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Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem

Authors Per Austrin , Kilian Risse

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LIPIcs.CCC.2023.31.pdf
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Author Details

Per Austrin
  • KTH Royal Institute of Technology, Stockholm, Sweden
Kilian Risse
  • EPFL, Lausanne, Switzerland

Funding

Supported by the Approximability and Proof Complexity project funded by the Knut and Alice Wallenberg Foundation.
  • Risse, Kilian: Supported by Swiss National Science Foundation project 200021-184656 “Randomness in Problem Instances and Randomized Algorithms”.

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Per Austrin and Kilian Risse. Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CCC.2023.31

Abstract

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}ⁿ → {0,1}, SoS requires degree Ω(s^{1-ε}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ⊈ P/poly.
We also show that for any 0 < α < 1 there are Boolean functions with circuit complexity larger than 2^{n^α} but SoS requires size 2^{2^Ω(n^α)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions.
Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof Complexity
  • Sum of Squares
  • Minimum Circuit Size Problem

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