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.
Research supported in part by the National Science Foundation under grant CCR-8957604. A full version is available upon request to lane@cs.rochester.edu.
Work done in part while visiting Friedrich Schiller Universität Jena and Universität Ulm.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
E. Allender. Limitations of the upward separation technique. Mathematical Systems Theory, 24(1):53–67, 1991.
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.
E. Allender and C. Wilson. Downward translations of equality. Theoretical Computer Science, 75(3):335–346, 1990.
K. Ambos-Spies. A note on complete problems for complexity classes. Information Processing Letters, 23:227–230, 1986.
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.
J. Balcázar and D. Russo. Immunity and simplicity in relativizations of probabilistic complexity classes. Theoretical Informatics and Applications (RAIRO), 22(2):227–244, 1988.
R. Book. Tally languages and complexity classes. Information and Control, 26:186–193, 1974.
R. Book and K. Ko. On sets truth-table reducible to sparse sets. SIAM Journal on Computing, 17(5):903–919, 1988.
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.
J. Díaz and J. Torán. Classes of bounded nondeterminism. Mathematical Systems Theory, 23(1):21–32, 1990.
M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
W. Gasarch. Oracles for deterministic versus alternating classes. SIAM Journal on Computing, 16(4):613–627, 1987.
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.
J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, 1977.
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.
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.
J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.
J. Hartmanis and H. Hunt. The LBA problem and its importance in the theory of computing. SIAM-AMS Proceedings, 7:1–26, 1974.
J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, 65(2/3):159–181, 1985.
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.
L. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299–322, 1989.
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.
L. Hemachandta and R. Rubinstein. Separating complexity classes with tally oracles. Theoretical Computer Science, 92(2):309–318, 1992.
J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.
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.
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.
C. Kintala and P. Fisher. Refining nondeterminism in relativized polynomial-time bounded computations. SIAM Journal on Computing, 9(1):46–53, 1980.
A. Kolmogorov. Three approaches for defining the concept of information quantity. Prob. Inform. Trans., 1:1–7, 1965.
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.
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.
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.
K. Regan. Provable complexity properties and constructive reasoning. Manuscript, Apr. 1989.
H. Rogers, Jr. The Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.
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.
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.
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.
L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.
L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23, 1976.
M. Zimand. On relativizations with a restricted number of accesses to the oracle set. Mathematical Systems Theory, 20:1–11, 1987.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hemachandra, L.A., Jha, S.K. (1993). Defying upward and downward separation. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_21
Download citation
DOI: https://doi.org/10.1007/3-540-56503-5_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56503-1
Online ISBN: 978-3-540-47574-3
eBook Packages: Springer Book Archive