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Every polynomial-time 1-degree collapses if and only if P = PSPACE

Published online by Cambridge University Press:  12 March 2014

Stephen A. Fenner ,
Stuart A. Kurtz  and
James S. Royer
Stephen A. Fenner
Affiliation:
Dept. of Computer Science, University of South Carolina, Columbia, SC 29208, USA, E-mail: [email protected]
Stuart A. Kurtz
Affiliation:
Dept. of Computer Science, University of Chicago, 1100 E. 58TH ST., Chicago, IL 60637-1581, USA, E-mail: [email protected]
James S. Royer
Affiliation:
Dept. of Elec. Eng. and Computer Science, Syracuse University, Syracuse, NY 13244, USA, E-mail: [email protected]
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Abstract.

A set A is m-reducible (or Karp-reducible) to B if and only if there is a polynomial-time computable function f such that, for all x, x ∈ A if and only if f(x) ∈ B. Two sets are:

• 1-equivalent if and only if each is m-reducible to the other by one-one reductions;

• p-invertible equivalent if and only if each is m-reducible to the other by one-one, polynomial-time invertible reductions; and

• p-isumorphic if and only if there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible.

In this paper we show the following characterization.

Theorem. The following are equivalent:

(a) P = PSPACE.

(b) Every two 1-equivalent sets are p-isomorphic.

(c) Every two p-invertible equivalent sets are p-isomorphic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 3 , September 2004 , pp. 713 - 741
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

[Ben89]Bennett, C., Time/space trade-offs for reversible computation, SIAM Journal on Computing, vol. 18 (1989), pp. 766–776.CrossRefGoogle Scholar
[Ber77]Berman, L., Polynomial reducibilities and complete sets, Ph.D. thesis, Cornell University, 1977.Google Scholar
[BH77]Berman, L. and Hartmanis, J., On isomorphism and density of NP and other complete sets, SIAM Journal on Computing, vol. 1 (1977), pp. 305–322.CrossRefGoogle Scholar
[Ded88]Dedekind, R., Was sind und was sollen die Zahlen?, Vieweg, 1888, English translation in [Ewa96], Vol. 2.Google Scholar
[Dow82]Dowd, M., Isomorphism of complete sets, Technical Report LCSR-TR-34, Laboratory for Computer Science Research, Rutgers University, Busch Campus, 1982.Google Scholar
[Ewa96]Ewald, W. (editor), From Kant to Hilbert: A sourcebook in the foundations of mathematics, Oxford University Press, 1996, in two volumes.Google Scholar
[Fer99]Ferreirós, J., Labyrinth of thought: A history of set theory and its role in modern mathematics, Birkhäuser, 1999.CrossRefGoogle Scholar
[GH89]Ganesan, K. and Homer, S., Complete problems and strong polynomial reducibilities, Proceedings of the symposium on theoretical aspects of computer science, Springer-Verlag, 1989, pp. 240–250.Google Scholar
[GJ79]Garey, M. and Johnson, D., Computers and intractability, W. H. Freeman and Company, 1979.Google Scholar
[GS84]Grollmann, J. and Selman, A., Complexity measures for public-key cryptosystems, Proceedings of the 25th annual ieee symposium on foundations of computer science, IEEE Computer Society, 1984, pp. 495–503.Google Scholar
[GS88]Grollmann, J. and Selman, A., Complexity measures for public-key cryptosystems, SIAM Journal on Computing, vol. 17 (1988), pp. 309–335.CrossRefGoogle Scholar
[HKR93]Homer, S., Kurtz, S., and Royer, J., On many-one and 1-truth-table complete sets, Theoretical Computer Science, vol. 115 (1993), pp. 383–389.CrossRefGoogle Scholar
[Ko85]Ko, K., On some natural complete operators, Theoretical Computer Science, vol. 37 (1985), pp. 1–30.CrossRefGoogle Scholar
[KLD87]Ko, K., Long, T., and Du, D., A note on one-way functions and polynomial-time isomorphisms, Theoretical Computer Science, vol. 47 (1987), pp. 263–276.CrossRefGoogle Scholar
[KMR90]Kurtz, S., Mahaney, S., and Royer, J., The structure of complete degrees, Complexity theory retrospective (Selman, A., editor), Springer-Verlag, 1990, pp. 108–146.CrossRefGoogle Scholar
[Moo82]Moore, G., Zermelo's axiom of choice: Its origins, development, and influence, Springer-Verlag, 1982.CrossRefGoogle Scholar
[Myh55]Myhill, J., Creative sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97–108.CrossRefGoogle Scholar
[Rog67]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1967, Reprinted, MIT Press, 1987.Google Scholar