## 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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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

### Keywords

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

### Subject Classification

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