
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 `` is to what is to '' 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 satisfying iff for all elements and show that their kernel is a congruence. Moreover, we introduce (proportional) analogies…
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