Quantaloidal Approach to Constraint Satisfaction

TL;DR

This paper introduces a novel categorical framework using quantaloids to generalize the constraint satisfaction problem (CSP), enabling a broader understanding of its structure and complexity.

Contribution

It formulates CSP concepts within the 2-category PFinSet as a quantaloid, allowing generalizations and complexity classification via polymorphisms in this abstract setting.

Findings

CSP concepts can be modeled in the quantaloid PFinSet.

Generalizations of CSP are possible within this categorical framework.

Complexity of optimization problems can be classified using polymorphisms.

Abstract

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category PFinSet of finite sets and sets of functions between them. The 2-category PFinSet is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.