In 1983, Allen presented an ingenious method for the representation and maintenance of temporal information in the presence of imprecise, uncertain, and relative knowledge about time of occurrence. He introduced 13 relations between his primitive "temporal intervals," providing for the expression of "any relationship which can hold between two intervals." The model, however, did not address the problem of temporally incomparable events, such as events occurring in a distributed system without a common clock. Lamport's interprocessor communication model furnishes an axiomatic system for describing such events and their possible relationships. This article demonstrates that Allen's temporal model can be subsumed in a more general model based on Lamport's axiomatics. It is further suggested that this extended model can provide the underpinnings of a temporal knowledge base containing time-dependent information measured by unsynchronized clocks or in relativistic space-time. In this model, the number of relations between intervals increases dramatically from Allen's 13 or Lamport's 2 or 3 to over 80. Within this context, a modification of Allen's algorithm for the maintenance of a temporal reasoning system is presented, thus permitting the advantages of such a system to extend to reasoning about a wider range of phenomena.

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Due to increasing interest in representation of temporal knowledge, automation of temporal reasoning, and analysis of distributed systems, literally dozens of temporal models have been proposed and explored during the last decade. Interval-based temporal models are especially appealing when reasoning about events with temporal extent but pose special problems when deducing possible relationships among events. The paper delves deeply into the structure of the set of atomic relations in a class of temporal interval models assumed to satisfy density and homogeneity properties. An order structure is imposed on the atomic relations of a given model allowing the characterization of the compositions of atomic relations (or even lattice intervals) as lattice intervals. By allowing the utilization of lattice intervals rather than individual relations, this apparently abstract result explicitly leads to a concrete approach which speeds up constraint propagation algorithms.

A practical method of reasoning about intervals in a branching-time model which is dense, unbounded, future-branching, without rejoining branches is presented. The discussion is based on heuristic constraint-propagation techniques using the relation algebra of binary temporal relations among the intervals over the branching-time model. This technique has been applied with success to models of intervals over linear time by Allen and others, and is of cubic-time complexity. To extend it to branching-time models, it is necessary to calculate compositions of the relations; thus, the table of compositions for the "atomic" relations is computed, enabling the rapid determination of the composition of arbitrary relations, expressed as disjunctions or unions of the atomic relations.