A Simple Obligation to Metric Interval Temporal Logic

2026-07-15

AI Summary

This work addresses the satisfiability problem for Metric Interval Temporal Logic (MITL). To overcome the state-space explosion commonly encountered by existing approaches, the paper proposes a novel method based on bounded obligation tracking. The approach explicitly maintains time-constrained obligations over timed words, extending the notion of obligation propagation from Linear Temporal Logic (LTL) with explicit temporal information. It further introduces mechanisms for obligation merging and pruning to ensure that only a bounded number of obligations are retained globally at any point. Integrated with symbolic region-based techniques, this method substantially simplifies the decision procedure, achieving both completeness and enhanced algorithmic clarity and practicality.

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.

Problem

• Metric Interval Temporal Logic
• satisfiability
• timed word
• obligations
• temporal logic

Innovation

• Metric Interval Temporal Logic
• obligation tracking
• redundancy elimination
• symbolic satisfiability
• timed words