A Model Checker for Interval Temporal Logic over Finite Structures (original) (raw)
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Interval temporal logics: a journey
We discuss a family of modal logics for reasoning about relational structures of intervals over (usually) linear orders, with modal operators associated with the various binary relations between such intervals, known as Allen's interval relations. The formulae of these logics are evaluated at intervals rather than points and the main effect of that semantic feature is substantially higher expressiveness and computational complexity of the interval logics as compared to point-based ones. Without purporting to provide a comprehensive survey of the field, we take the reader to a journey through the main developments in it over the past 10 years and outline some landmark results on expressiveness and (un)decidability of the satisfiability problem for the family of interval logics.
A Note on Ultimately-Periodic Finite Interval Temporal Logic Model Checking
2020
In this paper, we deal with the ultimately-periodic finite interval temporal logic model checking problem. The problem has been shown to be in PTIME for full Halpern and Shoham’s interval temporal logic (HS for short) over finite models, as well as for the HS fragment featuring a modality for Allen relation meets and metric constraints over non-sparse ultimately-periodic (finite) models, that is, over ultimately-periodic models whose representation is polynomial in their size. Here, we generalize the latter result to the case of sparse ultimately-periodic models.
Information and Computation, 2018
Some temporal properties of reactive systems, such as actions with duration, accomplishments, and temporal aggregations, which are inherently interval-based, can not be properly dealt with by the standard, point-based temporal logics LTL, CTL and CTL*, as they give a state-by-state account of system evolution. Conversely, interval temporal logics-which feature intervals, instead of points, as their primitive entities-are highly expressive formalisms for temporal representation and reasoning that naturally allow one to deal with them. In this paper, we study the model checking (MC) problem for Halpern and Shoham's modal logic of time intervals (HS), interpreted on Kripke structures, under the homogeneity assumption, according to which a proposition letter holds over a finite computation path (interval) if and only if it holds at all of its states. HS is the best known interval-based temporal logic, which has one modality for each of the 13 possible ordering relations between pairs of intervals (the so-called Allen's relations), apart from equality. We focus on the MC problem for some HS fragments featuring modalities for (a subset of) Allen's relations meet, met-by, started-by, and finished-by, showing that it is in P NP , a class to which other pointbased logics (e.g., CTL+ and FCTL) are known to belong. Additionally, we provide some complexity lower bounds to the problem. All the algorithms we propose can be efficiently implemented by means of a polynomial-time procedure which iteratively invokes a SATsolver, enabling us to directly exploit the great speed of SAT-solvers.
Two-sorted Point-Interval Temporal Logics
Electronic Notes in Theoretical Computer Science, 2011
There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as particular, duration-less intervals. Here we develop explicitly two-sorted point-interval temporal logical framework whereby time instants (points) and time periods (intervals) are considered on a par, and the perspective can shift between them within the formal discourse. We focus on fragments involving only modal operators that correspond to the inter-sort relations between points and intervals. We analyze their expressiveness, comparative to interval-based logics, and the complexity of their satisfiability problems. In particular, we identify some previously not studied and potentially interesting interval logics.
Interval temporal logics over strongly discrete linear orders: Expressiveness and complexity
Theoretical Computer Science, 2014
ABSTRACT Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. Their computational behavior mainly depends on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders as well as over its relevant subclasses (the class of finite linear orders, ZZ, NN, and Z−Z−). We classify them in terms of both their relative expressive power and their complexity, which ranges from NP-completeness to non-primitive recursiveness.
Interval Temporal Logics over Strongly Discrete Linear Orders: the Complete Picture
Interval temporal logics provide a general framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders. We classify them in terms of both their relative expressive power and their complexity. We show that there are exactly 44 expressively different decidable fragments, whose complexity ranges from NP to EXPSPACE. In addition, we identify some new undecidable fragments (all the remaining HS fragments were already known to be undecidable over strongly discrete linear orders). We conclude the paper by an analysis of the specific case of natural numbers, whose behavior slightly differs from that of the whole class of strongly discrete linear orders. The number of decidable fragments over natural numbers raises ...
The Dark Side of Interval Temporal Logic: Sharpening the Undecidability Border
Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undesirability. Identifying minimal undecidable interval logics is thus a natural and important issue in the research agenda in the area. The decidability status of a logic often depends on the class of models (in our case, the class of interval structures)in which it is interpreted. In this paper, we have identified several new minimal undecidable logics amongst the fragments of Halpern-Shoham logic HS, including the logic of the overlaps relation, over the classes of all and finite linear orders, as well as the logic of the meet and subinterval relations, over the class of dense linear orders. Together with previous undecid ability results, this work contributes to delineate the border of the dark side of interval temporal logics quite sharply.