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LEARNING EQUIVALENCE RELATIONS ON POLISH SPACES

Part of: Set theory Theory of computing Computability and recursion theory

Published online by Cambridge University Press:  03 February 2025

DINO ROSSEGGER Open the ORCID record for DINO ROSSEGGER [Opens in a new window] ,
THEODORE SLAMAN Open the ORCID record for THEODORE SLAMAN [Opens in a new window]  and
TOMASZ STEIFER Open the ORCID record for TOMASZ STEIFER [Opens in a new window]
DINO ROSSEGGER*
Affiliation:
INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY TECHNISCHE UNIVERSITÄT WIEN VIENNA, AUSTRIA
THEODORE SLAMAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA, USA E-mail: slaman@math.berkeley.edu
TOMASZ STEIFER
Affiliation:
UNIVERSIDAD CATÓLICA DE CHILE & INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH POLISH ACADEMY OF SCIENCES WARSAW, POLAND E-mail: tsteifer@ippt.pan.pl
*
E-mail: dino.rossegger@tuwien.ac.at
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Abstract

We investigate natural variations of behaviourally correct learning and explanatory learning—two learning paradigms studied in algorithmic learning theory—that allow us to “learn” equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviourally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\boldsymbol {\Pi }^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

Keywords

computational learning theoryBorel equivalence relationequivalence relation

MSC classification

Primary: 03E15: Descriptive set theory 68Q32: Computational learning theory
Secondary: 03D80: Applications of computability and recursion theory
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 19
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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