Logical compactness and constraint satisfaction problems

TL;DR

This paper explores the relationship between the complexity of constraint satisfaction problems and logical compactness hypotheses, revealing that more complex problems align with stronger set-theoretic assumptions.

Contribution

It establishes a hierarchy linking the complexity of CSPs to the strength of associated logical compactness hypotheses, connecting computational complexity with foundational set theory.

Findings

NP-complete CSPs correspond to ultrafilter axiom-based compactness hypotheses

Simpler CSPs relate to compactness hypotheses provable from ZF axioms

A hierarchy of logical strength mirrors the complexity hierarchy of CSPs

Abstract

We investigate a correspondence between the complexity hierarchy of constraint satisfaction problems and a hierarchy of logical compactness hypotheses for finite relational structures. It seems that the harder a constraint satisfaction problem is, the stronger the corresponding compactness hypothesis is. At the top level, the NP-complete constraint satisfaction problems correspond to compactness hypotheses that are equivalent to the ultrafilter axiom in all the cases we have investigated. At the bottom level, the simplest constraint satisfaction problems correspond to compactness hypotheses that are readily provable from the axioms of Zermelo and Fraenkel.