Huet-Coquand’s CONSTRUCTIONS. Our goal is to develop a constructive logic programming tool for specification and modular verification of programs in any imperative concurrent language, and for the extraction of concurrent programs from constructive proofs. Subsequent papers will introduce analogous logics for declarative and functional concurrent languages. (shrink)

Buddhist and Vedic Studies. A Miscellany. O.H. De A. Wijesekera. Motilal Banarsidass, Delhi 1994. xv, 361 pp. Rs 275.

We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert (...) axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic. (shrink)