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Defying upward and downward separation

  • Lane A. Hemachandra
  • Sudhir K. Jha
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)

Abstract

Upward and downward separation results link the collapse of small and large classes, and are a standard tool in complexity theory. We study the limitations of upward and downward separation.

We show that the exponential-time limited nondeterminism hierarchy does not robustly possess downward separation. We show that probabilistic classes do not robustly possess upward separation. Though NP is known [19] to robustly possess upward separation, we show that NP does not robustly possess upward separation with respect to strong (immunity) separation. On the other hand, we provide a structural sufficient condition for upward separation.

Keywords

Turing Machine Complexity Class SIAM Journal Kolmogorov Complexity Information Processing Letter 
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.
    E. Allender. Limitations of the upward separation technique. Mathematical Systems Theory, 24(1):53–67, 1991.Google Scholar
  2. 2.
    E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521–539, 1992.Google Scholar
  3. 3.
    E. Allender and C. Wilson. Downward translations of equality. Theoretical Computer Science, 75(3):335–346, 1990.Google Scholar
  4. 4.
    K. Ambos-Spies. A note on complete problems for complexity classes. Information Processing Letters, 23:227–230, 1986.Google Scholar
  5. 5.
    T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.Google Scholar
  6. 6.
    J. Balcázar and D. Russo. Immunity and simplicity in relativizations of probabilistic complexity classes. Theoretical Informatics and Applications (RAIRO), 22(2):227–244, 1988.Google Scholar
  7. 7.
    R. Book. Tally languages and complexity classes. Information and Control, 26:186–193, 1974.Google Scholar
  8. 8.
    R. Book and K. Ko. On sets truth-table reducible to sparse sets. SIAM Journal on Computing, 17(5):903–919, 1988.Google Scholar
  9. 9.
    D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Technical Report CS-017/91, Università di Roma “La Sapienza,” Dipartimento di Matematica, Rome, Italy, Feb. 1991.Google Scholar
  10. 10.
    J. Díaz and J. Torán. Classes of bounded nondeterminism. Mathematical Systems Theory, 23(1):21–32, 1990.Google Scholar
  11. 11.
    M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.Google Scholar
  12. 12.
    W. Gasarch. Oracles for deterministic versus alternating classes. SIAM Journal on Computing, 16(4):613–627, 1987.Google Scholar
  13. 13.
    J. Geske. A note on almost-everywhere complexity, bi-immunity and nondeterministic space. In Advances in Computing and Information: Proceedings of the 1990 International Conference on Computing and Information, pages 112–116. Canadian Scholars' Press, May 1990.Google Scholar
  14. 14.
    J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, 1977.Google Scholar
  15. 15.
    Y. Gurevich. Algebras of feasible functions. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 210–214. IEEE Computer Society Press, Nov. 1983.Google Scholar
  16. 16.
    Y. Han, L. Hemachandra, and T. Thierauf. Threshold computation and cryptographic security. Technical Report TR-443, University of Rochester, Department of Computer Science, Rochester, NY, Nov. 1992.Google Scholar
  17. 17.
    J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.Google Scholar
  18. 18.
    J. Hartmanis and H. Hunt. The LBA problem and its importance in the theory of computing. SIAM-AMS Proceedings, 7:1–26, 1974.Google Scholar
  19. 19.
    J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, 65(2/3):159–181, 1985.Google Scholar
  20. 20.
    L. Hemachandra. Counting in Structural Complexity Theory. PhD thesis, Cornell University, Ithaca, NY, May 1987. Available as Cornell Department of Computer Science Technical Report TR87-840.Google Scholar
  21. 21.
    L. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299–322, 1989.Google Scholar
  22. 22.
    L. Hemachandra, S. Jain, and N. Vereshchagin. Banishing robust Turing completeness. In Proceedings of Logic at Tver '92: Symposium on Logical Foundations of Computer Science, pages 168–197. Springer-Verlag Lecture Notes in Computer Science #620, July 1992.Google Scholar
  23. 23.
    L. Hemachandta and R. Rubinstein. Separating complexity classes with tally oracles. Theoretical Computer Science, 92(2):309–318, 1992.Google Scholar
  24. 24.
    J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.Google Scholar
  25. 25.
    R. Impagliazzo and G. Tardos. Decision versus search problems in super-polynomial time. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 222–227. IEEE Computer Society Press, October/November 1989.Google Scholar
  26. 26.
    D. Johnson. A catalog of complexity classes. In J. V. Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 2, pages 67–161. MIT Press/Elsevier, 1990.Google Scholar
  27. 27.
    C. Kintala and P. Fisher. Refining nondeterminism in relativized polynomial-time bounded computations. SIAM Journal on Computing, 9(1):46–53, 1980.Google Scholar
  28. 28.
    A. Kolmogorov. Three approaches for defining the concept of information quantity. Prob. Inform. Trans., 1:1–7, 1965.Google Scholar
  29. 29.
    W. Kowalczyk. Some connections between representability of complexity classes and the power of formal reasoning systems. In Proceedings of the 11th Symposium on Mathematical Foundations of Computer Science, pages 364–369. Springer-Verlag Lecture Notes in Computer Science #176, 1984.Google Scholar
  30. 30.
    M. Li and P. Vitanyi. Applications of Kolmogorov complexity in the theory of computation. In A. Selman, editor, Complexity Theory Retrospective, pages 147–203. Springer-Verlag, 1990.Google Scholar
  31. 31.
    A. Meyer and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, pages 125–129, 1972.Google Scholar
  32. 32.
    K. Regan. Provable complexity properties and constructive reasoning. Manuscript, Apr. 1989.Google Scholar
  33. 33.
    H. Rogers, Jr. The Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.Google Scholar
  34. 34.
    A. Selman. A note on adaptive vs. nonadaptive reductions to NP. Technical Report 90-20, State University of New York at Buffalo Department of Computer Science, Buffalo, NY, Sept. 1990.Google Scholar
  35. 35.
    V. Sewelson. A Study of the Structure of NP. PhD thesis, Cornell University, Ithaca, NY, Aug. 1983. Available as Cornell Department of Computer Science Technical Report #83-575.Google Scholar
  36. 36.
    M. Sipser. On relativization and the existence of complete sets. In Proceedings of the 9th International Colloquium on Automata, Languages, and Programming. Springer-Verlag Lecture Notes in Computer Science #140, 1982.Google Scholar
  37. 37.
    L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.Google Scholar
  38. 38.
    L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23, 1976.Google Scholar
  39. 39.
    M. Zimand. On relativizations with a restricted number of accesses to the oracle set. Mathematical Systems Theory, 20:1–11, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Lane A. Hemachandra
    • 1
  • Sudhir K. Jha
    • 1
  1. 1.Department of Computer ScienceUniversity of RochesterRochester

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