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Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions

Authors Michael Saks, Rahul Santhanam

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Michael Saks
  • Rutgers University, Piscataway, NJ, USA
Rahul Santhanam
  • University of Oxford, UK

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Michael Saks and Rahul Santhanam. Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 26:1-26:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


The fundamental Minimum Circuit Size Problem is a well-known example of a problem that is neither known to be in 𝖯 nor known to be NP-hard. Kabanets and Cai [Kabanets and Cai, 2000] showed that if MCSP is NP-hard under "natural" m-reductions, superpolynomial circuit lower bounds for exponential time would follow. This has triggered a long line of work on understanding the power of reductions to MCSP. Nothing was known so far about consequences of NP-hardness of MCSP under general Turing reductions. In this work, we consider two structured kinds of Turing reductions: parametric honest reductions and natural reductions. The latter generalize the natural reductions of Kabanets and Cai to the case of Turing-reductions. We show that NP-hardness of MCSP under these kinds of Turing-reductions imply superpolynomial circuit lower bounds for exponential time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity classes
  • Minimum Circuit Size Problem
  • Turing reductions
  • circuit lower bounds


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