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# AC^0[p] Lower Bounds Against MCSP via the Coin Problem

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## Acknowledgements

This work was partly carried out while many of the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. We also thank Chris Umans and Ronen Shaltiel for helpful discussions in the early stages of this project (during the Dagstuhl 2018 workshop on "Algebraic Methods in Complexity"). We thank Eric Allender and Shuichi Hirahara for their comments, and special thanks to Eric for pointing us to the paper of Dančik [Vladimir Dančik, 1996] and the discussion of various circuit and formula complexity measures for constant-depth circuit models. We are grateful to our anonymous reviewers for helpful comments on this paper.

## Cite As

Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal. AC^0[p] Lower Bounds Against MCSP via the Coin Problem. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.66

## Abstract

Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p >= 2 (the circuit class AC^0[p]). Namely, we show that MCSP requires d-depth AC^0[p] circuits of size at least exp(N^{0.49/d}), where N=2^n is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N-bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2+N^{-0.49}, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC^0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC^0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC^1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC^0 circuit with MCSP-oracle gates.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
• Theory of computation → Problems, reductions and completeness
##### Keywords
• Minimum Circuit Size Problem (MCSP)
• circuit lower bounds
• AC0[p]
• coin problem
• hybrid argument
• MKTP
• biased random boolean functions

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