Checking Interval Properties of Computations

Model checking is a powerful method widely explored in formal verification. Given a model of a system, e.g., a Kripke structure, and a formula specifying its expected behaviour, one can verify whether the system meets the behaviour by checking the formula against the model. Classically, system behaviour is expressed by a formula of a temporal logic, such as LTL and the like. These logics are "point-wise" interpreted, as they describe how the system evolves state-by-state. However, there are relevant properties, such as those constraining the temporal relations between pairs of temporally extended events or involving temporal aggregations, which are inherently "interval-based", and thus asking for an interval temporal logic. In this paper, we give a formalization of the model checking problem in an interval logic setting. First, we provide an interpretation of formulas of Halpern and Shoham's interval temporal logic HS over finite Kripke structures, which allows one to check interval properties of computations. Then, we prove that the model checking problem for HS against finite Kripke structures is decidable by a suitable small model theorem, and we provide a lower bound to its computational complexity.Comment: In Journal: Acta Informatica, Springer Berlin Heidelber, 201

Complexity of ITL model checking: some well-behaved fragments of the interval logic HS

Model checking has been successfully used in many computer science fields, including artificial intelligence, theoretical computer science, and databases. Most of the proposed solutions make use of classical, point-based temporal logics, while little work has been done in the interval temporal logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS over finite Kripke structures (under the homogeneity assumption) and an EXPSPACE model checking procedure for two meaningful fragments of it have been proposed. In this paper, we show that more efficient model checking procedures can be developed for some expressive enough fragments of HS

Model Checking Well-Behaved Fragments of HS: The (Almost) Final Picture

Model checking is one of the most powerful and widespread tools for system verification with applications in many areas of computer science and artificial intelligence. The large majority of model checkers deal with properties expressed in point-based temporal logics, such as LTL and CTL. However, there exist relevant properties of systems which are inherently interval-based. Model checking algorithms for interval temporal logics (ITLs) have recently been proposed to check interval properties of computations. As the model checking problem for full Halpern and Shoham\u2019s ITL (HS for short) turns out to be decidable, but computationally heavy, research has focused on its well-behaved fragments. In this paper, we provide an almost final picture of the computational complexity of model checking for HS fragments with modalities for (a subset of) Allen\u2019s relations meets, met by, starts, and end

Interval vs. Point Temporal Logic Model Checking: an Expressiveness Comparison

Model checking is a powerful method widely explored in formal verification to check the (state-transition) model of a system against desired properties of its behaviour. Classically, properties are expressed by formulas of a temporal logic, such as LTL, CTL, and CTL*. These logics are "point-wise" interpreted, as they describe how the system evolves state-by-state. On the contrary, Halpern and Shoham\u27s interval temporal logic (HS) is "interval-wise" interpreted, thus allowing one to naturally express properties of computation stretches, spanning a sequence of states, or properties involving temporal aggregations, which are inherently "interval-based". In this paper, we study the expressiveness of HS in model checking, in comparison with that of the standard logics LTL, CTL, and CTL*. To this end, we consider HS endowed with three semantic variants: the state-based semantics, introduced by Montanari et al., which allows branching in the past and in the future, the linear-past semantics, allowing branching only in the future, and the linear semantics, disallowing branching. These variants are compared, as for their expressiveness, among themselves and to standard temporal logics, getting a complete picture. In particular, HS with linear (resp., linear-past) semantics is proved to be equivalent to LTL (resp., finitary CTL*)

Satisfiability and Model Checking for the Logic of Sub-Intervals under the Homogeneity Assumption

The expressive power of interval temporal logics (ITLs) makes them really fascinating, and one of the most natural choices as specification and planning language. However, for a long time, due to their high computational complexity, they were considered not suitable for practical purposes. The recent discovery of several computationally well-behaved ITLs has finally changed the scenario. In this paper, we investigate the finite satisfiability and model checking problems for the ITL D featuring the sub-interval relation, under the homogeneity assumption (that constrains a proposition letter to hold over an interval if and only if it holds over all its points). First we prove that the satisfiability problem for D, over finite linear orders, is PSPACE-complete; then we show that its model checking problem, over finite Kripke structures, is PSPACE-complete as well. The paper enrich the set of tractable interval temporal logics with a meaningful representative.Comment: arXiv admin note: text overlap with arXiv:1901.0388

