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Randomness and Intractability in Kolmogorov Complexity

Author Igor Carboni Oliveira

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Igor Carboni Oliveira
  • Department of Computer Science, University of Oxford, UK

Acknowledgements

I am grateful to Ján Pich, Eric Allender, Ryan Williams, Shuichi Hirahara, Michal Koucký, Rahul Santhanam, and Jan Krajíček for discussions. Part of this work was completed while the author was visiting the Simons Institute for the Theory of Computing. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 615075.

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Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.32

Abstract

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability. This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time. Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • computational complexity
  • randomness
  • circuit lower bounds
  • Kolmogorov complexity

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