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Short lists with short programs in short time


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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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).

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  • List-approximator
  • Kolmogorov complexity
  • Online matching
  • Expander graph

Subject Classification

  • 68Q17
  • 68Q30
  • 03D15
  • 03D25
  • 05C70
  • 05C85