Advertisement

Theory of Computing Systems

, Volume 61, Issue 4, pp 1440–1450 | Cite as

Short lists with short programs from programs of functions and strings

  • Nikolay VereshchaginEmail author
Article
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)

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|−εO(1) strings. Here ε is an arbitrary function of p.

Keywords

Gödel numbering Kolmogorov numbering Optimal Gödel numbering Short list problem 

Notes

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.

References

  1. 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)Google Scholar
  2. 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. 3.
    Rogers, H. Jr.: The Theory of Recursive Functions and Effective Computability. MIT Press (1987)Google Scholar
  4. 4.
    Schnorr, C. P.: Optimal enumerations and optimal Gödel numberings. Mathematical Systems Theory 8(2), 182–191 (1975)CrossRefzbMATHGoogle Scholar
  5. 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, 2014Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations