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The logarithmic alternation hierarchy collapses: \(A\Sigma _2^\mathcal{L} = A\Pi _2^\mathcal{L}\)

  • Klaus-Jörn Lange
  • Birgit Jenner
  • Bernd Kirsig
Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)

Abstract

We show that \(A\Sigma _2^\mathcal{L}\) coincides with \(A\Pi _2^\mathcal{L}\). Essentially this is done by Hausdorff reducing the \(A\Sigma _2^\mathcal{L}\)-complete set (GAP¢Co-GAP)(∃) to the question whether of two vectors A and B of n components, A contains more “solvable” components, i.e. components which are contained in GAP, than B. Moreover, we show \(A\Sigma _2^\mathcal{L} = \mathcal{L}_{hd} (N\mathcal{L})\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Klaus-Jörn Lange
    • 1
  • Birgit Jenner
    • 2
  • Bernd Kirsig
    • 2
  1. 1.TU München, Inst. für InformatikMünchen 2
  2. 2.Fachbereich InformatikUniv. HamburgHamburg 13

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