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The Complexity of Boolean Formula Minimization

  • David Buchfuhrer
  • Christopher Umans
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be \(\Sigma_2^P\)-complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth-2 variant was only shown to be \(\Sigma_2^P\)-complete in 1998 [Uma98], and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is \(\Sigma_2^P\)-complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is \(\Sigma_2^P\)-complete under Turing reductions.

Keywords

Positive Instance Boolean Formula Negative Instance Boolean Circuit Equivalent Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • David Buchfuhrer
    • 1
  • Christopher Umans
    • 1
  1. 1.Computer Science DepartmentCalifornia Institute of TechnologyPasadena

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