Combining linear-time temporal logic with constructiveness and paraconsistency

Abstract

It is known that linear-time temporal logic (LTL), which is an extension of classical logic, is useful for expressing temporal reasoning as investigated in computer science. In this paper, two constructive and bounded versions of LTL, which are extensions of intuitionistic logic or Nelson's paraconsistent logic, are introduced as Gentzen-type sequent calculi. These logics, IB [ l ] and PB [ l ] , are intended to provide a useful theoretical basis for representing not only temporal (linear-time), but also constructive, and paraconsistent (inconsistency-tolerant) reasoning. The time domain of the proposed logics is bounded by a fixed positive integer. Despite the restriction on the time domain, the logics can derive almost all the typical temporal axioms of LTL. As a merit of bounding time, faithful embeddings into intuitionistic logic and Nelson's paraconsistent logic are shown for IB [ l ] and PB [ l ] , respectively. Completeness (with respect to Kripke semantics), cut–elimination, normalization (with respect to natural deduction), and decidability theorems for the newly defined logics are proved as the main results of this paper. Moreover, we present sound and complete display calculi for IB [ l ] and PB [ l ] . In [P. Maier, Intuitionistic LTL and a new characterization of safety and liveness, in: Proceedings of Computer Science Logic 2004, in: Lecture Notes in Computer Science, vol. 3210, Springer-Verlag, Berlin, 2004, pp. 295–309] it has been emphasized that intuitionistic linear-time logic (ILTL) admits an elegant characterization of safety and liveness properties. The system ILTL, however, has been presented only in an algebraic setting. The present paper is the first semantical and proof-theoretical study of bounded constructive linear-time temporal logics containing either intuitionistic or strong negation.