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Short lists with short programs from programs of functions and strings

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Abstract

Let {φ p } be an optimal Gödel numbering of the family of computable functions (in Schnorr’s sense), where p ranges over binary strings. Assume that a list of strings L(p) is computable from p and for all p contains a φ-program for φ p whose length is at most ε bits larger than the length of the shortest φ-programs for φ p . We show that for infinitely many p the list L(p) must have 2|p|−ε−O(1) strings. Here ε is an arbitrary function of p.

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Notes

  1. The term “optimal Gödel numbering” was introduced by Schnorr [4]. Teutsch and Zimand [2] call optimal Gödel numberings Kolmogorov numberings. However Kolmogorov has neither introduced nor studied them.

References

  1. Bauwens, B., Makhlin, A., Vereshchagin, N., Zimand, M.: Short lists with short programs in short time. In: Proceedings 28th IEEE Conference on Computational Complexity (CCC). ECCC report TR13-007, pp. 98–108. Stanford, CA (2013)

  2. Teutsch, J., Zimand, M.: On approximate decidability of minimal programs. Available from arXiv:1409.0496 and http://people.cs.uchicago.edu/~teutsch/papers/teutschpubs.html (2014)

  3. Rogers, H. Jr.: The Theory of Recursive Functions and Effective Computability. MIT Press (1987)

  4. Schnorr, C. P.: Optimal enumerations and optimal Gödel numberings. Mathematical Systems Theory 8(2), 182–191 (1975)

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  5. Shen, A.: A talk on some open problems in Kolmogorov complexity. The talk was delivered on a meeting during the IMS program “Algorithmic Randomness” (IMS, University of Singapore, 2–30 June 2014) around June 20, 2014

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Acknowledgments

The author is sincerely grateful to Alexander Shen for asking the question and hearing the preliminary version of the proof of the result. The author is grateful to Jason Teutsch for the idea of how to omit the use of the fixed point theorem. The author thanks anonymous referees for helpful remarks. The author is also grateful to the hospitality of the IMS of the University of Singapore.

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Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    Nikolay Vereshchagin

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  1. Nikolay Vereshchagin
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Correspondence to Nikolay Vereshchagin.

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This article is part of the Topical Collection on Special Issue on Computability, Complexity and Randomness (CCR 2015)

The work was done while visiting IMS (University of Singapore), the program “Algorithmic Randomness”, 2–30 June 2014. The work was in part supported by the Russian Academic Excellence Project ‘5-100’ and by the RFBR grant 16-01-00362.

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Vereshchagin, N. Short lists with short programs from programs of functions and strings. Theory Comput Syst 61, 1440–1450 (2017). https://doi.org/10.1007/s00224-017-9773-x

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  • DOI: https://doi.org/10.1007/s00224-017-9773-x

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