Partially Punctual Metric Temporal Logic is Decidable

Metric Temporal Logic \mathsf{MTL}[\until_I,\since_I] 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 $\mathsf{MTL}$ called $\mathsf{MITL}$ is decidable. In this paper, we sharpen this decidability result by showing that the partially punctual fragment of $\mathsf{MTL}$ (denoted $\mathsf{PMTL}$) 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 $\mathsf{PMTL}$ to the decidable logic \mathsf{MTL}[\until_I]. The first reduction uses simple projections, while the second reduction uses a novel technique of temporal projections with oversampling. We study the trade-off between the two reductions: while the second reduction allows the introduction of extra action points in the underlying model, the equisatisfiable \mathsf{MTL}[\until_I] formula obtained is exponentially succinct than the one obtained via the first reduction, where no oversampling of the underlying model is needed. We also show that $\mathsf{PMTL}$ is strictly more expressive than the fragments \mathsf{MTL}[\until_I,\since] and \mathsf{MTL}[\until,\since_I]

A History of Until

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A until B holds at the current time instant w iff either B holds at w or there exists a time instant w' in the future at which B holds and such that A holds in all the time instants between the current one and w'. This "ambivalent" nature poses a significant challenge when attempting to give deduction rules for until. In this paper, in contrast, we make explicit this duality of until to provide well-behaved natural deduction rules for linear-time logics by introducing a new temporal operator that allows us to formalize the "history" of until, i.e., the "internal" universal quantification over the time instants between the current one and w'. This approach provides the basis for formalizing deduction systems for temporal logics endowed with the until operator. For concreteness, we give here a labeled natural deduction system for a linear-time logic endowed with the new operator and show that, via a proper translation, such a system is also sound and complete with respect to the linear temporal logic LTL with until.Comment: 24 pages, full version of paper at Methods for Modalities 2009 (M4M-6