Loop-Type Sequent Calculi for Temporal Logic

Abstract

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been invented. In this paper, a sound and complete loop-type sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ for PLTL with the temporal operators “next” and “henceforth always” ( $${\mathbf{PLTL}}^{n,a}$$ ) is considered at first. We prove that all rules of $$\mathbf{G} _\text {L}{} \mathbf{T} $$ are invertible and that the structural rules of weakening and contraction, as well as the rule of cut, are admissible in $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . We describe a decision procedure for $${\mathbf{PLTL}}^{n,a}$$ based on the introduced calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ . Afterwards, we introduce a sound and complete sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} ^\mathcal {U}$$ for PLTL with the temporal operators “next” and “until”.