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General random sequences and learnable sequences

Published online by Cambridge University Press:  12 March 2014

C. P. Schnorr  and
P. Fuchs
C. P. Schnorr
Affiliation:
Universität Frankfurt, Fachbereich Mathematik, Federal Republic of Germany
P. Fuchs
Affiliation:
Universität Frankfurt, Fachbereich Mathematik, Federal Republic of Germany
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Abstract

We formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequence z is learnable (super-learnable, resp.) if and only if there is a computable probability measure p such that p is Schnorr (Martin-Löf, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 3 , September 1977 , pp. 329 - 340
Copyright
Copyright © Association for Symbolic Logic 1977

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References

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