Partially Punctual Metric Temporal Logic is Decidable

TL;DR

This paper proves that the partially punctual fragment of Metric Temporal Logic (PMTL) is decidable over strictly monotonic finite point-wise time, using two reductions to the decidable logic MTL[until_I], and explores its expressiveness and trade-offs.

Contribution

It introduces the decidability of PMTL over finite point-wise time and presents two satisfiability-preserving reductions to MTL[until_I], including a novel technique of temporal projections with oversampling.

Findings

PMTL is decidable over finite point-wise time.

Two reductions to MTL[until_I] are provided, one using projections and the other using oversampling.

PMTL is more expressive than certain existing MTL fragments.

Abstract

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 $\mathsf{MTL}[\until,\since_I]$.