Partial cut elimination for propositional discrete linear time temporal logic

Jūratė Sakalauskaitė

doi:10.15388/lmr.2010.63

Abstract

We consider propositional discrete linear time temporal logic with future and past operators of time. For each formula ϕ of this logic, we present Gentzen-type sequent calculus Gr(ϕ) with a restricted cut rule. We sketch a proof of the soundness and the completeness of the sequent calculi presented. The completeness is proved via construction of a canonical model.

Highlights

One of the simplest temporal logics which is still widely applicable is the propositional discrete linear time temporal logic LTL. This logic is an extension of propositional logic with two future operators: and U
Each state is a set of primitive propositions
Sakalauskaitė for each formula φ of LTL−, we present Gentzen-type sequent calculus Gr(φ) with a restricted cut rule

Summary

Introduction

We consider an extension of LTL with past temporal operators: W (weak yesterday) and S (since). The aim of this paper is to present the Gentzen-type sequent calculi for LTL−. Sakalauskaitė for each formula φ of LTL−, we present Gentzen-type sequent calculus Gr(φ) with a restricted cut rule. The main results of this paper are the following: 1)we prove the soundness of Gr(φ) using Hilberttype calculus for LTL−; 2)we sketch the proof of the completeness of Gr(φ). We recall the Hilbert-type calculus HLTL− for LTL− [7]. (See [7].) For each formula φ of LTL−, φ is provable in HLTL− iff φ is globally valid Proposition 1 [Soundness and completeness of HLTL−]. (See [7].) For each formula φ of LTL−, φ is provable in HLTL− iff φ is globally valid

Preliminaries

Gentzen-type sequent calculi with restricted cut rule

Soundness of sequent calculi

Findings

Construction of canonical model