Interval vs. Point Temporal Logic Model Checking: an Expressiveness Comparison

In recent years, model checking with interval temporal logics is emerging as a viable alternative to model checking with standard point-based temporal logics, such as LTL, CTL, CTL*, and the like. The behavior of the system is modeled by means of (finite) Kripke structures, as usual. However, while temporal logics which are interpreted \u201cpoint-wise\u201d describe how the system evolves state-by-state, and predicate properties of system states, those which are interpreted \u201cinterval-wise\u201d express properties of computation stretches, spanning a sequence of states. A proposition letter is assumed to hold over a computation stretch (interval) if and only if it holds over each component state (homogeneity assumption). A natural question arises: is there any advantage in replacing points by intervals as the primary temporal entities, or is it just a matter of taste? In this article, we study the expressiveness of Halpern and Shoham\u2019s interval temporal logic (HS) in model checking, in comparison with those of LTL, CTL, and CTL*. To this end, we consider three semantic variants of HS: the state-based one, introduced by Montanari et al. in [30, 34], that allows time to branch both in the past and in the future, the computation-tree-based one, that allows time to branch in the future only, and the trace-based variant, that disallows time to branch. These variants are compared among themselves and to the aforementioned standard logics, getting a complete picture. In particular, we show that HS with trace-based semantics is equivalent to LTL (but at least exponentially more succinct), HS with computation-tree-based semantics is equivalent to finitary CTL*, and HS with state-based semantics is incomparable with all of them (LTL, CTL, and CTL*)

Interval temporal logic model checking: The border between good and bad HS fragments

The model checking problem has thoroughly been explored in the context of standard point-based temporal logics, such as LTL, CTL, and CTL 17, whereas model checking for interval temporal logics has been brought to the attention only very recently. In this paper, we prove that the model checking problem for the logic of Allen\u2019s relations started-by and finished-by is highly intractable, as it can be proved to be EXPSPACE-hard. Such a lower bound immediately propagates to the full Halpern and Shoham\u2019s modal logic of time intervals (HS). In contrast, we show that other noteworthy HS fragments, namely, Propositional Neighbourhood Logic extended with modalities for the Allen relation starts (resp., finishes) and its inverse started-by (resp., finished-by), turn out to have\u2014maybe unexpectedly\u2014the same complexity as LTL (i.e., they are PSPACE-complete), thus joining the group of other already studied, well-behaved albeit less expressive, HS fragments

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

Interval temporal logic model checking based on track bisimilarity and prefix sampling

Since the late 80s, LTL and CTL model checking have been extensively applied in various areas of computer science and AI. Even thoughtheyprovedthemselvestobe quitesuccessfulin manyapplication domains,therearesomerelevanttemporalconditionswhichareinherently “interval based” (this is the case, for instance, with telic statements like “theastronautmustwalkhomeinanhour”andtemporalaggregationslike “the average speed of the rover cannot exceed the established threshold”) and thus cannot be properly modelled by point-based temporal logics. In general, to check interval properties of the behavior of a system, one needs to collect information about states into behavior stretches, which amounts to interpreting each finite sequence of states as an interval and to suitably defining its labelling on the basis of the labelling of the states that compose it. In orderto deal with these properties,a model checking framework based on Halpern and Shoham’s interval temporal logic (HS for short) and its fragments has been recently proposed and systematically investigated in the literature. In this paper, we give an original proof of EXPSPACE membership of the model checking problem for the HS fragment AAbarBBbarE (resp.,AAbarEBEbar)ofAllen’sintervalrelationsmeets,met-by,started-by (resp., finished-by),starts,andfinishes. The proofexploits track bisimilarity and prefix sampling, and it turns out to be much simpler than the previously known one. In addition, it improves some upper bounds

Interval Temporal Logic Model Checking Based on Track Bisimilarity and Prefix Sampling

Since the late 80s, LTL and CTL model checking have been extensively applied in various areas of computer science and AI. Even though they proved themselves to be quite successful in many application domains, there are some relevant temporal conditions which are inher- ently \interval based" (this is the case, for instance, with telic statements like \the astronaut must walk home in an hour" and temporal aggrega- tions like \the average speed of the rover cannot exceed the established threshold") and thus cannot be properly modelled by point-based tem- poral logics. In general, to check interval properties of the behavior of a system, one needs to collect information about states into behavior stretches, which amounts to interpreting each nite sequence of states as an interval and to suitably dening its labelling on the basis of the labelling of the states that compose it. In order to deal with these properties, a model checking framework based on Halpern and Shoham's interval temporal logic (HS for short) and its fragments has been recently proposed and systematically investigated in the literature. In this paper, we give an original proof of EXPSPACE membership of the model checking problem for the HS fragment AABBE (resp., AAEBE) of Allen's interval relations meets, met-by, started-by (resp., nished-by), starts, and nishes. The proof exploits track bisimi- larity and prex sampling, and it turns out to be much simpler than the previously known one. In addition, it improves some upper bounds