On the Completeness and Decidability of the Horn‐Like Fragment of the First‐Order Linear Temporal Logic

Abstract

We prove the completeness and decidability of the Horn‐like sequents, specifically, the so‐called D2‐sequents (of the first‐order linear temporal logic) considered in the author's paper [Lith. Math. J., 41(3), 266–281 (2001)]. In this paper, with the help of the infinitary calculus GLω, grounded by the author in his earlier papers, for D2‐sequents we construct a D2Sat calculus of the so‐called saturated type consisting of decidable deductive procedures replacing the omega‐rule for the “always” operator. In the present paper, in order to prove the completeness and decidability of the calculus D2Sat, we construct the so‐called invariant decidable calculus D2IN. We prove the equivalence of the calculi D2IN, D2Sat, and G Lω ** for the so‐called saturated D2‐sequents. From this equivalence, by reducing an arbitrary D2‐sequent to a saturated D2‐sequent, and also from the ω‐completeness of the G Lω ** calculus and decidability of the invariant calculus D2IN, we deduce the completeness and decidability of the calculus D2Sat in the class of D2‐sequents.