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Local Reductions

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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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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Authors and Affiliations

  1. Northeastern University, Boston, MA, USA

    Hamid Jahanjou & Emanuele Viola

  2. UCLA, Los Angeles, CA, USA

    Eric Miles

Authors
  1. Hamid Jahanjou
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  2. Eric Miles
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  3. Emanuele Viola
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Correspondence to Eric Miles .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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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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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_61

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