MTL with Bounded Variability: Decidability and Complexity

Abstract

Abstract. This paper investigates the properties of Metric Temporal Logic (MTL) over models in which time is dense but phenomena are constrained to have bounded variability. Contrary to the case of generic dense-time behaviors, MTL is proved to be fully decidable over models with bounded variability, if the variability bound is given. In these decidable cases, MTL complexity is shown to match that of simpler decidable logics such as MITL. On the contrary, MTL is undecidable if all behaviors with variability bounded by some generic constant are considered, but with an undecidability degree that is lower than in the case of generic behaviors.

MSO logics for weighted timed automata

“…The restriction to timed languages with bounded variability is a reasonable assumption as practically any system can only handle a bounded number of tasks within a time unit. Recently, another positive decidability result concerning MTL model checking was shown for this particular class of timed languages [18]. Let M ⊆ T Σ + be a set of timed words.…”

“…Hence, every finite word is non-Zeno and thus this notion is weaker than that of bounded variability. The restriction of being non-Berkeley for some positive real number δ has been introduced by Furia and Rossi [18] and means that between any two events more than δ time units must pass. For a comparison between these three restrictions see the paper of Furia and Rossi.…”

Bounded Variability of Metric Temporal Logic

“…Bounded variability is a natural semantic restriction over dense time, which has been applied to various formalisms including timed automata [27], duration calculus [10], and, in our previous work, MTL [12]. Recently, we also applied it to LTL over discrete time; while it is obvious that every discrete-time model has bounded variability (given by the fixed duration associated with one discrete time unit), in [14,15] we showed how the LTL validity problem can be simplified under the assumption that only v < V change events happen every V discrete time steps.…”

“…Proof. We showed that the MTL satisfiability problem over BR ≥0 [v/V ] is decidable for generic v/V [12, Corollary 1]; and that we can encode in MTL the bounded variability constrain as well as its complement [12,Section 4.3].…”

Automata-based Verification of Linear Temporal Logic Models with Bounded Variability

“…Some of the ideas used in our work are inspired-and named after-similar notions introduced for dense time models; in particular, Pnueli operators [18] and bounded variability itself [19], [20].…”