New ways for developing proof theories for first-order multi modal logics

Abstract

Most of the nonclassical logics, temporal logics, process logics etc., which have been used for the specification and verification of processes are essentially extensions of modal logics. In this paper a quite complex first-order many-sorted multi modal logic (MM-Logic) with modal operators referring to a basic branching accessibility relation, its reflexive, transitive and reflexive-transitive closure, indexed modal operators, ‘eventually’ operators, ‘until’ operators and built-in equality is defined. It can serve as temporal, action, process or epistemic logic in various applications. The main purpose of this paper, however, is to demonstrate the development of a proof theory using the translation (into predicate logic) and refutation (with predicate logic resolution and paramodulation) paradigm. MM-Logic formulae are first translated into an intermediate logic called Context Logic (CL) and then with the standard translator from CL into an order-sorted predicate logic where a standard theorem prover can be used. The CL translation mechanism which simplifies the development of proof theories for complex nonclassical logics is briefly described.

This work was supported by the Sonderforschungsbereich 314 of the German Science Foundation (DFG) and the ESPRIT Project 1033, FORMAST, of the European Community. The original paper was written during a research stay at the Automated Reasoning Project of the Australian National University, Canberra.