Local reduction

We reduce non-deterministic time T >= 2(n) to a 3SAT instance phi quasilinear size | phi| = T center dot log(O(1)) Tsuch that there is an explicit NC0 circuit Cthat encodes fin the following way: on input a (log| phi|)-bit index i, Coutputs the ith clause of f. The previous best result was Cin NC1. Even in the simpler setting of polynomial size (| phi| = poly(T)), the previous best result was Cin AC(0). More generally, for any time T= nand parameter r= nwe obtain | phi| = max(T, 2n/ r) center dot (n logT) O(1), and each output bit of Cis a decision tree of depth O(logr). As an application, we tighten Williams' connection between satisfiability algorithms and circuit lower bounds (STOC 2010; SIAM J. Comput. 2013). (C) 2018 Elsevier Inc. All rights reserved.