On the Decidability of Metric Temporal Logic

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 nonprimitive 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. The question of the decidability of MTL over infinite words remains open.

Controller Synthesis for MTL Specifications

“…We may omit the subscript I when it is equal to R ≥0 . In this paper we consider the so-called pointwise semantics, and thus interpret MTL over timed words [22]. Given a (finite or infinite) timed word σ = (a 1 , τ 1 )(a 2 , τ 2 ) .…”

“…The rules for atoms, negation, and conjunction are standard. For the until modality, following [22], we give a strict-future interpretation as follows:…”

“…where negation is only applied to actions a ∈ Σ. We then define the fragment of MTL, called Safety-MTL [22], consisting of those MTL formulas in positive normal form that only include instances of the constrained until operator U I in which interval I has bounded length. Note that no restriction is placed on the dual-until operator.…”

“…We encode the executions of S (i.e. the paths of G(S) from (s 0 , ε)) [22] by the set L correct of timed words (a 1 , t 1 )(a 2 , t 2 ) · · · over {m?, m! | m ∈ M } such that: Reduction to the control problem.…”

“…For instance, we can write a formula ψ = (p → ♦ =1 q), which expresses that a request p is always followed one time unit later by a response q. The interest in this logic has encountered a great soar in the last year, since Ouaknine and Worrell proved that the model-checking and the satisfiability problems for this logic are decidable [22], as soon as they are interpreted using a pointwise semantics over finite timed words. It is worth noticing that MTL, like most real-time logics, can be interpreted either using a pointwise semantics (the system is observed through events), or using a continuous semantics (the system is observed at any point in time).…”