TR19-018 Authors: Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Avishay Tal

Publication: 18th February 2019 23:30

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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.