Arc-Consistency for Continuous Variables

Davis 1] has investigated the properties of the Waltz propagation algorithm with interval labels in continuous domains. He shows that in most cases, the algorithm does not achieve arc consistency, and furthermore is subject to innnite iterations. In this paper, I show that the main reason for Davis' negative results lies in the way he formulates the propagation rule for the Waltz algorithm. For binary constraints, I propose a diierent propagation rule and show that it guarantees arc consistency upon quiescence of the propagation. Generalizations to n-ary constraints are possible but involve more complex geometry. Arc consistency guarantees a minimal network only when the constraint graph is a tree. I show that the new formulation of the propagation algorithm rules out the possibility of innnite iterations for all tree-structured constraint networks, and thus obtain a general and reliable algorithm for arc consistency in continous domains.