Generalization-baed similarity
Christian Antić
TL;DR
This paper develops a qualitative notion of similarity across algebras grounded in sets of generalizations, formalizing a maximal joint-property concept abunarrow_\mathfrak{(A,B)} b and the resulting a\approx b relation. It demonstrates that similarity interacts with structure in nuanced ways: reflexive and symmetric but not generally transitive, and only partially compatible with homomorphisms; it remains compatible with isomorphisms via first-isomorphism-type results. The authors introduce k,ℓ-fragments to obtain tractable, syntactic specializations and show how, in finite/unary and word/tree domains, similarity can be computed using automata-theoretic techniques. They further show that fundamental mathematical relations (modular arithmetic, Green’s relations, and conjugacy) are instances of similarity under appropriate fragments, and provide a logical interpretation via model-theoretic types. Finally, the work sketches practical avenues for theoretical CS and AI, including program synthesis, neural-symbolic integration, and analogical reasoning, highlighting the potential for cross-domain transfer guided by algebraic similarity.
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.