Abstract
We give a self-reduction for the Circuit Evaluation problem (CircEval) and prove the following consequences.
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Amplifying size–depth lower bounds. If CircEval has Boolean circuits of n k size and n 1−δ depth for some k and δ, then for every \({\epsilon > 0}\), there is a δ′ > 0 such that CircEval has circuits of \({n^{1 + \epsilon}}\) size and \({n^{1- \delta^{\prime}}}\) depth. Moreover, the resulting circuits require only \({\tilde{O}(n^{\epsilon})}\) 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).
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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.
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Lipton, R.J., Williams, R. Amplifying circuit lower bounds against polynomial time, with applications. comput. complex. 22, 311–343 (2013). https://doi.org/10.1007/s00037-013-0069-5
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DOI: https://doi.org/10.1007/s00037-013-0069-5