On the Density of the Probabilistic Polynomial Classes

  • José D. P. Rolim


There are many results on which intractable problems can be solved quickly using randomized algorithms, starting with the classical paper by Rabin [10]. Comparisons of probabilistic and deterministic performances have stimulated several possibilities for randomization by considering the structure of the probabilistic classes [8]. The study of expected complexity has been developed from two basic points of view [13]. In the first one, the so called distributional approach, the input probability must be known and the theory is developed under these input assumptions. The second one, called the randomized approach, allows stochastic moves in the computations. In this paper, we study a distributional randomized approach. By defining density for probabilistic machines, we consider probability on the inputs and we adopt the probabilistic Turing machine as our formal model of computation.


Error Probability Turing Machine Complexity Class Computation Path Uniform Probability Distribution 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Ajtai and A. Wigderson. Deterministic simulation of probabilistic constant depth circuits. In Proceedings 26th Annual Symposium on Foundations of Computer Science, pages 11–19, Portland, Oregon, 1985.Google Scholar
  2. [2]
    M. Blum. A machine-independent theory of the complexity of recursive functions. J. ACM, 14(2), 1967.Google Scholar
  3. [3]
    R. Book. Tally languages and complexity classes. Information and Control, 26:186–193, 1974.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    J. Gill. Computational complexity of probabilistic turing machines. SIAM Journal of Computing, (6):675–695, 1977.MATHCrossRefGoogle Scholar
  5. [5]
    J. T. Gill. Probabilistic Turing Machines and Complexity of Computation. PhD thesis, Dept. of Mathematics, University of California, Berkeley, 1972.Google Scholar
  6. [6]
    J. Hartmanis. On sparse sets in NP-P. Information Processing Letters, 16:55–60, 1983.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    J. W. Hunt. Topics in Probabilistic Complexity. PhD thesis, Stanford University, 1979.Google Scholar
  8. [8]
    D. S. Johnson. The NP-completeness column: An ongoing guide. Journal of Algorithms, (5):433–447, 1984.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    K. Ko and D. Moore. Completeness, approximation and density. SIAM Journal of Computing, 10:787–796, 1981.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    M. O. Rabin. Probabilistic algorithms. In J. F. Traub, editor, Algorithms and Complexity: New Directions and Recent Results, pages 21–39. Academic Press, New York, 1976.Google Scholar
  11. [11]
    J. Rolim and S. Greibach. On the IO-complexity and approximation languages. Information Processing Letters, 28(1):27–31, 1988.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    E. S. Santos. Probabilistic turing machines and computability. In Proceedings American Mathematical Society, 1969.Google Scholar
  13. [13]
    A. C. Yao. Probabilistic computation: Toward a unified measure of complexity. In Proceedings 18th Annual Symposium on Foundations of Computer Science, pages 222–227, 1977.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • José D. P. Rolim
    • 1
  1. 1.Centre Universitaire d’InformatiqueUniversité de GenèveGenèveSwitzerland

Personalised recommendations