The CSP Dichotomy, the Axiom of Choice, and Cyclic Polymorphisms

TL;DR

This paper explores the complexity of infinite CSPs, linking their computational difficulty to set-theoretic principles like the Axiom of Choice, and characterizes the role of cyclic polymorphisms in this context.

Contribution

It establishes a connection between the algebraic properties of structures and the strength of compactness principles over ZF, revealing how cyclic polymorphisms influence the logical strength of CSP solutions.

Findings

CSP dichotomy extends to infinite structures with set-theoretic implications.

Structures without cyclic polymorphisms relate to the Boolean Prime Ideal Theorem.

Structures with cyclic polymorphisms correspond to weaker set-theoretic principles.

Abstract

We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems -- established already in the finite case -- presents itself as the strength of the corresponding De Bruijin-Erd\H{o}s-type compactness theorem over ZF. More precisely, if $\mathcal{D}$ is a structure, let $K_\mathcal{D}$ stand for the following statement: for every structure $\mathcal{X}$ if every finite substructure of $\mathcal{X}$ admits a solution to $\mathcal{D}$, then so does $\mathcal{X}$. We prove that if $\mathcal{D}$ admits no cyclic polymorphism, and thus it is NP-complete by the CSP Dichotomy Theorem, then $K_\mathcal{D}$ is equivalent to the Boolean Prime Ideal Theorem (BPI) over ZF. Conversely, we also show that if $\mathcal{D}$ admits a cyclic polymorphism, and thus it is in P, then $K_\mathcal{D}$ is strictly weaker than BPI.