The Role of Commutativity in Constraint Propagation Algorithms

TL;DR

This paper introduces a unified framework for understanding various constraint propagation algorithms, emphasizing the role of commutativity, and demonstrates that several well-known algorithms are instances of a single generic algorithm.

Contribution

It presents a general iteration framework for constraint propagation algorithms and shows how key algorithms are specific instances within this framework, highlighting the importance of commutativity.

Findings

Unified explanation of multiple algorithms

Demonstrates equivalence of algorithms via commutativity

Simplifies understanding of constraint propagation methods

Abstract

Constraint 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 {\tt AC-3}, {\tt PC-2}, {\tt DAC} and {\tt 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 \citeyear{Apt99b}.