Revisiting MITL to Fix Decision Procedures

TL;DR

This paper corrects the semantics and decision procedures for MITL, establishing their complexity and introducing a richer fragment with more efficient decision procedures, supported by formal proofs.

Contribution

It identifies errors in previous MITL decision procedures, proposes correct semantics, and provides new complexity results and a richer fragment with improved decision procedures.

Findings

Previous MITL procedures are incorrect due to semantics issues.

Correct semantics lead to EXPSPACE-complete decision procedures.

A new fragment MITL_{WI} has PSPACE-complete decision procedures.

Abstract

Metric Interval Temporal Logic (MITL) is a well studied real-time, temporal logic that has decidable satisfiability and model checking problems. The decision procedures for MITL rely on the automata theoretic approach, where logic formulas are translated into equivalent timed automata. Since timed automata are not closed under complementation, decision procedures for MITL first convert a formula into negated normal form before translating to a timed automaton. We show that, unfortunately, these 20-year-old procedures are incorrect, because they rely on an incorrect semantics of the R operator. We present the right semantics of R and give new, correct decision procedures for MITL. We show that both satisfiability and model checking for MITL are EXPSPACE-complete, as was previously claimed. We also identify a fragment of MITL that we call MITL_{WI} that is richer than MITL_{0,\infty}, for which we show that both satisfiability and model checking are PSPACE-complete. Many of our results have been formally proved in PVS.