Galois theory for analogical classifiers

TL;DR

This paper develops a Galois theory framework for analogical classifiers based on analogical proportions, providing a formal understanding of classifiers that preserve analogy relations, with applications to Boolean domains.

Contribution

It introduces a Galois theory for analogical classifiers, linking formal models of analogy with classes of analogy-preserving functions, and characterizes these classifiers in Boolean domains.

Findings

Established a Galois correspondence for analogical classifiers.

Explicitly characterized closed sets of classifiers for Boolean analogies.

Demonstrated the framework's usefulness in Boolean domains.

Abstract

Analogical proportions are 4-ary relations that read "A is to B as C is to D". Recent works have highlighted the fact that such relations can support a specific form of inference, called analogical inference. This inference mechanism was empirically proved to be efficient in several reasoning and classification tasks. In the latter case, it relies on the notion of analogy preservation. In this paper, we explore this relation between formal models of analogy and the corresponding classes of analogy preserving functions, and we establish a Galois theory of analogical classifiers. We illustrate the usefulness of this Galois framework over Boolean domains, and we explicitly determine the closed sets of analogical classifiers, i.e., classifiers that are compatible with the analogical inference, for each pair of Boolean analogies.