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On complexity classes and algorithmically random languages

Extended abstract
  • Ronald V. Book
  • Jack H. Lutz
  • Klaus W. Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)

Abstract

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Löf. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to the following result:
  1. (i)

    P = NP if and only if there exists an algorithmically random set that is ≤ btt P -hard for NP.

     
  2. (ii)

    P = PSPACE if and only if there exists an algorithmically random set that is ≤ btt P -hard for PSPACE.

     
  3. (iii)

    The polynomial-time hierarchy collapses if and only if there exists k>0 such that some algorithmically random set is σ k P -hard for PH.

     
  4. (iv)

    PH = PSPACE if and only if there exists a algorithmically random set that is PH-hard for PSPACE.

     

Keywords

Turing Machine Complexity Class Random Oracle Finite Variation Oracle Separation 
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 1992

Authors and Affiliations

  • Ronald V. Book
    • 1
  • Jack H. Lutz
    • 2
  • Klaus W. Wagner
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of Computer ScienceIowa State UniversityAmesUSA
  3. 3.Institut für InformatikUniversität WürzburgWürzburgGermany

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