Journal of Symbolic Logic

General Random Sequences and Learnable Sequences

C. P. Schnorr and P. Fuchs

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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 sequences $z$ is learnable (super-learnable, resp.) if and only if there is a computable probability measure $p$ such that $p$ is Schnorr (Martin-Lof, 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.

Article information

Source
J. Symbolic Logic, Volume 42, Issue 3 (1977), 329-340.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183740009

Mathematical Reviews number (MathSciNet)
MR495206

Zentralblatt MATH identifier
0376.02026

JSTOR
links.jstor.org

Citation

Schnorr, C. P.; Fuchs, P. General Random Sequences and Learnable Sequences. J. Symbolic Logic 42 (1977), no. 3, 329--340. https://projecteuclid.org/euclid.jsl/1183740009


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