Proportoids

TL;DR

This paper develops a mathematical foundation for analogical proportions, introducing proportoids and studying their properties, homomorphisms, and analogies to advance the theory of analogical reasoning in AI.

Contribution

It introduces the concept of proportoids, formalizes homomorphisms and analogies, and explores their properties to deepen the mathematical understanding of analogical proportions.

Findings

Defined proportoids with axiomatic relations

Characterized homomorphisms and their kernels as congruences

Developed methods to compute partial analogies

Abstract

Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning, which itself is at the core of artificial intelligence. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated -- in the tradition of the ancient Greeks -- by Yves Lepage two decades ago. More precisely, we first introduce the name ``proportoid'' for sets endowed with a 4-ary analogical proportion relation satisfying a suitable set of axioms. We then study study different kinds of proportion-preserving mappings and relations and their properties. Formally, we define homomorphisms of proportoids as mappings $\mathsf H$ satisfying $a:b::c:d$ iff $\mathsf Ha: \mathsf Hb:: \mathsf Hc: \mathsf Hd$ for all elements and show that their kernel is a congruence. Moreover, we introduce (proportional) analogies as mappings $\mathsf A$ satisfying $a:b:: \mathsf Aa: \mathsf Ab$ for all elements $a$ and $b$ in the source domain and show how to compute partial analogies. We then introduce a number of useful relations between functions (including homomorphisms and analogies) on proportoids and study their properties. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions.