Discrete Linear Temporal Logic Research Articles

Discrete linear temporal logic with current time point clusters, deciding algorithms

The paper studies the logic TL(N\Box+-wC) – logic of discrete linear time with current time point clusters. Its language uses modalities \Diamond+ (possible in future) and \Diamond- (possible in past) and special temporal operations, – \Box+w (weakly necessary in future) and \Box-w (weakly necessary in past). We proceed by developing an algorithm recognizing theorems of TL(N\Box+-wC), so we prove that TL(N\Box+-wC) is decidable. The algorithm is based on reduction of formulas to inference rules and converting the rules in special reduced normal form, and, then, on checking validity of such rules in models of singleexponential size in the rules. Also we consider the admissibility problem for TL(N\Box+-wC) and show how to reduce the problem for admissibility to the decidability of TL(N\Box+-wC) itself using definable universal modalities.

Logical consecutions in discrete linear temporal logic

AbstractWe 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.

Clausal temporal resolution

In this article, we examine how clausal resolution can be applied to a specific, but widely used, nonclassical logic, namely discrete linear temporal logic. Thus, we first define a normal form for temporal formulae and show how arbitrary temporal formulae can be translated into the normal form, while preserving satisfiability. We then introduce novel resolution rules that can be applied to formulae in this normal form, provide a range of examples, and examine the correctness and complexity of this approach. Finally, we describe related work and future developments concerning this work.