Classifying different criteria for learning algebraic structures

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 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 Ex-learning in. Such characterizations give a powerful method to understand whether a family of structures is learnable with respect to the desired criterion.