On the decidability and complexity of Metric Temporal Logic over finite words

TL;DR

This paper proves that the satisfiability and model-checking problems for Metric Temporal Logic over finite words are decidable, with some problems having non-primitive recursive complexity, challenging previous beliefs.

Contribution

It establishes the decidability of MTL satisfiability and model-checking over finite words, including safety fragments over infinite words, contradicting earlier claims.

Findings

MTL satisfiability over finite words is decidable.

Model checking MTL over finite words is decidable.

Safety fragment model checking over infinite words is decidable.

Abstract

Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also consider the model-checking problem for MTL: whether all words accepted by a given Alur-Dill timed automaton satisfy a given MTL formula. We show that this problem is decidable over finite words. Over infinite words, we show that model checking the safety fragment of MTL--which includes invariance and time-bounded response properties--is also decidable. These results are quite surprising in that they contradict various claims to the contrary that have appeared in the literature.