## Non-uniform ACC circuit lower bounds (2010)

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### BibTeX

@TECHREPORT{Williams10non-uniformacc,

author = {Ryan Williams},

title = {Non-uniform ACC circuit lower bounds},

institution = {},

year = {2010}

}

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### Abstract

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have non-uniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasi-polynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have non-uniform ACC circuits of 2no(1) size. The lower bound gives an exponential size-depth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depth-d ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth-3 polynomial size circuits made out of only MOD6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.