Decidable Relationships between Consistency Notions for Constraint Satisfaction Problems

Abstract

Abstract. We define an abstract pebble game that provides game interpretations for essentially all known consistency algorithms for constraint satisfaction problems including arc-consistency, (j, k)-consistency, k-consistency, k-minimality, and refinements of arc-consistency such as peek arc-consistency and singleton arc-consistency. Our main result is that for any two instances of the abstract pebble game where the first satisfies the additional condition of being stacked, there exists an algorithm to decide whether consistency with respect to the first implies consistency with respect to the second. In particular, there is a decidable criterion to tell whether singleton arc-consistency with respect to a given constraint language implies k-consistency with respect to the same constraint language, for any fixed k. We also offer a new decidable criterion to tell whether arc-consistency implies satisfiability which pulls from methods in Ramsey theory and looks more amenable to generalization.

One Hierarchy Spawns Another

“…If P ĺ G, then G " P and we are done. Otherwise G has bounded tree depth by Lemma 3.18 (3). By Proposition 3.12 there is d P N such that G has stack depth d.…”

“…Such algorithms are typically efficient and simple heuristics that can detect inconsistency (that is, that an instance is a no instance) and are based on local reasoning. Identifying cases of these problems where such algorithms provide a sound and complete decision procedure has been a central theme in the tractability theory of these problems (see for example [31,16,4,9,7,3,10]).…”

Arc consistency and friends

“…Atserias and Weyer [5] gave a uniform treatment of AC, PAC, SAC, and general consistency. Among other results, they show that it can be decided, given a constraint language and any pair of the previous consistency methods, whether it is true that the set of instances that passes one of the consistency tests coincides with the set of instances that passes the other.…”

“…Checking for consistency is a primary reasoning technique for the practical solution of the CSP, and has been studied theoretically from many viewpoints [22,2,4,1,3,6,5]. The most basic and simplest form of consistency is arc consistency, which algorithmically involves performing inferences concerning the set of feasible values for each variable.…”

One hierarchy spawns another

“…Such algorithms are typically efficient and simple heuristics that can detect inconsistency (that is, that an instance is a no instance) and are based on local reasoning. Identifying cases of these problems where such algorithms provide a sound and complete decision procedure has been a central theme in the tractability theory of these problems (see for example [3,4,7,9,13,26]). …”