Reasoning about time

Reasoning about time is essential for applications in artificial intelligence and in many other disciplines. Given certain explicit relationships between a set of events, we would like to have the ability to infer additional relationships which are implicit in those given. For example, the transitivity of “before” and “contains” may allow us to infer information regarding the sequence of events. Such inferences are essential in story understanding, planning and causal reasoning. There are a great number of practical problems in which one is interested in constructing a time line where each particular event or phenomenon corresponds to an interval representing its duration. These include seriation in archeology, behavioral psychology, temporal reasoning, scheduling, and combinatorics. Other applications arise in non-temporal context, for example, in molecular biology, arrangement of DNA segments along a linear DNA chain involves similar problems. Interval consistency problems deal with events, each of which is assumed to be an interval on the real line or on any other linearly ordered set, and reasoning about such intervals when the precise topological relationships between them is unknown or only partially specified. In Golumbic and Shamir (1991), we relate the two notions of interval algebra from the temporal reasoning community and interval graphs from the combinatorics community, obtaining new algorithmic complexity results of interest to both disciplines. Several versions of the satisfiability, minimum labeling and all consistent solutions problems for temporal (interval) data are investigated. The satisfiability question is shown to be NP-Complete even when restricting the possible interval relationships to subsets of the relations intersection and precedence only. On the other hand, we give efficient algorithm for several other restrictions of the problem.