Partially Punctual Metric Temporal Logic is Decidable

Abstract

Metric Temporal Logic MTL[UI, SI] is one of the most studied real time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints I. Henzinger et al., in their seminal paper showed that the non-punctual fragment of MTL called MITL is decidable. In this paper, we sharpen this decidability result by showing that the partially punctual fragment of MTL (denoted PMTL) is decidable over strictly monotonic finite point wise time. In this fragment, we allow either punctual future modalities, or punctual past modalities, but never both together. We give two satisfiability preserving reductions from PMTL to the decidable logic MTL[UI]. The first reduction uses simple projections, while the second reduction uses a novel technique of temporal projections with oversampling. We study the tradeoff between the two reductions: while the second reduction allows the introduction of extra action points in the underlying model, the equisatisfiable MTL[UI] formula obtained is exponentially more succinct than the one obtained via the first reduction, where no oversampling of the underlying model is needed. We also show that PMTL is strictly more expressive than the fragments MTL[UI, S] and MTL[U, SI].