Generalized Intervals: A Guided Tour

Generalized intervals are nite ordered sequences of time points (Allen's calculus is the special case of ordered pairs). In this paper, we show why generalized intervals are good candidates for reasoning about complex events (with more than two crucial time points): Binary relations between them can be easily encoded; the conversion and composition operations on disjunc-tive relations provide them with a structure of a relation algebra; although the whole calculus is not tractable in general, there exists a subclass of disjunc-tive relations, which is easily characterized in geometric terms, which is tractable (in Allen's cases, this subclass coincides with the ORD-Horn class); for binary temporal networks on this subclass, consistency is decid-able in cubic time by testing path-consistency; moreover , a scenario can be computed in cubic time without backtrack (in quadratic time for consistent networks). Finally, the strong theory of n-intervals (generalized intervals with n time points) has a unique countable model (up to isomorphism), which implies its decid-ability. In a word, most of the pleasant properties of Allen's calculus hold in this generalized framework.