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)


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.”


Function Class Recursive Function Inductive Inference Total Function Computational Learn Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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