The Network Satisfaction Problem for Relation Algebras with at most 4 Atoms

TL;DR

This paper extends the classification of the network satisfaction problem for relation algebras with up to 4 atoms, showing it is always either efficiently solvable or NP-hard, by constructing specific algebra representations.

Contribution

It generalizes previous results to include non-representable algebras with 4 atoms, establishing a dichotomy for the NSP complexity.

Findings

NSP for these algebras is in P or NP-hard

Constructs universal and normal representations for these algebras

Extends classification to non-representable algebras with 4 atoms

Abstract

Andr\'eka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. There are relation algebras with 4 atoms that are not representable, and there are many results in the literature about representations and non-representability of relation algebras with at most 4 atoms. We extend the result of Hirsch and Cristiani to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.