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On the Complexity of Learning Programs

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Unity of Logic and Computation (CiE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13967))

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Abstract

Given a computable sequence of natural numbers, it is a natural task to find a Gödel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning theory this problem is well studied from several perspectives and one question studied there is for which sequences this problem is at least learnable in the limit. Here we study the problem on all computable sequences and we classify the Weihrauch complexity of it. For this purpose we can, among other methods, utilize the amalgamation technique known from learning theory. As a benchmark for the classification we use closed and compact choice problems and their jumps on natural numbers, which correspond to induction and boundedness principles, as they are known from the Kirby-Paris hierarchy in reverse mathematics. We provide a topological as well as a computability-theoretic classification, which reveal some significant differences.

Vasco Brattka is supported by the National Research Foundation of South Africa.

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Authors and Affiliations

  1. Faculty of Computer Science, Universität der Bundeswehr München, Neubiberg, Germany

    Vasco Brattka

  2. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, South Africa

    Vasco Brattka

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  1. Vasco Brattka
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Correspondence to Vasco Brattka .

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Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Gianluca Della Vedova

  2. Kutaisi International University, Kutaisi, Georgia

    Besik Dundua

  3. University of Wisconsin, Madison, WI, USA

    Steffen Lempp

  4. University of Göttingen, Göttingen, Germany

    Florin Manea

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Brattka, V. (2023). On the Complexity of Learning Programs. In: Della Vedova, G., Dundua, B., Lempp, S., Manea, F. (eds) Unity of Logic and Computation. CiE 2023. Lecture Notes in Computer Science, vol 13967. Springer, Cham. https://doi.org/10.1007/978-3-031-36978-0_14

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  • DOI: https://doi.org/10.1007/978-3-031-36978-0_14

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  • Online ISBN: 978-3-031-36978-0

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