Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

  • Arfst Nickelsen
  • Till Tantau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2138)

Abstract

Polynomial time partial information classes are extensions of the class P of languages decidable in polynomial time. A partial information algorithm for a language A computes, for fixed n ∈ ℕ, on input of words x 1,...,x n a set P of bitstrings, called a pool, such that χA(x 1,...,x n ) ∈ P, where P is chosen from a family \( \mathcal{D} \) of pools. A language A is in \( P\left[ \mathcal{D} \right] \), if there is a polynomial time partial information algorithm which for all inputs (x 1,... x n ) outputs a pool \( \mathcal{P} \in \mathcal{D} \) with χa(x 1,..., x n ) ∈ P. Many extensions of P studied in the literature, including approximable languages, cheatability, p-selectivity and frequency computations, form a class \( P\left[ \mathcal{D} \right] \) for an appropriate family \( \mathcal{D} \).

We characterise those families \( \mathcal{D} \) for which \( P\left[ \mathcal{D} \right] \) is closed under certain polynomial time reductions, namely bounded truth-table, truth-table, and Turing reductions. We also treat positive reductions. A class \( P\left[ \mathcal{D} \right] \) is presented which strictly contains the class P-sel of p-selective languages and is closed under positive truth-table reductions.

Keywords

structural complexity partial information polynomial time reductions verboseness p-selectivity positive reductions 

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References

  1. 1.
    A. Amir, R. Beigel, and W. Gasarch. Some connections between bounded query classes and non-uniform complexity. In Proc. 5th Structure in Complexity Theory, 1990.Google Scholar
  2. 2.
    A. Amir and W. Gasarch. Polynomial terse sets. Inf. and Computation, 77, 1988.Google Scholar
  3. 3.
    R. Beigel. Query-Limited Reducibilities. PhD thesis, Stanford University, 1987.Google Scholar
  4. 4.
    R. Beigel. Bounded queries to SAT and the boolean hierarchy. Theoretical Comput. Sci., 84(2), 1991.Google Scholar
  5. 5.
    R. Beigel, W. Gasarch, and E. Kinber. Frequency computation and bounded queries. In Proc. 10th Structure in Complexity Theory, 1995.Google Scholar
  6. 6.
    R. Beigel, M. Kummer, and F. Stephan. Quantifying the amount of verboseness. In Proc. Logical Found. of Comput. Sci., volume 620 of LNCS. Springer, 1992.CrossRefGoogle Scholar
  7. 7.
    R. Beigel, M. Kummer, and F. Stephan. Approximable sets. In Proc. 9th Structure in Complexity Theory, 1994.Google Scholar
  8. 8.
    R. Beigel, M. Kummer, and F. Stephan. Approximable sets. Inf. and Computation, 120(2), 1995.Google Scholar
  9. 9.
    L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets. SIAM J. Comput., 6(2):305–322, 1977.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Goldsmith, D. Joseph, and P. Young. Using self-reducibilities to characterize polynomial time. Inf. and Computation, 104(2):288–308, 1993.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Goldsmith, D. A. Joseph, and P. Young. Using self-reducibilities to characterize polynomial time. Technical Report CS-TR-88-749, University of Wisconsin, Madison, 1988.Google Scholar
  12. 12.
    L. Hemaspaandra, A. Hoene, and M. Ogihara. Reducibility classes of p-selective sets. Theoretical Comput. Sci., 155:447–457, 1996.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    L. A. Hemaspaandra, Z. Jiang, J. Rothe, and O. Watanabe. Polynomial-time multi-selectivity. J. of Universal Comput. Sci., 3(3), 1997.Google Scholar
  14. 14.
    M. Hinrichs and G. Wechsung. Time bounded frequency computations. In Proc. 12th Conf. on Computational Complexity, 1997.Google Scholar
  15. 15.
    A. Hoene and A. Nickelsen. Counting, selecting, and sorting by query-bounded machines. In Proc. STACS 93, volume 665 of LNCS. Springer, 1993.Google Scholar
  16. 16.
    K.-I. Ko. On self-reducibility and weak p-selectivity. J. Comput. Syst. Sci., 26:209–221, 1983.MATHCrossRefGoogle Scholar
  17. 17.
    J. Köbler. On the structure of low sets. In Proc. 10th Structure in Complexity Theory, pages 246–261. IEEE Computer Society Press, 1995.Google Scholar
  18. 18.
    R. E. Ladner, N. A. Lynch, and A. L. Selman. A comparison of polynomial time reducibilities. Theoretical Comput. Sci., 1(2):103–123, Dec. 1975.Google Scholar
  19. 19.
    A. Nickelsen. On polynomially \( \mathcal{D} \)-verbose sets. In Proc. STACS 97, volume 1200 of LNCS, pages 307–318. Springer, 1997.CrossRefGoogle Scholar
  20. 20.
    A. Selman. P-selective sets, tally languages and the behaviour of polynomial time reducibilities on NP. Math. Systems Theory, 13:55–65, 1979.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    A. L. Selman. Reductions on NP and p-selective sets. Theoretical Comput. Sci., 19:287–304, 1982.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    P. Young. On semi-cylinders, splinters, and bounded-truth-table reducibility. Trans. of the AMS, 115:329–339, 1965.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Arfst Nickelsen
    • 1
  • Till Tantau
    • 1
  1. 1.Fakultät für Elektrotechnik und InformatikTechnische Universität BerlinBerlinGermany

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