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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

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

This document contains definitions of a wide range of concepts specific to and widely used within temporal databases. In addition to providing definitions, the document also includes separate explanations of many of the defined concepts. Two sets of criteria are included. First, all included concepts were required to satisfy four relevance criteria, and, second, the naming of the concepts was resolved using a set of evaluation criteria. The concepts are grouped into three categories: concepts of general database interest, of temporal database interest, and of specialized interest. This document is a digest of a full version of the glossary 1 . In addition to the material included here, the full version includes substantial discussions of the naming of the concepts.The consensus effort that lead to this glossary was initiated in Early 1992. Earlier status documents appeared in March 1993 and December 1992 and included terms proposed after an initial glossary appeared in SIGMOD Record in September 1992. The present glossary subsumes all the previous documents. It was most recently discussed at the "ARPA/NSF International Workshop on an Infrastructure for Temporal Databases," in Arlington, TX, June 1993, and is recommended by a significant part of the temporal database community. The glossary meets a need for creating a higher degree of consensus on the definition and naming of temporal database concepts.

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This article deals with the problem of providing Kowalski and Sergot's event calculus. extended with context dependency, with an efficient implementation in a logic programming framework. Despite a widespread recognition that a positive solution to efficiency issues is necessary to guarantee the computational feasibility of existing approaches to temporal reasoning, the problem of analyzing the complexity of temporal reasoning programs has been largely overlooked. This article provides a mathematical analysis of the efficiency of query and update processing in the event calculus and defines a cached version of the calculus that (i) moves computational complexity from query to update processing and (ii) features an absolute improvement of performance. because query processing in the event calculus costs much more than update processing in the proposed cached version.

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Abstract. In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations "meets", "met by", "begun by", and "begins" of Allen's interval algebra (AĀBB logic). AĀBB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by/begins" relations and propositional neighborhood logic AĀ, in its many variants (including metric ones). We prove that the satisfiability problem for AĀBB, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AĀBB turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AĀBB is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, and R.