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


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.


Gödel numbering Kolmogorov numbering Optimal Gödel numbering Short list problem 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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