Galois correspondence for counting quantifiers

TL;DR

This paper introduces new closure operators for relations in the context of counting CSPs, explores their impact on complexity classification, and characterizes related clones and co-clones, linking algebraic and logical perspectives.

Contribution

It defines max-implementation and max-quantification, analyzes their effects on counting CSP complexity, and establishes a Galois correspondence for k-existential quantification.

Findings

Approximation-preserving reductions are maintained under new closure operators.

Partial clones closed under k-existential quantification are characterized by k-subset surjectivity.

Boolean max-co-clones are described in terms of max-implementations.

Abstract

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.