Path Consistency in a Network of Non-Convex Intervals

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 Introduction Common sense knowledge about the occurrence of events is often incomplete. The kinds of incompleteness of interest here is expressed in the following sentences: 1. John sometimes drives to Orlando. 2. John's ten trips to Orlando this month twice preceded his phoning Phyllis. 3. Some of John's ten trips to Orlando preceded her phoning Phyllis. The first example exhibits number incompleteness: it is not clear how many driving events there are, hence how many periods of time are involved, but that at least one occurs. The second example illustrates distribution uncertainty ; we don't know which phoning intervals overlapped with which meetings. The third example exhibits , in addition to distribution uncertainty, what can be called participation uncertainty; it is not known how often the relation between the trips and the phonings occurred. Uncertainty of this kind is removed by adding quantitative information. Compare (l)-(3) with "John's ten trips to Orlando this month twice preceded his phoning Phyllis: the first time and the third time". Adding the first quantity eliminates number uncertainty; the second quantity ("twice") eliminates participation uncertainty; the last two quantities together eliminate distribution uncertainty. Temporal knowledge incompleteness makes it difficult to build a reasoner that adequately deals with time. In this paper, the concern is to formalize a representation which addresses the problem of reasoning with number, distribution, and participation uncertainty. To accomplish this aim the remaining sections 1. Define a class of binary temporal relations …