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Learning Families of Algebraic Structures from Text

Nikolay Bazhenov, Ekaterina Fokina, Dino Rossegger, Alexandra Soskova, Stefan Vatev

TL;DR

A model-theoretic characterization of the learnability from text for classes of structures shows that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $\Sigma_2$ sentences.

Abstract

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $Σ_2$ sentences.

Learning Families of Algebraic Structures from Text

TL;DR

A model-theoretic characterization of the learnability from text for classes of structures shows that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary sentences.

Abstract

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary sentences.
Paper Structure (9 sections, 20 theorems, 12 equations)

This paper contains 9 sections, 20 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The Formal Framework
  3. Cantor-Continuous Embeddings
  4. Scott-Continuous Embeddings
  5. Characterization of $\mathbf{TxtEx}$-Learnability
  6. $\mathbf{TxtEx}$-Complete Classes
  7. Applications
  8. Further Discussion
  9. Acknowledgements.

Key Result

Proposition 1

A function $\Psi: 2^\omega \to 2^\omega$ is Cantor-continuous if and only if there exists a Turing operator $\Phi_e$ and a set $A \in 2^\omega$ such that $\Psi(X) = \Phi_e(A\oplus X)$ for all $X \in 2^\omega$.

Theorems & Definitions (39)

  • Definition 1: bazhenov_turing_2021
  • Definition 2
  • Remark 1
  • Remark 2
  • Proposition 1: Folklore
  • Definition 3
  • Theorem 1: Pullback Theorem knight_turing_2007
  • Corollary 1: Non-effective Pullback Theorem
  • Definition 4: Case case_enumeration_1971
  • Proposition 2: Folklore
  • ...and 29 more