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Loop-Type Sequent Calculi for Temporal Logic

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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”.

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Correspondence to R. Alonderis.

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Alonderis, R., Pliuškevičius, R., Pliuškevičienė, A. et al. Loop-Type Sequent Calculi for Temporal Logic. J Autom Reasoning (2020). https://doi.org/10.1007/s10817-020-09544-1

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Keywords

  • Temporal logics
  • Sequent calculi
  • Derivation loops