<title>Atomic temporal interval relations in branching time: calculation and application</title>

Abstract

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.

Solving Infinite-Domain CSPs Using the Patchwork Property

“…There are interesting examples of structures A that do not have PP. An eminent example is the Branching Time Algebra (BTA) [4] which has been used, for example, in planning [30], as the basis for temporal logics [35], and as the basis for a generalisation of Allen's Interval Algebra [65]. We note that, in particular, the complexity of the branching variant of Allen's Interval Algebra has recently gained attraction [10,39].…”

“…The relation to BD is more complex, since there are problems that are covered by BD but not by our algorithm. In some cases our algorithm is not applicable to a CSP directly, but one can find an equivalent problem that has the patchwork property via homogenisation -one example is the Branching Time Algebra [4] -the details are discussed later on. However, there are cases where homogenisation is not applicable, so the exact dividing line is unfortunately unclear.…”

Lattice structure of temporal interval relations

“…Density is chosen mainly to avoid the situation illustrated by the O relation above which gives rise to an infinite number of different relations O n. The structure of such infinite relation algebras is under study. The development of particular exam- [3]. In the case of relativistic space-time, Rodriguez [25,27] encountered 82 atomic interval relations.…”

“…Additional lines of research have extended the basic linear-time model using intervals to 3-D Space [ 13,21 ], branching time [3], and relativistic space-time [24,4,5,25] utilizing relation algebra [17] and an axiomatic framework developed by Lamport to study concurrent systems [18,6]. To implement constraint propagation, however, one must pre-calculate (or be able to calculate efficiently on the fly) the compositions of the basic atomic or irreducible temporal relations: compositions of more general relations are then computed as the disjunction of the compositions of the atomic components.…”