A Model Checking Procedure for Interval Temporal Logics based on Track Representatives

Model checking is commonly recognized as one of the most effective tools for system verification. While it has been systematically investigated in the context of classical, point-based temporal logics, it is still largely unexplored in the interval logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham\u2019s modal logic of time intervals HS, interpreted over finite Kripke structures, has been proposed, together with a proof of the EXPSPACE-hardness of the problem. In this paper, we devise an EXPSPACE model checking procedure for two meaningful HS fragments. It exploits a suitable contraction technique that allows one to replace sufficiently long tracks of a Kripke structure by equivalent shorter ones

Axiomatisation and decidability of multi-dimensional Duration Calculus

AbstractThe Shape Calculus is a spatio-temporal logic based on an n-dimensional Duration Calculus tailored for the specification and verification of mobile real-time systems. After showing non-axiomatisability, we give a complete embedding in n-dimensional interval temporal logic and present two different decidable subsets, which are important for tool support and practical use

Crossing the Undecidability Border with Extensions of Propositional Neighborhood Logic over Natural Numbers

Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen's relations, meets and met by). Recently, it has been shown that PNL interpreted over several classes of linear orders, including natural numbers, is decidable (NEXPTIME-complete) and that some of its natural extensions preserve decidability. Most notably, this is the case with PNL over natural numbers extended with a limited form of metric constraints and with the future fragment of PNL extended with modal operators corresponding to Allen's relations begins, begun by, and before. This paper aims at demonstrating that PNL and its metric version MPNL, interpreted over natural numbers, are indeed very close to the border with undecidability, and even relatively weak extensions of them become undecidable. In particular, we show that (i) the addition of binders on integer variables ranging over interval lengths makes the resulting hybrid extension of MPNL undecidable, and (ii) a very weak first-order extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of first-order formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar first-order extensions of point-based temporal logics)

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

A Road Map of Interval Temporal Logics and Duration Calculi

We survey main developments, results, and open problems on interval temporal logics and duration calculi. We present various formal systems studied in the literature and discuss their distinctive features, emphasizing on expressiveness, axiomatic systems, and (un)decidability results

A verification system for interval-based specification languages with its application to simulink

Ph.DDOCTOR OF PHILOSOPH