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International Computing and Combinatorics Conference

COCOON 2001: Computing and Combinatorics pp 49–58Cite as

Algebraic Properties for P-Selectivity

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2108))

Abstract

Karp and Lipton, in their seminal 1980 paper, introduced the notion of advice (nonuniform) complexity, which since has been of central importance in complexity theory. Nonetheless, much remains unknown about the optimal advice complexity of classes having polynomial advice complexity.

In particular, let P-sel denote the class of all P-selective sets [23] For the nondeterministic advice complexity of P-sel, linear upper and lower bounds are known [10]. However, for the deterministic advice complexity of P-sel, the best known upper bound is quadratic [13], and the best known lower bound is the linear lower bound inherited from the nondeterministic case. This paper establishes an algebraic sufficient condition for P-sel to have a linear upper bound: If all P-selective sets are associatively P-selective then the deterministic advice complexity of P-sel is linear. (The weakest previously known sufficient condition was P = NP.) Relatedly, we prove that every associatively P-selective set is commutatively, associatively P-selective.

Supported in part by grants NSF-CCR-9322513 and NSF-INT-9815095/DAAD-315-PPP-gü-ab.

Supported in part by grant NSF-INT-9815095/DAAD-315-PPP-gü-ab. Work donewhile visiting the University of Rochester under a NATO Postdoctoral Science Fellowship from the DAAD’s “Gemeinsames Hochschulsonderprogramm III von Bund und Ländern” program.

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References

  1. E. Allender and L. Hemachandra Lower bounds for the low hierarchy. Journal of the ACM, 39(1):234–251, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Balcázar and U. Schöning Logarithmic advice classes. Theoretical Computer Science, 99(2):279–290, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Hartmanis and Y. Yesha Computation times of NP sets of different densities. Theoretical Computer Science, 34:17–32, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman P-selective sets and reducing search to decision vs. self-reducibility. Journal of Computer and System Sciences, 53(2):194–209, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Hemaspaandra, H. Hempel, and A. Nickelsen Algebraic properties for deterministic and nondeterministic selectivity. In Preparation.

    Google Scholar 

  6. L. Hemaspaandra, H. Hempel, and A. Nickelsen Algebraic properties for deterministic selectivity. Technical Report TR-746, Department of Computer Science, University of Rochester, Rochester, NY, April 2001.

    Google Scholar 

  7. L. Hemaspaandra, A. Hoene, A. Naik, M. Ogiwara, A. Selman, T. Thierauf, and J. Wang Nondeterministically selective sets. International Journal of Foundations of Computer Science, 6(4):403–416, 1995.

    Article  MATH  Google Scholar 

  8. L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman Computing solutions uniquely collapses the polynomial hierarchy. SIAM Journal on Computing, 25(4):697–708, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Hemaspaandra, M. Ogihara, and G. Wechsung Reducing the number of solutions of NP functions. In Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science, pages 394–404. Springer-Verlag Lecture Notes in Computer Science #1893, August/September 2000.

    Google Scholar 

  10. L. Hemaspaandra and L. Torenvliet Optimal advice. Theoretical Computer Science, 154(2):367–377, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Hemaspaandra, M. Zaki, and M. Zimand Polynomial-time semi-rankable sets. In Journal of Computing and Information, 2(1), Special Issue: Proceedings of the 8th International Conference on Computing and Information, pages 50–67, 1996. CD-ROM ISSN 1201-8511/V2/#1.

    MathSciNet  Google Scholar 

  12. R. Karp and R. Lipton Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302-309. ACM Press, April 1980. An extended version has also appeared as: Turing machines that take advice, L’Enseignement Mathématique, 2nd series, 28, 1982, pages 191–209.

    MATH  MathSciNet  Google Scholar 

  13. K. Ko On self-reducibility and weak P-selectivity. Journal of Computer and System Sciences, 26:209–221, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Ko On helping by robust oracle machines. Theoretical Computer Science, 52:15–36, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Ko and U. Schöning On circuit-size complexity and the low hierarchy in NP. SIAM Journal on Computing, 14(1):41–51, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Köbler and O. Watanabe New collapse consequences of NP having small circuits. SIAM Journal on Computing, 28(1):311–324, 1998.

    Article  MATH  Google Scholar 

  17. J. Moon Topics on Tournaments. Holt, Rinehart, and Winston, 1968.

    Google Scholar 

  18. A. Naik, J. Rogers, J. Royer, and A. Selman A hierarchy based on output multiplicity. Theoretical Computer Science, 207(1):131–157, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Naik and A. Selman Adaptive versus nonadaptive queries to NP and P-selective sets. Computational Complexity, 8(2):169–187, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Nickelsen Polynomial-Time Partial Information Classes. PhD thesis, Technische Universität Berlin, Berlin, Germany, 2000.

    Google Scholar 

  21. M. Ogihara Functions computable with limited access to NP. Information Processing Letters, 58:35–38, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  22. U. Schöning A low and a high hierarchy within NP. Journal of Computer and System Sciences, 27:14–28, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Selman P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55–65, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Selman Some observations on NP real numbers and P-selective sets. Journal of Computer and System Sciences, 23:326–332, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Selman Analogues of semirecursive sets and effective reducibilities to the study of NP complexity. Information and Control, 52:36–51, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Selman Reductions on NP and P-selective sets. Theoretical Computer Science, 19(3):287–304, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  27. Till Tantau, March 2001. Personal Communication.

    Google Scholar 

  28. S. Toda On polynomial-time truth-table reducibilities of intractable sets to Pselective sets. Mathematical Systems Theory, 24:69–82, 1991.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Rochester, Rochester, NY, 14627-0226, USA

    Lane A. Hemaspaandra

  2. Institut für Informatik, Friedrich-Schiller-Universität Jena, D-07743, Jena, Germany

    Harald Hempel⋆

  3. Fachbereich Informatik, Technische Universität Berlin, D-10587, Berlin, Germany

    Arfst Nickelsen

Authors
  1. Lane A. Hemaspaandra
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  2. Harald Hempel⋆
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  3. Arfst Nickelsen
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Massachusetts, One University Avenue, Lowell, MA, 01854, USA

    Jie Wang

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© 2001 Springer-Verlag Berlin Heidelberg

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Hemaspaandra, L.A., Hempel⋆, H., Nickelsen, A. (2001). Algebraic Properties for P-Selectivity. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_6

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  • DOI: https://doi.org/10.1007/3-540-44679-6_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42494-9

  • Online ISBN: 978-3-540-44679-8

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