Abstract
We reduce non-deterministic time \(T \ge 2^n\) to a 3SAT instance \(\phi \) of quasilinear size \(|\phi | = T \cdot \log ^{O(1)} T\) such that there is an explicit circuit C that on input an index i of \(\log |\phi |\) bits outputs the ith 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 \(|\phi | = \mathrm {poly}(T)\) the previous best result was C in AC\(^0\).
More generally, for any time \(T \ge n\) and parameter \(r \le n\) we obtain \(\log _2 |\phi | = \max (\log T, n/r) + O(\log n) + O(\log \log T)\) 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).
Supported by NSF grants CCF-0845003, CCF-1319206.
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Jahanjou, H., Miles, E., Viola, E. (2015). Local Reductions. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_61
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