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Classifying different criteria for learning algebraic structures

Nikolay Bazhenov, Vittorio Cipriani, Sanjay Jain, Luca San Mauro, Frank Stephan

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.

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 -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 -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 -learning in [6]. Such characterizations give a powerful method to understand whether a family of structure is learnable with respect to the desired criterion.

Paper Structure

This paper contains 13 sections, 30 theorems, 23 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sequences and structures
  4. The learning paradigm
  5. Infinitary logic
  6. $E$-learnability
  7. Characterizing learnabilities in terms of $\Sigma_{1}^{\mathrm{inf}}$-formulas
  8. $\Sigma_{1}^{\mathrm{inf}}$-(strong) antichains
  9. $\Sigma_{1}^{\mathrm{inf}}$-partial orders
  10. non-U-shaped and decisive learning
  11. Characterizing learnabilities in terms of $\Sigma_{2}^{\mathrm{inf}}$-formulas
  12. $\mathbf{PL}$-learnability under learning reducibility
  13. Conclusions

Key Result

Proposition 2.4

$\{\omega,\omega^*\}$ is $\mathbf{Ex}$-learnable.

Figures (1)

  • Figure 1: The learning paradigm considered in this paper. On the left-hand-side the picture refers to learn reducibility while the right-hand-side refers to finitary learn reducibility: The arrows represent (finite) learn reducibility in the direction of the arrow. For $n \in \{1,2\}$$\Sigma_{n}^{\mathrm{inf}}$-a., -s.a., and -p. denote respectively $\Sigma_{n}^{\mathrm{inf}}$-antichains, -strong antichains and partial orders as defined in \ref{['definition:sigmainfnposet']}.

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Theorem 2.9
  • ...and 53 more