On the Uniform Learnability of Approximations to Non-recursive Functions

  • Frank Stephan
  • Thomas Zeugmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1720)

Abstract

Blum and Blum (1975) showed that a class \( \mathcal{B} \) of suitable recursive approximations to the halting problem is reliably EX-learnable. These investigations are carried on by showing that \( \mathcal{B} \) is neither in NUM nor robustly EX-learnable. Since the definition of the class \( \mathcal{B} \) is quite natural and does not contain any self-referential coding, \( \mathcal{B} \) serves as an example that the notion of robustness for learning is quite more restrictive than intended.

Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes \( \mathcal{U} \)(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Additionally, it is proved that \( \mathcal{U} \)(A) is neither in NUM nor robustly EX-learnable provided A is part of a recursively inseparable pair, A is simple but not hypersimple or A is neither recursive nor high. These results provide more evidence that there is still some need to find an adequate notion for “naturally learnable function classes.”

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Frank Stephan
    • 1
  • Thomas Zeugmann
    • 2
  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany
  2. 2.Department of InformaticsKyushu UniversityKasugaJapan

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