Analogical proportions II
Christian Antić
TL;DR
This work advances the mathematical foundations of analogical proportions within a universal-algebra framework by formalizing a justification-based approach and proving a Homomorphism Theorem that aligns proportions with structure-preserving maps. It develops fragment-based analyses (k,ℓ-fragments, especially the monolinear $(1,1)$) across arithmetic (additive/multiplicative), word, and tree domains, revealing precise characterizations such as difference proportions, geometric proportions, and word/tree-specific rules. The paper further connects analogical proportions with anti-unification in trees and finite algebras, providing algorithms via tree automata for deciding and computing proportions in finite settings and sketching pathways to broader infinite contexts. Collectively, these results enlarge the practical toolbox for analogical reasoning in AI, logic programming, and linguistic applications, with clear avenues for future work on linear fragments, infinite structures, and automatic representations.
Abstract
Analogical reasoning is the ability to detect parallels between two seemingly distant objects or situations, a fundamental human capacity used for example in commonsense reasoning, learning, and creativity which is believed by many researchers to be at the core of human and artificial general intelligence. Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. It is the purpose of this paper to further develop the mathematical theory of analogical proportions within that framework as motivated by the fact that it has already been successfully applied to logic program synthesis in artificial intelligence.