Constraint Propagation Algorithms

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 this algebra have been put to use in several ways. For example, the planning system of Allen and Koomen [1983] relies heavily on the temporal algebra to perform reasoning about the ordering of actions. Elegant approaches such as this one may be compromised, however, by computational characteristics of the interval algebra. This paper concerns itself with the computational aspects of Allen's algebra, and of two variants of a simpler algebra of time points. Our perspective here is primarily computation-theoretic. We approach the problem of temporal representation by asking questions of complexity and tractability. In this light, this paper establishes some formal results about these temporal algebras. In brief these results are: • Determining consistency of statements in the interval algebra is NP-hard, as is determining the deductive closure of these statements. Allen's polynomial-time constraint propagation algorithm for deductive closure is thus incomplete. • We define a restricted form of the interval algebra, concerned with measuring the relative durations of events. This algebra can be formulated in terms of a time point algebra without disequality (≠). Allen's propagation algorithm is sound and complete for this fragment, and operates in O(n3) time and O(n2) space. • We also define a broader interval algebra fragment, corresponding to the time point algebra with ≠. A variant propagation algorithm performs closure in this fragment in O(n4) time. Throughout the paper, we consider how these formal results affect practical Artificial Intelligence programs. The Interval Algebra Allen's interval algebra has been described in detail in [Allen 1983]. In brief, the elements of the algebra are relations that may exist between intervals of time. Because the algebra allows for indefiniteness in temporal relations, it admits many possible relations between intervals (213 in fact). But all of these relations can be expressed as vectors of definite simple relations, of which there are only thirteen. The thirteen simple A B A BEFORE B B AFTER A A B A MEETS B B MET-BY A A B A OVERLAPS B B OVERLAPPED-BY A A B A STARTS B B STARTED-BY A A B A DURING B B CONTAINS A A B A ENDS B B ENDED-BY A