A new characterization of BPP

  • Stathis Zachos
  • Hans Heller
Session 3 Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 181)


The complexity class BPP contains languages that can be solved in polynomial time with bounded error probability. It is shown that a language L is in BPP iff (x∈L ↔ ∃my ∀z P(x,y,z)) and (x∉L ↔ ∀y ∃mz⌍P(x,y,z)) for a polynomial time predicate P and for |y|, |z| ≦ poly(|x|). The formula ∃myP(y) with the random quantifier ∃m means that the probability Pr({y| P(y)}) <1/2 + ɛ for a fixed ɛ. Note that the weaker conditions ∃ y ∀zP(x,y,z) and ∀ y ∃ z⌍P(x,y,z) are complementary and thus decide whether x∈L. Some of the consequences of the characterization of BPP are that various probabilistic polynomial time hierarchies collapse as well as that probabilistic oracles do not add anything to the computing power of classes as low as Σ 2 P . For example, Σ 2 P, BPP = Σ 2 P , where Σ 2 P, BPP is the class σ 2 P relativized to BPP.


Probalistic algorithms polynomial time complexity classes oracles polynomial hierarchies 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Stathis Zachos
    • 1
  • Hans Heller
    • 2
  1. 1.Dept. of Comp. and Inf. Sci.Brooklyn College of the City University of New YorkBrooklyn
  2. 2.Techn. Universität MünchenMünchenW-Germany

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