Angelo Montanari

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.

Temporal representation and reasoning in artificial intelligence: Issues and approaches

Time is one of the most relevant topics in AI. It plays a major role in several areas, ranging from logical foundations to applications of knowledge‐based systems. In this paper, we survey a wide range of research in temporal representation and reasoning, without committing ourselves to the point of view of any specific application. The organization of the paper follows the commonly recognized division of the field in two main subfields: reasoning about actions and change, and reasoning about temporal constraints. We give an overview of the basic issues, approaches, and results in these two areas, and outline relevant recent developments. Furthermore, we briefly analyze the major emerging trends in temporal representation and reasoning as well as the relationships with other well‐established areas, such as temporal databases and logic programming.

Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions

Abstract In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

EFFICIENT TEMPORAL REASONING IN THE CACHED EVENT CALCULUS

This article deals with the problem of providing Kowalski and Sergots 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.

Decidable and Undecidable Fragments of Halpern and Shoham's Interval Temporal Logic: Towards a Complete Classification

Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allens relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allens relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.

An Optimal Decision Procedure for Right Propositional Neighborhood Logic

Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.

Dealing with different time granularities in formal specifications of real-time systems

The article presents a formalization of the notion of time granularity within a logical language for specifying real-time systems. It provides the specifier with the ability of dealing with different time granularities within a single specification. That is, it allows the specifier to describe the behavior and the properties of a system and its environment with respect to different time scales and to switch among them in a suitable way. The extended logical formalism is then embedded in an object oriented structure that enhances both the expressive power and the executability of the specification language. With regard to expressiveness, it enables one to subdivide a single specification of the system and its environment into different part and to provide a number of specifications of them at different levels of abstraction, each one referring to a different time granularity. With regard to executability, it allows one to verify the consistency and the adequacy of specifications at each step of their incremental development. It also suggests an enlargement of the notions of verification and validation that takes into account the stratified structure of the object oriented specifications.

Tableaux for Logics of Subinterval Structures over Dense Orderings

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.

The dark side of interval temporal logic: marking the undecidability border

Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undecidability. Identifying minimal undecidable interval logics is thus a natural and important issue in that research area. In this paper, we identify several new minimal undecidable logics amongst the fragments of Halpern and Shoham’s logic HS, including the logic of the overlaps relation, over the classes of all finite linear orders and all linear orders, as well as the logic of the meets and subinterval relations, over the classes of all and dense linear orders. Together with previous undecidability results, this work contributes to bringing the identification of the dark side of interval temporal logics very close to the definitive picture.

Embedding time granularity in a logical specification language for synchronous real-time systems

Abstract Formal methods have proved to be highly beneficial in the requirements specification phase of software production and are particularly valuable in the development of real-time applications (the most critical software systems). Unfortunately, most common specification languages are inadequate for real-time applications because they lack a quantitative representation of time. In this paper, we define a logical language to specify the temporal constraints of the wide-ranging class of real-time systems whose components have dynamic behaviours regulated by very different time constants. We motivate the need for allowing the consistent treatment of different time scales in formal specifications of these systems with the purpose of enhancing the naturalness and practical usability of the notation. The logical specification language is based on a revised version of the specification language TRIO. We first present the features of the basic logical language; then, we semantically and axiomatically define its granularity extension in a topological logic framework. Finally, we show some examples of its application.