We reduce non-deterministic time T ≥ 2 n to a 3SAT instance φ of size |φ | = T ·log O(1) T such that there is an explicit circuit C that on input an index i of log|φ| bits outputs the ith clause, and each output bit of C depends on O(1) inputs bits. The previous best result was C in NC 1. Even in the simpler setting of |φ | = poly(T) the previous best result was C in AC 0. More generally, for any time T ≥ n and parameter r ≤ n we obtain log 2|φ | = max(logT,n/r)+O(logn)+O(loglogT) and each output bit of C is a decision tree of depth O(logr). As an application, we simplify the proof of Williams ’ ACC 0 lower bound, and tighten his connection between satisfiability algorithms and lower bounds
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