Logical consecutions in discrete linear temporal logic
Vladimir Vladimirovich Rybakov · Journal of Symbolic Logic · 2005
Abstract We investigate logical consequence in temporal logics in terms of logical consecutions, i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be ‘correct’ in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logicLDTL, consisting of all formulas valid in the frame 〈L≤, ≥〉 of all integer numbers, is the prime object of our investigation. We describe consecutions admissible inLDTLin a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible inLDTL. Also, we note that using the same technique it follows that the linear temporal logicL(N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logicsLDTLandL(N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (known fact), and the given deciding algorithms are explicit.