Local reductions

We reduce non-deterministic time $T \ge 2^n$ to a 3SAT instance $ϕ$ of quasilinear size $|ϕ| = T \cdot \log^{O(1)} T$ such that there is an explicit circuit $C$ that on input an index $i$ of $\log |ϕ|$ bits outputs the $i$th clause, and each output bit of $C$ depends on $O(1)$ input bits. The previous best result was $C$ in NC$^1$. Even in the simpler setting of polynomial size $|ϕ| = \poly(T)$ the previous best result was $C$ in AC$^0$. More generally, for any time $T \ge n$ and parameter $r \leq n$ we obtain $\log_2 |ϕ| = \max(\log T, n/r) + O(\log n) + O(\log\log T)$ and each output bit of $C$ is a decision tree of depth $O(\log r)$. As an application, we tighten Williams' connection between satisfiability algorithms and circuit lower bounds (STOC 2010; SIAM J. Comput. 2013).

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