A Simple Obligation to Metric Interval Temporal Logic

Abstract

Satisfiability of Metric Interval Temporal Logic (MITL) is a widely investigated subject. In this work, we present a new, and arguably simpler, approach for MITL satisfiability, based on an idea of tracking time-constrained obligations along a word. To check whether a Linear Temporal Logic (LTL) formula is true at a position of a word, it is natural to generate certain obligations that need to be satisfied at a later point. For instance, $a\mathcal{U}b$ (with strict Until semantics) is true at position $i$ if either $b$ or the set $\{a,a\mathcal{U}b\}$ is true at $i+1$. We enhance this idea in the context of MITL by introducing a notion of time inside these obligations. However, a naïve procedure could lead to more and more obligations getting generated along the word, with no bound on the number. We propose a simple mechanism to eliminate or merge redundant obligations. For MITL, this mechanism ensures that only a bounded number of obligations are maintained along the entire timed word. We develop this observation into a symbolic procedure for MITL satisfiability using regions.

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