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We give a self-reduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. • Amplifying Size-Depth Lower Bounds. If CircEval has Boolean circuits of n k size and n1−δ depth for some k and δ, then for every ε> 0, there is a δ ′> 0 such that CircEval has circuits of n1+ε size and n1−δ ′ depth. Moreover, the resulting circuits require only Õ(nε) bits of non-uniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound). • Lower Bounds for Quantified Boolean Formulas. Let c, d> 1 and e < 1 satisfy c < (1 − e + d)/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n c], or the Circuit Evaluation problem cannot be solved with circuits of n d size and n e depth. This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n c-time uniform NC circuits, for all c < 2.

Year: 2014

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