Generalization-baed similarity

TL;DR

This paper introduces a novel algebraic and logical framework for defining similarity based on generalizations, with potential applications in AI and computer science.

Contribution

It develops a new similarity notion from first principles using universal algebra and model theory, linking it to mathematical relations and logic.

Findings

Mathematically appealing properties of the similarity measure

Modeling fundamental mathematical relations

Embedding into first-order logic

Abstract

Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can model fundamental relations occurring in mathematics and be naturally embedded into first-order logic via model-theoretic types. Finally, we sketch some potential applications to theoretical computer science and artificial intelligence.