Proving SAT does not have small circuits with an application to the two queries problem

  • Authors:

  • Lance Fortnow;A. Pavan;Samik Sengupta

  • Affiliations:

  • Department of Computer Science, University of Chicago, Chicago, IL 60637, USA;Department of Computer Science, Iowa State University, Ames, IA 50011, USA;Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY 14260, USA

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2008

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Abstract

We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P^N^P^[^1^]=P^N^P^[^2^], then the polynomial-time hierarchy collapses to S"2^p@?@S"2^p@?@P"2^p. Even showing that the hierarchy collapsed to @S"2^p remained open prior to this paper.