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DOI: 10.4230/LIPIcs.MFCS.2017.1
Does Looking Inside a Circuit Help?

Authors: Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, and Shadab Romani

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
The Black-Box Hypothesisstates that any property of Boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an upper bound on its circuit size. If this hypothesis is true, then P neq NP. We focus on the consequences of the hypothesis being false, showing that (under general conditions on the structure of a counterexample) it implies a non-trivial algorithm for CSAT. More specifically, we show that if there is a property F of boolean functions such that F has high sensitivity on some input function f of subexponential circuit complexity (which is a sufficient condition for F being a counterexample to the Black-Box Hypothesis), then CSAT is solvable by a subexponential-size circuit family. Moreover, if such a counterexample F is symmetric, then CSAT is in Ppoly. These results provide some evidence towards the conjecture (made in this paper) that the Black-Box Hypothesis is false if and only if CSAT is easy.
Cite as

Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Pierre McKenzie, and Shadab Romani. Does Looking Inside a Circuit Help?. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{impagliazzo_et_al:LIPIcs.MFCS.2017.1,
  author =	{Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina and McKenzie, Pierre and Romani, Shadab},
  title =	{{Does Looking Inside a Circuit Help?}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{1:1--1:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.1},
  URN =		{urn:nbn:de:0030-drops-80975},
  doi =		{10.4230/LIPIcs.MFCS.2017.1},
  annote =	{Keywords: Black-Box Hypothesis, Rice's theorem, circuit complexity, SAT, sensitivity of boolean functions, decision tree complexity}
}
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