Bounded variability of metric temporal logic

Deciding validity of Metric Temporal Logic (MTL) formulas is generally very complex and even undecidable over dense time domains; bounded variability is one of the several restrictions that have been proposed to bring decidability back. A temporal model has bounded variability if no more than v events occur over any time interval of length V, for constant parameters v and V. Previous work has shown that MTL validity over models with bounded variability is less complex—and often decidable—than MTL validity over unconstrained models. This paper studies the related problem of deciding whether an MTL formula has intrinsic bounded variability, that is whether it is satisfied only by models with bounded variability. The results of the paper are mainly negative: over dense time domains, the problem is mostly undecidable (even if with an undecidability degree that is typically lower than deciding validity); over discrete time domains, it is decidable with the same complexity as deciding validity. As a partial complement to these negative results, the paper also identifies MTL fragments where deciding bounded variability is simpler than validity, which may provide for a reduction in complexity in some practical cases

LNCS

Imprecision in timing can sometimes be beneficial: Metric interval temporal logic (MITL), disabling the expression of punctuality constraints, was shown to translate to timed automata, yielding an elementary decision procedure. We show how this principle extends to other forms of dense-time specification using regular expressions. By providing a clean, automaton-based formal framework for non-punctual languages, we are able to recover and extend several results in timed systems. Metric interval regular expressions (MIRE) are introduced, providing regular expressions with non-singular duration constraints. We obtain that MIRE are expressively complete relative to a class of one-clock timed automata, which can be determinized using additional clocks. Metric interval dynamic logic (MIDL) is then defined using MIRE as temporal modalities. We show that MIDL generalizes known extensions of MITL, while translating to timed automata at comparable cost

A Logical Characterization of Timed (non-)Regular Languages

CLTLoc (Constraint LTL over clocks) is a quantifier-free extension of LTL allowing variables behaving like clocks over real numbers. CLTLoc is in PSPACE [9] and its satisfiability can polynomially be reduced to a SMT problem, allowing a feasible implementation of a decision procedure. We used CLTLoc to capture the semantics of metric temporal logics over continuous time, such as Metric Interval Temporal Logic (MITL), resulting in the first successful implementation of a tool for checking MITL satisfiability [7]. In this paper, we assess the expressive power of CLTLoc, by comparing it with various temporal formalisms over dense time.When interpreted over timed words, CLTLoc is equivalent to Timed Automata. We also define a monadic theory of orders, extending the one introduced by Kamp, which is expressively equivalent to CLTLoc. We investigate a decidable extension with an arithmetical next operator, which allows the expression of timed non-ω-regular languages