Abstract
Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been considered in the literature. Cutfree Gentzen-type sequent calculi are convenient tools for backward proof-search search of formulas and sequents. In this paper we present a cut-free Gentzen type sequent calculus for PLTL with the operator
Highlights
Propositional linear temporal logic (PLTL) is used in computer science for specification and verification of programs [2, 6]
A sequent S is called derivable in loop-type sequent calculus (LTSC) ( ⊢ S in notation), iff it is axiomatically derivable or there exists a backward proof-search tree V (S) such that: 1) each leaf of V (S) is an axiom or a terminal sequent of a derivation loop with some eventuality formula and 2) each connected component in V (S) has a common eventuality formula
We have introduced and considered the Gentzen-type sequent calculus LTSC
Summary
Propositional linear temporal logic (PLTL) is used in computer science for specification and verification of programs [2, 6]. We consider the loop-type sequent calculus (LTSC) for PLTL with temporal operators “” and “until”. Infinitary sequent calculi containing ω-type induction rule. The saturated calculus contains (instead of induction-like rules) some non-logical axioms indicating the saturation of proof-search process. The loop-type sequent calculi (as saturated calculi) for temporal, mutual belief and dynamic logics were considered in [10]. 5. A cut-free and invariant-free sequent calculus for PLTL is presented in [5]. A cut-free and invariant-free sequent calculus for PLTL is presented in [5] This calculus has the new operator “unless”, and do not retain the sub-formula property. The loop-type sequent calculus introduced in the present paper has not been considered in the literature before.
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