Sets computable in polynomial time on average

  • Rainer Schuler
  • Tomoyuki Yamakami
Session 7B: Complexity Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)


In this paper, we discuss the complexity and properties of the sets which are computable in polynomial-time on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomial-time on average for every reasonable (i.e., polynomial-time computable) distribution on the instances. Let PP-comp denote the class of all those sets which are computable in polynomial-time on average for every polynomial-time computable distribution on the instances. It is known that P ⊂ PP-comp ⊂ E. In this paper, we show that PP-comp is not contained in DTIME(2cn) for any constant c and that it lacks some basic structural properties: for example, it is not closed under many-one reducibility or for the existential operator. From these results, it follows that PP-comp contains P-immune sets but no P-bi-immune sets; it is not included in P/cn for any constant c; and it is different from most of the well-known complexity classes, such as UP, NP, BPP, and PP. Finally, we show that, relative to a random oracle, NP is not included in PP-comp and PP-comp is not in PSPACE with probability 1.


Turing Machine Complexity Core Random Oracle Average Computation Time Existential Operator 
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 1995

Authors and Affiliations

  • Rainer Schuler
    • 1
  • Tomoyuki Yamakami
    • 2
  1. 1.Abteilung Theoretische InformatikUniversität UlmUlmGermany
  2. 2.Department of Computer ScienceUniversity of TorontoTorontoCanada

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