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Combining Self-reducibility and Partial Information Algorithms

  • André Hernich
  • Arfst Nickelsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

A partial information algorithm for a language A computes, for some fixed m, for input words x 1, ..., x m a set of bitstrings containing χ A (x 1,...,x m ). E.g., p-selective, approximable, and easily countable languages are defined by the existence of polynomial-time partial information algorithms of specific type. Self-reducible languages, for different types of self-reductions, form subclasses of PSPACE.

For a self-reducible language A, the existence of a partial information algorithm sometimes helps to place A into some subclass of PSPACE. The most prominent known result in this respect is: P-selective languages which are self-reducible are in P [9].

Closely related is the fact that the existence of a partial information algorithm for A simplifies the type of reductions or self-reductions to A. The most prominent known result in this respect is: Turing reductions to easily countable languages simplify to truth-table reductions[8].

We prove new results of this type. We show:
  1. 1

    Self-reducible languages which are easily 2-countable are in P. This partially confirms a conjecture of [8].

     
  2. 2

    Self-reducible languages which are (2m – 1,m)-verbose are truth-table self-reducible. This generalizes the result of [9] for p-selective languages, which are (m + 1,m)-verbose.

     
  3. 3

    Self-reducible languages, where the language and its complement are strongly 2-membership comparable, are in P. This generalizes the corresponding result for p-selective languages of [9].

     
  4. 4

    Disjunctively truth-table self-reducible languages which are 2-membership comparable are in UP.

     

Topic: Structural complexity.

Keywords

Normal Form Polynomial Time Boolean Function Leaf Node Partial Information 
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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References

  1. 1.
    Agrawal, M., Arvind, V.: Polynomial time truth-table reductions to P-selective sets. In: Proc. 9th Structure in Complexity Theory (1994)Google Scholar
  2. 2.
    Amir, A., Beigel, R., Gasarch, W.: Some connections between bounded query classes and non-uniform complexity. In: Proc. 5th Struct. in Complexity Th. (1990)Google Scholar
  3. 3.
    Amir, A., Gasarch, W.: Polynomial terse sets. Inf. and Computation 77 (1988)Google Scholar
  4. 4.
    Bab, S., Nickelsen, A.: One query reducibilities between partial information classes. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 404–415. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Beigel, R.: Query-Limited Reducibilities. PhD thesis, Stanford University (1987)Google Scholar
  6. 6.
    Beigel, R., Fortnow, L., Pavan, A.: Membership comparable and p-selective sets. Technical Report 2002-006N, NEC Research Institute (2002)Google Scholar
  7. 7.
    Beigel, R., Kummer, M., Stephan, F.: Quantifying the amount of verboseness. In: Nerode, A., Taitslin, M.A. (eds.) LFCS 1992. LNCS, vol. 620, Springer, Heidelberg (1992)CrossRefGoogle Scholar
  8. 8.
    Beigel, R., Kummer, M., Stephan, F.: Approximable sets. Inf. and Computation 120(2) (1995)Google Scholar
  9. 9.
    Buhrman, van Helden, Torenvliet: P-selective self-reducible sets: A new characterization of P. In: Conference on Computational Complexity, vol. 8 (1993)Google Scholar
  10. 10.
    Buhrman, H., Torenvliet, L., van Emde Boas, P.: Twenty questions to a P-selector. Information Processing Letters 4(4), 201–204 (1993)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Du, D.-Z., Ko, K.I.: Theory of Computational Complexity. Wiley, Chichester (2000)zbMATHGoogle Scholar
  12. 12.
    Gasarch, W.: Bounded queries in recursion theory: A survey. In: Proc. 6th Structure in Complexity Theory (1991)Google Scholar
  13. 13.
    Goldsmith, J., Joseph, D., Young, P.: Self-reducible, P-selective, near-testable, and P-cheatable sets: The effect of internal structure on the complexity of a set. In: 2nd Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos (1987)Google Scholar
  14. 14.
    Hemaspaandra, L., Hoene, A., Ogihara, M.: Reducibility classes of p-selective sets. tcs 155, 447–457 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hemaspaandra, L., Jiang, Z., Rothe, J., Watanabe, O.: Polynomial-time multi-selectivity. J. of Universal Comput. Sci. 3(3) (1997)Google Scholar
  16. 16.
    Hemaspaandra, L., Torenvliet, T.: Theory of semi-feasible Alg. Springer, Heidelberg (2002)Google Scholar
  17. 17.
    Hinrichs, M., Wechsung, G.: Time bounded frequency computations. In: Proc. 12th Conf. on Computational Complexity (1997)Google Scholar
  18. 18.
    Hoene, A., Nickelsen, A.: Counting, selecting, and sorting by query-bounded machines. In: Enjalbert, P., Wagner, K.W., Finkel, A. (eds.) STACS 1993. LNCS, vol. 665, Springer, Heidelberg (1993)Google Scholar
  19. 19.
    Ko, K.: On self-reducibility and weak p-selectivity. Journal of Computer and System Sciences 26, 209–221 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Köbler, J.: On the structure of low sets. In: Proc. 10th Structure in Complexity Theory, pp. 246–261. IEEE Computer Society Press, Los Alamitos (1995)CrossRefGoogle Scholar
  21. 21.
    Meyer, A.R., Paterson, M.S.: With what frequency are apparently intractabel problems difficult? Technical Report MIT/LCS/TM-126, MIT, Cambridge (1979)Google Scholar
  22. 22.
    Nickelsen, A.: On polynomially \(\mathcal{D}\)-verbose sets. In: Proc. STACS 1997. LNCS, vol. 1200, pp. 307–318. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Nickelsen, A.: Partial information and special case algorithms. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 573–584. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. 24.
    Nickelsen, A.: Polynomial Time Partial Information Classes. W&T Verlag, 2001. Dissertation, TU Berlin (1999), also available at http://www.tal.cs.tu-berlin.de/nickelsen/
  25. 25.
    Nickelsen, A., Tantau, T.: Closure of polynomial time partial information classes under polynomial time reductions. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 299–310. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  26. 26.
    Nickelsen, A., Tantau, T.: Partial information classes. Complexity Theory Column, SIGACT News 34 (2003)Google Scholar
  27. 27.
    Ogihara, M.: Polynomial-time membership comparable sets. sicomp 24(5), 1068–1081 (1995)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Selman, A.: P-selective sets, tally languages and the behaviour of polynomial time reducibilities on NP. Math. Systems Theory 13, 55–65 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sivakumar, D.: On membership comparable sets. Journal of Computer and System Sciences 59(2), 270–280 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Toda, S.: On polynomial-time truth-table reducibility of intractable sets to p-selective sets. Mathematical Systems Theory 24, 69–82 (1991)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • André Hernich
    • 1
  • Arfst Nickelsen
    • 2
  1. 1.Humboldt-Universität zu BerlinGermany
  2. 2.Technische Universität BerlinGermany

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