Analogical proportions II

TL;DR

This paper advances the mathematical theory of analogical proportions within an algebraic framework, highlighting its potential applications in logic program synthesis and AI reasoning.

Contribution

It further develops the algebraic theory of analogical proportions, extending previous work and demonstrating its applicability to logic programming in AI.

Findings

Extended the algebraic framework of analogical proportions.

Applied the framework to logic program synthesis.

Enhanced understanding of analogical reasoning in AI.

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.