Functional Clones and Expressibility of Partition Functions

TL;DR

This paper explores the structure and properties of functional clones, which are sets of non-negative functions relevant to counting problems, revealing their lattice organization and relationships.

Contribution

It identifies a specific sublattice of functional clones and analyzes their relationships and properties within the context of counting CSPs.

Findings

Identified a sublattice of functional clones.

Analyzed relationships among clones.

Connected clones to counting CSPs.

Abstract

We study functional clones, which are sets of non-negative pseudo-Boolean functions (functions $\{0,1\}^k\to\mathbb{R}_{\geq 0}$) closed under (essentially) multiplication, summation and limits. Functional clones naturally form a lattice under set inclusion and are closely related to counting Constraint Satisfaction Problems (CSPs). We identify a sublattice of interesting functional clones and investigate the relationships and properties of the functional clones in this sublattice.