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Search-To-Decision Reductions for Kolmogorov Complexity

Authors Noam Mazor, Rafael Pass

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Author Details

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

Funding

  • Mazor, Noam: Research partly supported by NSF CNS-2149305 and DARPA under Agreement No. HR00110C0086.
  • Pass, Rafael: Supported in part by AFOSR Award FA9550-23-1-0387, AFOSR Award FA9550-23-1-0312, and an Algorand Foundation grant. 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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Noam Mazor and Rafael Pass. Search-To-Decision Reductions for Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.34

Abstract

A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s.
In this work, we consider the more "robust" version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t,|x|)-time program of length s+O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^{ε n} for any ε > 0 and additionally only queries its oracle on "thresholds" s of size s+O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^{α s} poly(|x|,t,s) for solving GapMINKT (given an instance (x,t,s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^{(α+ε) s} poly(|x|,t,s).
Our second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS'20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime.
A central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Kolmogorov complexity
  • search to decision

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