Complete and Terminating Tableau for the Logic of Proper Subinterval Structures over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work

Decidability of the interval temporal logic ABBar over the natural numbers

In this paper, we focus our attention on the interval temporal logic of the Allen's relations "meets", "begins", and "begun by" (ABBar for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties,such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for ABBar over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACE-completeness of the proble

Undecidability of the Logic of Overlap Relation over Discrete Linear Orderings

5The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham’s interval modal logic HS. Still, decidability is the rule for the fragments of HS with only one modal operator, based on an Allen’s relation. In this paper, we show that the logic O of the Overlap relation, when interpreted over discrete linear orderings, is an exception. The proof is based on a reduction from the undecidable octant tiling problem. This is one of the sharpest undecidability result for fragments of HS.openopenBRESOLIN Davide; DELLA MONICA Dario; GORANKO Valentin; MONTANARI Angelo; SCIAVICCO GuidoBresolin, Davide; DELLA MONICA, Dario; Goranko, Valentin; Montanari, Angelo; Sciavicco, Guid

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

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

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

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: 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

Verification of real-time systems: improving tool support

We address a number of limitations of Timed Automata and real-time model-checkers, which undermine the reliability of formal verification. In particular, we focus on the model-checker Uppaal as a representative of this technology. Timelocks and Zeno runs represent anomalous behaviours in a timed automaton, and may invalidate the verification of safety and liveness properties. Currently, model-checkers do not offer adequate support to prevent or detect such behaviours. In response, we develop new methods to guarantee timelock-freedom and absence of Zeno runs, which improve and complement the existent support. We implement these methods in a tool to check Uppaal specifications. The requirements language of model-checkers is not well suited to express sequence and iteration of events, or past computations. As a result, validation problems may arise during verification (i.e., the property that we verify may not accurately reflect the intended requirement). We study the logic PITL, a rich propositional subset of Interval Temporal Logic, where these requirements can be more intuitively expressed than in model-checkers. However, PITL has a decision procedure with a worst-case non-elementary complexity, which has hampered the development of efficient tool support. To address this problem, we propose (and implement) a translation from PITL to the second-order logic WS1S, for which an efficient decision procedure is provided by the tool MONA. Thanks to the many optimisations included in MONA, we obtain an efficient decision procedure for PITL, despite its non-elementary complexity. Data variables in model-checkers are restricted to bounded domains, in order to obtain fully automatic verification. However, this may be too restrictive for certain kinds of specifications (e.g., when we need to reason about unbounded buffers). In response, we develop the theory of Discrete Timed Automata as an alternative formalism for real-time systems. In Discrete Timed Automata, WS1S is used as the assertion language, which enables MONA to assist invariance proofs. Furthermore, the semantics of urgency and synchronisation adopted in Discrete Timed Automata guarantee, by construction, that specifications are free from a large class of timelocks. Thus, we argue that well-timed specifications are easier to obtain in Discrete Timed Automata than in Timed Automata and most other notations for real-time systems