Abstract: This paper revises and expands upon a paper presented by two of the present authors at AAAI 1986 [Vilain & Kautz 1986]. As with the original, this revised document considers computational aspects of intervalbased and point-based temporal representations. Computing the consequences of temporal assertions is shown to be computationally intractable in the interval-based representation, but not in the point-based one. However, a fragment of the interval language can be expressed using the point language and benefits from the tractability of the latter. The present paper departs from the original primarily in correcting claims made there about the point algebra, and in presenting some closely related results of van Beek [1989]. The representation of time has been a recurring concern of Artificial Intelligence researchers. Many representation schemes have been proposed for temporal reasoning; of these, one of the most attractive is James Allen's algebra of temporal intervals [Allen 1983]. This representation scheme is particularly appealing for its simplicity and for its ease of implementation with constraint propagation algorithms. Reasoners based on

The Trains project is an effort to build a conversationally proficient planning assistant. A key part of the project is the construction of the Trains system, which provides the research platform for a wide range of issues in natural language understanding, mixedinitiative planning systems, and representing and reasoning about time, actions and events. Four years have now passed since the beginning of the project. Each year we have produced a demonstration system that focused on a dialog that illustrates particular aspects of our research. The commitment to building complete integrated systems is a significant overhead on the research, but we feel it is essential to guarantee that the results constitute real progress in the field. This paper describes the goals of the project, and our experience with the effort so far. This paper is to appear in the Journal of Experimental and Theoretical AI, 1995. The TRAINS project has been funded in part by ONR/ARPA grant N00014-92-J-1512, U.S. Air ...

issues remain essentially the same. One of the most crucial problems in any computer system that involves representing the world is the representation of time. This includes applications such as databases, simulation, expert systems and applications of Artificial Intelligence in general. In this brief paper, I will give a survey of the basic techniques available for representing time, and then talk about temporal reasoning in a general setting as needed in AI applications. Quite different representations of time are usable depending on the assumptions that can be made about the temporal information to be represented. The most crucial issue is the degree of certainty one can assume. Can one assume that a time stamp can be assigned to each event, or barring that, that the events are fully ordered? Or can we only assume that a partial ordering of events is known? Can events be simultaneous? Can they overlap in time and yet not be simultaneous? If they are not instantaneous, do we know the durations of events? Different answers to each of these questions allow very different representations of time. I. Representations Based on Dating Schemes A good representation of time for instantaneous events, if it is possible, is using an absolute dating system. This involves time stamping each event with an absolute real-time, say taken off the system clock

The notion of time is ubiquitous in any activity that requires intelligence. In particular, several important notions like change, causality, action are described in terms of time. Therefore, the representation of time and reasoning about time is of crucial importance for many Artificial Intelligence systems. Specifically during the last 10 years, it has been attracting the attention of many AI researchers. In this survey, the results of this work are analysed. Firstly, Temporal Reasoning is defined. Then, the most important representational issues which determine a Temporal Reasoning approach are introduced: the logical form on which the approach is based, the ontology (the units taken as primitives, the temporal relations, the algorithms that have been developed,. . . ) and the concepts related with reasoning about action (the representation of change, causality, action,. . . ). For each issue the different choices in the literature are discussed. 1 Introduction The notion of time i...

We describe an e#ective generic method for solving constraint problems, based on Tarski's relation algebra, using path-consistency as a pruning technique. Weinvestigate the performance of this method on interval constraint problems. Time performance is a#ected strongly by the path-consistency calculations, whichinvolve the calculation of compositions of relations. Weinvestigate various methods of tuning composition calculations, and also path-consistency computations. Space performance is a#ected by the branching factor during search. Reducing this branching factor depends on the existence of `nice' subclasses of the constraint domain. Finally,we survey the statistics of consistency properties of interval constraint problems. Problems of up to 500 variables may be solved in expected cubic time. Evidence is presented that the `phase transition' occurs in the range 6 # n:c # 15, where n is the numberofvariables, and c is the ratio of non-trivial constraints to possible constra...

Time is an important aspect of information in medical domains. In this paper, we adopt Allen's influential interval algebra framework for representing temporal information. The interval algebra allows the representation of indefinite and incomplete information which is necessary in many applications. However, answering interesting queries in this framework has been shown to be almost assuredly intractable. We show that when the representation language is sufficiently restricted we can develop efficient algorithms for answering interesting classes of queries including: (i) determining whether a formula involving temporal relations between events is possibly true and necessarily true; and (ii) answering aggregation questions where the set of all events that satisfy a formula are retrieved. We also show, by examining applications of the interval algebra discussed in the literature, that our restriction on the representation language often is not overly restrictive in practice. 1 Introduct...

Reasoning about time often involves incomplete information about periods and their relationships. Varieties of incompleteness include uncertainty about the number of objects involved, the distribution of a set of temporal relations among these objects, and what can be called the participation of a set of objects in a temporal relation. A solution to the problem of representing and reasoning about incomplete temporal information of these kinds is forthcoming if a restricted class of non-convex intervals (called n-tntervals) is added to the temporal domain of discourse. An n-interval corresponds to the common sense notion of a recurring period of time with a (possibly) unspecified number of occurrences. In this paper, we formalize a representation for temporal reasoning problems using n-intervals. The language of the framework is restricted in such a way that tractable techniques from constraint satisfaction can be applied. Specifically, it is demonstrated how the problem of determining path-consistency in a network of binary n-interval relations can be solved. 1

Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tied-in with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The model-theoretic notion of interpretation is used to generalise known complexity results to a range of other algebraic logics. In particular a number of relation algebras are shown to have intractable network satisfaction problems. 1 Introduction A basic problem in theoretical computing and applied logic is to select and evaluate the ideal formalism to represent and reason about a given application. Many different formalisms are adopted: classical first-order logic, modal and temporal logics (either...

Temporal reasoning is widely used in AI, especially for natural language processing. Existing methods for temporal reasoning are extremely expensive in time and space, because complete graphs are used. We present an approach of temporal reasoning for expert systems in technical applications that reduces the amount of time and space by using sequence graphs. A sequence graph consists of one or more sequence chains and other intervals that are connected only loosely with these chains. Sequence chains are based on the observation that in technical applications many events occur sequentially. The uninterrupted execution of technical processes for a long time is characteristic for technical applications. To relate the first intervals in the application with the last ones makes no sense. In sequence graphs only these relations are stored that are needed for further propagation. In contrast to other algorithms which use incomplete graphs, no information is lost and the reduction of complexity is significant. Additionally, the representation is more transparent, because the "flow" of time is modelled.

interpretation of four 89-relations between I and J) 89 d ! = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total function R I \ThetaJ is one-one.g 89! = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total function.g 89 d = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total relation [8i; i 0 : I; 8j : JR I \ThetaJ (i; j) R I \ThetaJ (i 0 ; j) ! i = i 0 ]g 89 = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total relationg 14 IJCAI Spatial and Temporal Workshop For recurrence relations based on simple correlation, R I \ThetaJ is instantiated to COR I \ThetaJ , and R 2 to a relation in CR. Thus, the quantifier construction creates a class of relations which places restrictions on the temporal order of the pairs of correlated intervals. What distinguishes the 89 class from other operators is the condition that R I \ThetaJ ` R 2 . To summarize: the construction of recurrence relations using quantification, and the rel...