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On Randomized Reductions to the Random Strings

Authors Michael Saks, Rahul Santhanam

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Author Details

Michael Saks
  • Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ, USA
Rahul Santhanam
  • Department of Computer Science, University of Oxford, UK

Funding

  • Saks, Michael: Supported in part by the Simons Foundation under Grant 332622.
  • Santhanam, Rahul: Partially funded by EPSRC New Horizons Grant EP/V048201/1.

Acknowledgements

We thank the anonymous reviewers for useful comments. The second author benefited from conversations with Shuichi Hirahara and Rahul Ilango.

Cite AsGet BibTex

Michael Saks and Rahul Santhanam. On Randomized Reductions to the Random Strings. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 29:1-29:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.29

Abstract

We study the power of randomized polynomial-time non-adaptive reductions to the problem of approximating Kolmogorov complexity and its polynomial-time bounded variants. As our first main result, we give a sharp dichotomy for randomized non-adaptive reducibility to approximating Kolmogorov complexity. We show that any computable language L that has a randomized polynomial-time non-adaptive reduction (satisfying a natural honesty condition) to ω(log(n))-approximating the Kolmogorov complexity is in AM ∩ coAM. On the other hand, using results of Hirahara [Shuichi Hirahara, 2020], it follows that every language in NEXP has a randomized polynomial-time non-adaptive reduction (satisfying the same honesty condition as before) to O(log(n))-approximating the Kolmogorov complexity. As our second main result, we give the first negative evidence against the NP-hardness of polynomial-time bounded Kolmogorov complexity with respect to randomized reductions. We show that for every polynomial t', there is a polynomial t such that if there is a randomized time t' non-adaptive reduction (satisfying a natural honesty condition) from SAT to ω(log(n))-approximating K^t complexity, then either NE = coNE or 𝖤 has sub-exponential size non-deterministic circuits infinitely often.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Kolmogorov complexity
  • randomized reductions

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