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TR19-018 | 18th February 2019 23:29

#### AC0[p] Lower Bounds against MCSP via the Coin Problem

TR19-018
Authors: Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Avishay Tal
Publication: 18th February 2019 23:30
Downloads: 1154
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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\geq 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.

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