Qualitative and quantitative time interval constraint networks

Constraint propagation algorithms are often used in Artificial Intelligence systems to perform inference about quantities [5]. On the other hand, many temporal reasoning systems, e.g. Allen's interval algebra [1] and Vilain and Kautz's point algebra [20], deal only with qualitative relations between time intervals or between time points. In this paper we propose a general framework for representing and reasoning about both qualitative and quantitative temporal constraints over time dependent events. Information about (possibly disjunctions of) qualitative relations between time intervals, e.g. during, meet, start, finish, . , as in Allen's interval algebra [1], or relations between time points, e.g. before, after, and equal, as in Vilain and Kautz's point algebra [20], can be expressed. At the same time, quantitative information, in particular, durations of events, references to absolute time, and metric information, can be represented and propagated in this unified framework. The framework called Time Interval Constraint Satisfaction Problem (TICSP) is proposed. TICSP consists of a set of time intervals, and a set of four-ary constraints expressing relationships between any two time intervals (four time points). These constraints are labeled by a set of vectors of open/closed metric bounds. The problem is to make the tightiest possible assertions about the relationships among all time intervals. A strong 3consistency algorithm is derived for the propagation of these temporal constraints. *Supported by the Fulbright scholarship. lSupported in part by t h e US Army research office grant DAAL03-90-G-0217. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1991 ACM 089791-382-5/91/0003/0239 $1.50 239 1 Introduct ion Representing and reasoning about temporal knowledge is essential for applications in various areas of computer science. Many recent research investigations, especially in artificial intellligence, deal more and more about temporal aspects of problems, e.g. in natural language processing [2], planning or scheduling [4, 17], common-sense reasoning [10, 18], and qualitative reasoning [21]. Research work involving temporal knowledge representation and temporal reasoning can be divided into two categories: qualitative and quantitative systems. Qualitative systems put the emphasis on the description of qualitative relations between time intervals or between time points, as in Allen's interval algebra [1], and Vilain and Kautz's point algebra [20]. For example, we may represent some instances like interval A occurred during interval B, or time point C occurred before or after time point D. These systems deal only with qualitative symbols, so numerical data, e.g. durations of events, references to absolute time, and metric information, are not representable, or they are handled in separated modules. For example, the duration reasoner reported by Allen [1, 3] represents and manipulates the information about relative durations and about topological relations between time intervals in different networks. Interfacing between two different networks is needed, making it inefficient (some of the difficulties were discussed by Ladkin [12]). Even in his duration reasoner, Allen did not consider how to represent quantitative information which is essential and often given in many applications, e.g. durations of tasks in scheduling problems. On the other hand, quantitative systems put the emphasis on quantitative relations between time points or between time intervals. The beginning and the ending time points can represent an interval of occurrence, and they are associated with absolute time values. Dean and McDermott [6] and Davis [5] proposed constraint networks of time points where quantities are labeled with distance intervals, however, they did not allow the repre. ,~entation nor the propagation of disconnected metric bounds. Davis used the convex hull of the disconnected sets of intervals; which is the smallest single interval containing all the disconnected intervals. His simplified system of using a single interval contains less restricted censtraints which may include some values that are inccnsistent with the initial constraints. Dechter et al. [16] proposed constraint networks, in which time points are represented as variables, and the temporal information is given by a set of unary and binary constraints ort these time points. A disjunction of metric bounds are representable, however, they did not mention about the representation of open metric bounds. Furthermore, they did not consider some of the problems raised by Allen [1] and Vilain and Kautz [20], in which qualitative relations between events are represented and reasoned. Rit [17] proposed a general framework for propagating temporal constraints over events, in which Allen's symbolic relations as well as durations can be represented in the same framework. He manipulated these constraints ir terms of sets of possible occurrences (SOPOs), which are expressed by a 2-D graphical representation of occurrences. The picture representation and manipulation a::e the main complicated procedures, since the shape o::~ the SOPOs might be very complicated (for example with a lot of holes). A constraint propagation algorithm defined by Rit [17] achieves only arc consistency. In this paper we extend the representation proposed by Dechter et al. [16] in such a way that we can represent and manipulate both qualitative and quantitative information among time points, as well as, among time intervals in a unified framework. The expressibility of the proposed language subsumes all the languages ment ~oned above, both qualitative and quantitative systems. We envision a temporal reasoning system (similar to [i[6]) that can represent different kinds of temporal information, and has a capability of reasoning or discoverillg new information. Therefore, this temporal reasoni:ag system basically consists of a temporal knowledge base, a routine to check its consistency, a query answertag mechanism, and an inference mechanism. The basic entities in our temporal knowledge base are events to ~hieh we assign temporal intervals, e.g. "I have a meeting" or "the switch is on". Each interval represents the time during which the corresponding event occurs. 'The temporal information might be relative, e.g. "El occurred after E2", or metric, e.g. "El had finished at least 4 minutes before E2 starts". Disjunctive sentences may be needed to express less specific or indefinite intormation, e.g. "E~ occurred during or before E2". Besides, we also allow references to absolute time, such as :[1:30 a.m., and durations of events, e.g. "El lasted at most three hours". Given these different kinds of temporal information, we want to answer queries such as: Is it possible that event Ex occurs at time t? What is the temporal relations between events E1 and E2? In this paper, we will address only the issues about the representation and the reasoning mechanism. This paper introduces a framework based on constraint satisfaction formalism for representing and reasoning about time dependents events. Temporal constraint networks are used to express relations among time points which are labeled by open/closed metric bounds. Four-ary constraints are defined to represent relations between time intervals. Section 2 proposes a formalization of time interval constraint satisfaction problems (TICSPs). Time interval constraint networks are presented in Section 3. Section 4 defines binary operations on temporal constraints, then a strong 3consistency algorithm is derived in Section 5. Section 6 provides concluding remarks. 2 A Formalizat ion of Temporal Constraint N e t w o r k s The basic entities in our temporal knowledge base are events. Each event is associated with a time interval, it occurs over. Each time interval is associated with its corresponding beginning and ending time points. Being able to specify durations of intervals in our model, we may have an interval with zero duration, this is corresponding to a time instant, i.e. the beginning and ending of this interval is at the same time point. This aspect of our time model is opposed to Allen's determination [1, 2] which based his theory only on non-zero duration intervals rather on time instants. However, many recent research investigations, e.g. Shoham [18], and Galton [9], have argued in the flavor of the necessity in representing both intervals and points on an equal footing, especially for reasoning correctly about continuous change. In this article, we will show its advantages, particularly in its expressibility. The definitions of our temporal constraint satisfaction model are based on similar definitions for the general Constraint Satisfaction Problem (CSP) proposed by Montanari [15]. Given a set of time intervals, and knowledge about temporal constraints on these intervals, the problem is to make explicit the tightiest possible assertions about the relationships among all the time intervals. We now make this somewhat more formal. A Time Interval Constraint Satisfaction Problem (TICSP) involves a set of time intervals, X1, . ,X,~. Each interval, Xi, is associated with its beginning time point, X_. L, and its ending time point, Xi, which are treated as variables, whose domains are the real numbers, ~. Allen [! ] defined 13 primitive relations between any pair of intervals, e.g. during, meet, and start. Dis-