A sound and complete proof system for a unified temporal logic

Abstract

Theorem proving is one of the most widely used approaches to the verification of computer systems, and its theoretical basis is generally a proof system for formal derivation of logic formulas. In this paper, we propose a sound and complete proof system for Propositional Projection Temporal Logic (PPTL) with indexed expressions, which is a unified temporal logic that subsumes the well used Linear Temporal Logic (LTL). First, the syntax, semantics and logic laws of PPTL that allows indexed expressions are introduced, and the representation of LTL constructs by PPTL formulas is illustrated. Then, the proof system for the logic is presented which consists of axioms and inference rules for the derivation of both basic constructs and indexed expressions. To show the capability of the proof system, several examples of formal proofs are provided. Finally, the soundness and completeness of the proof system are demonstrated.