A Simple Obligation to Metric Interval Temporal Logic

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 na\"ive 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.