An efficient tableau for reasoning over general linear time

Since propositional linear time temporal logic (PLTL) was introduced for reasoning over linear discrete time, it has become a popular reasoning tool for the verification of hardware and software. Reasoning applications for discrete-time temporal logics such as PLTL consist of temporal databases, temporal reasoning in natural language processing, temporal planning, and temporal reasoning in medicine. Checking satisfiability of a temporal logic formula is a fundamental reasoning task. The tableau-based approach is one of the most popular techniques for providing decision procedures. The advantage of the tableau approach is providing a corresponding model of the formula when checking the satisfiability.

Practical reasoning aids for dense-time temporal logics are not common despite a range of potential applications from the verification of hybrid systems to artificial intelligence. There have been recent suggestions that mosaics can provide implementable tableau-style decision procedures for various linear time temporal logics beyond the standard discrete natural numbers model of time.

In this thesis, we implement the first-known tableau reasoner for general linear time with Kamp’s until and since, and also provide some heuristics to speed up the reasoner. Then, we extend the established idea of mosaic tableaux by introducing the abstract methodology of partial mosaics. Each partial mosaic is able to represent many mosaics. This can reduce the running time of building a tableau. We define partial mosaics, partial mosaic-based tableaux, and algorithms for building tableaux. In the last part, we extend the partial mosaic technique from general linear time to dense time and analyze the effects on performance.