Algebraic anti-unification
Christian Antić
TL;DR
This work introduces algebraic anti-unification in the setting of general algebras, defining minimally general generalizations $a\Uparrow_{\mathfrak{(A,B)}} b$ as the semantic counterpart to traditional syntactic anti-unification. The approach builds on the semantic generalization order $\sqsubseteq_{\mathfrak{(A,B)}}$ defined via $\downarrow_\mathfrak A s$ and $\downarrow_\mathfrak B s$, and develops a rich theory including generalization types, characteristic generalizations, homomorphisms, and extensive treatment of monounary and finite/unary algebras. It shows that in the finite/unary cases generalizations are computable (via automata and tree automata) and extends the framework to fragments and set-wise generalizations, with applications to similarity and analogical proportions. The paper outlines future directions for decidability, complexity, bilingual extensions, and practical implementations to support AI systems requiring robust abstraction and analogical reasoning.
Abstract
Abstraction is key to human and artificial intelligence as it allows one to see common structure in otherwise distinct objects or situations and as such it is a key element for generality in AI. Anti-unification (or generalization) is \textit{the} part of theoretical computer science and AI studying abstraction. It has been successfully applied to various AI-related problems, most importantly inductive logic programming. Up to this date, anti-unification is studied only from a syntactic perspective in the literature. The purpose of this paper is to initiate an algebraic (i.e. semantic) theory of anti-unification within general algebras. This is motivated by recent applications to similarity and analogical proportions.