Compact Quantitative Theories of Convex Algebras

TL;DR

This paper introduces compact quantitative equational theories, proves the compactness of a specific convex algebra theory, and uses it to derive other theories related to distances on probability distributions.

Contribution

It defines compact quantitative equational theories and demonstrates the compactness of a key convex algebra theory, enabling new axiomatizations of distribution distances.

Findings

The theory of interpolative barycentric algebras is compact.

Compact theories can axiomatize distances on probability distributions.

Framework facilitates deriving new convex algebra theories.

Abstract

We introduce the concept of compact quantitative equational theory. A quantitative equational theory is defined to be compact if all its consequences are derivable by means of finite proofs. We prove that the theory of interpolative barycentric (also known as convex) quantitative algebras of Mardare et. al. is compact. This serves as a paradigmatic example, used to obtain other compact quantitative equational theories of convex algebras, each axiomatizing some distance on finitely supported probability distributions.