Abstract
We reduce non-deterministic time T≥2n to a 3SAT instance ϕ of quasilinear size |ϕ|=T⋅logO(1)T such that there is an explicit NC0 circuit C that encodes ϕ in the following way: on input a (log|ϕ|)-bit index i, C outputs the ith clause of ϕ. The previous best result was C in NC1. Even in the simpler setting of polynomial size (|ϕ|=poly(T)), the previous best result was C in AC0.More generally, for any time T≥n and parameter r≤n we obtain |ϕ|=max(T,2n/r)⋅(nlogT)O(1), and each output bit of C is 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).
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