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The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False

Authors Noam Mazor, Rafael Pass

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LIPIcs.ITCS.2024.80.pdf
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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Kolmogorov complexity
  • perebor conjecture
  • function inversion

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Abstract

The Perebor (Russian for "brute-force search") conjectures, which date back to the 1950s and 1960s are some of the oldest conjectures in complexity theory. The conjectures are a stronger form of the NP ≠ P conjecture (which they predate) and state that for "meta-complexity" problems, such as the Time-bounded Kolmogorov complexity Problem, and the Minimum Circuit Size Problem, there are no better algorithms than brute force search.
In this paper, we disprove the non-uniform version of the Perebor conjecture for the Time-Bounded Kolmogorov complexity problem. We demonstrate that for every polynomial t(⋅), there exists of a circuit of size 2^{4n/5+o(n)} that solves the t(⋅)-bounded Kolmogorov complexity problem on every instance. 
Our algorithm is black-box in the description of the Universal Turing Machine U employed in the definition of Kolmogorov Complexity and leverages the characterization of one-way functions through the hardness of the time-bounded Kolmogorov complexity problem of Liu and Pass (FOCS'20), and the time-space trade-off for one-way functions of Fiat and Naor (STOC'91). We additionally demonstrate that no such black-box algorithm can have circuit size smaller than 2^{n/2-o(n)}.
Along the way (and of independent interest), we extend the result of Fiat and Naor and demonstrate that any efficiently computable function can be inverted (with probability 1) by a circuit of size 2^{4n/5+o(n)}; as far as we know, this yields the first formal proof that a non-trivial circuit can invert any efficient function.

Cite As Get BibTex

Noam Mazor and Rafael Pass. The Non-Uniform Perebor Conjecture for Time-Bounded Kolmogorov Complexity Is False. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.80

Author Details

Noam Mazor
  • Cornell Tech, New York, NY, USA
Rafael Pass
  • Tel Aviv University, Israel
  • Cornell Tech, New York, NY, USA

Funding

  • Mazor, Noam: Research partly supported by NSF CNS-2149305 and NSF CNS-2128519.
  • Pass, Rafael: Supported in part by NSF Award CNS 2149305, AFOSR Award FA9550-18-1-0267, AFOSR Award FA9550-23-1-0387, a JP Morgan Faculty Award and an Algorand Foundation award. This material is based upon work supported by DARPA under Agreement No. HR00110C0086. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government, DARPA, AFOSR or the Algorand Foundation.

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