The Role of Commutativity in Constraint Propagation Algorithms

Constraing propagation algorithms form an important part of most of the constraint programming systems. We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In this framework we proceed in two steps. First, we introduce a generic iteration algorithm on partial orderings and prove its correctness in an abstract setting. Then we instantiate this algorithm with specific partial orderings and functions to obtain specific constraint propagation algorithms. In particular, using the notions commutativity and semi-commutativity, we show that the AC-3, PC-2, DAC, and DPC algorithms for achieving (directional) arc consistency and (directional) path consistency are instances of a single generic algorithm. The work reported here extends and simplifies that of Apt [1999a].