Two remarks on the power of counting

  • Christos H. Papadimitriou
  • Stathis K. Zachos
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 145)


The relationship between the polynomial hierarchy and Valiant's class #P is at present unknown. We show that some low portions of the polynomial hierarchy, namely deterministic polynomial algorithms using an NP oracle at most a logarithmic number of times, can be simulated by one #P computation. We also show that the class of problems solvable by polynomial-time nondeterministic Turing machines which accept whenever there is an odd number of accepting computations is idempotent, that is, closed under usage of oracles from the same class.


Counting problems oracle computation polynomial hierarchy parity problems machine simulation 


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  1. [An]
    D. Angluin "On counting problems and the polynominal-time hierarchy", Theoretical Computer Science 12 (1980), pp. 161–173Google Scholar
  2. [GJ]
    M.R. Garey, D.S. Johnson Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, 1979.Google Scholar
  3. [Gi]
    J. Gill "Computational complexity of probabilistic Turing machines", SIAM J. Computing 6 (1977), pp. 675–695.Google Scholar
  4. [LP]
    H.R. Lewis, C.H. Papadimitriou Elements of the Theory of Computation, Prentice-Hall, 1981.Google Scholar
  5. [Pa]
    C.H. Papadimitriou "The complexity of unique solutions", Proc. 13th FOCS Conference, 1982, to appear.Google Scholar
  6. [PY]
    C.H. Papadimitriou, M. Yannakakis "The complexity of facets (and some facets of complexity", Proc. 14th STOC, pp. 255–260, 1982. Also to appear in JCSS.Google Scholar
  7. [Si]
    J. Simon "On the difference between one and many" Proc. 4th Intern. Colloguium on Automata, Languages and Programming, pp. 480–491, 1977.Google Scholar
  8. [St]
    L.J. Stockmeyer "The polynamial-time hierarchy", Theoretical Computer Science, 3 (1977), pp. 1–22.Google Scholar
  9. [Va1]
    L.G. Valiant "The complexity of computing the permanent", Theoretical Computer Science, 8 (1979), pp. 181–201.Google Scholar
  10. [Va2]
    L.G. Valiant, private communication, August 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Christos H. Papadimitriou
    • 1
  • Stathis K. Zachos
    • 1
  1. 1.Laboratory of Computer Science, M.I.T.CambridgeUSA

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