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Short lists with short programs in short time

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Abstract

Given a machine U, a c-short program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any standard Turing machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a \({\operatorname{O}\bigl({\mathrm{log}}|x|\bigr)}\)-short program for x. We also show that there exists a computable function that maps every x to a list of size |x|2 containing a \({\operatorname{O}\bigl(1\bigr)}\)-short program for x. This is essentially optimal because we prove that for each such function there is a c and infinitely many x for which the list has size at least c|x|2. Finally we show that for some standard machines, computable functions generating lists with 0-short programs must have infinitely often list sizes proportional to 2|x|.

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Authors and Affiliations

  1. Department of Computer Science, National Research University Higher School of Economics, Kochnovskiy Proezd 3, Moscow, Russia, 125319

    Bruno Bauwens & Nikolay Vereshchagin

  2. Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, Russia, 119234

    Anton Makhlin

  3. Department of Computer and Information Sciences, Towson University, 8000 York Road, Towson, MD, 21252, USA

    Marius Zimand

Authors
  1. Bruno Bauwens
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  2. Anton Makhlin
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  3. Nikolay Vereshchagin
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  4. Marius Zimand
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Correspondence to Bruno Bauwens.

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Bauwens, B., Makhlin, A., Vereshchagin, N. et al. Short lists with short programs in short time. comput. complex. 27, 31–61 (2018). https://doi.org/10.1007/s00037-017-0154-2

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  • DOI: https://doi.org/10.1007/s00037-017-0154-2

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