Synthesizing partial orders given comparability information: Partitive sets and slack in critical path networks

Abstract

A problem which arises in the analysis of reaction times is considered. Suppose a task requires the completion of a set of mental activities which can be represented as a partially ordered set of arcs in a critical path network, but the network is unknown. If one knows for every pair of activities whether the pair is comparable (i.e., sequential) or incomparable (i.e., concurrent), then a partial order for the activities can be constructed with a procedure known as the transitive orientation algorithm. It is known that only two partial orders are possible, one the converse of the other, unless some proper nonsingleton partitive subset of the activities is not stable. Two main results are presented: First, a set of arcs generating a weakly connected subnetwork is partitive if and only if the subnetwork has a unique source, a unique sink, and no vertex of attachment other than these. Second, a simple relationship between partitive sets of activities and the slack between comparable activities is presented. These issues are important for the uniqueness problem in the synthesis of critical path networks representing cognitive tasks.