Classifying different criteria for learning algebraic structures

TL;DR

This paper explores various criteria for learning algebraic structures, introducing a new framework called E-learnability, and provides syntactic characterizations to understand learnability under different paradigms.

Contribution

It introduces the concept of E-learnability and offers syntactic characterizations for different learning criteria in algebraic structures, expanding the theoretical understanding of learnability.

Findings

E-learnability provides a unified framework for analyzing learning criteria.

Syntactic characterizations help determine learnability of structure families.

The approach uses descriptive-set-theoretic tools to compare learning paradigms.

Abstract

In the last years there has been a growing interest in the study of learning problems associated with algebraic structures. The framework we use models the scenario in which a learner is given larger and larger fragments of a structure from a given target family and is required to output an hypothesis about the structure's isomorphism type. So far researchers focused on $\mathbf{Ex}$-learning, in which the learner is asked to eventually stabilize to the correct hypothesis, and on restrictions where the learner is allowed to change the hypothesis a fixed number of times. Yet, other learning paradigms coming from classical algorithmic learning theory remained unexplored. We study the "learning power" of such criteria, comparing them via descriptive-set-theoretic tools thanks to the novel notion of $E$-learnability. The main outcome of this paper is that such criteria admit natural syntactic characterizations in terms of infinitary formulas analogous to the one given for $\mathbf{Ex}$-learning in [6]. Such characterizations give a powerful method to understand whether a family of structure is learnable with respect to the desired criterion.