Journal of Symbolic Logic

General Random Sequences and Learnable Sequences

C. P. Schnorr and P. Fuchs

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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.

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

First available in Project Euclid: 6 July 2007

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Schnorr, C. P.; Fuchs, P. General Random Sequences and Learnable Sequences. J. Symbolic Logic 42 (1977), no. 3, 329--340.

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