Computational Aspects of Proofs in Modal Logic

Various modal logics seem well suited for developing models of knowledge, belief, time, change, causality, and other intensional concepts. Most such systems are related to the classical Lewis systems, and thereby have a substantial body of conventional proof theoretical results. However, most of the applied literature examines modal logics from a semantical point of view, rather than through proof theory. It appears arguments for validity are more clearly stated in terms of a semantical explanation, rather than a classical proof-theoretic one. We feel this is due to the inability of classical proof theories to adequately represent intensional aspects of modal semantics. This thesis develops proof theoretical methods which explicitly represent the underlying semantics of the modal formula in the proof. We initially develop a Gentzen style proof system which contains semantic information in the sequents. This system is, in turn, used to develop natural deduction proofs. Another semantic style proof representation, the modal expansion tree is developed. This structure can be used to derive either Gentzen style or Natural Deduction proofs. We then explore ways of automatically generating MET proofs, and prove sound and complete heuristics for that procedure. These results can be extended to most propositional system using a Kripke style semantics and a fist order theory of the possible worlds relation. Examples are presented for standard T, S4, and S5 systems, systems of knowledge and belief, and common knowledge. A computer program which implements the theory is briefly examined in the appendix. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-85-55. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/658 COMPUTATIONAL ASPECTS OF PROOFS IN MODAL LOGIC Gregory Donald Hager MS-CIS-85-55 Department Of Computer and Information Science Moore School University of Pennsylvania Philadelphia, PA 191 04