Abstract
The counting class C=P, which captures the notion of “exact counting”, while extremely powerful under various nondeterministic reductions, is quite weak under polynomial-time deterministic reductions. We discuss the analogies between NP and co-C=P, which allow us to derive many interesting results for such deterministic reductions to co-C=P. We exploit these results to obtain some interesting oracle separations. Most importantly, we show that there exists an oracleA such that\( \oplus P^A \nsubseteq P^{C_ = P^A } \) and\(BPP^A \nsubseteq P^{C_ = P^A } \) Therefore, techniques that would prove that C=P and PP are polynomial-time Turing equivalent, or that C=P is polynomial-time Turing hard for the polynomial-time hierarchy, would not relativize.
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References
L. Babai, E-Mail and the Unexpected Power of Interaction,Proceedings of the 5th Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1990, pp. 30–44.
J. L. Balcázar, J. Diaz, and J. Gabarró,Structural Complexity Theory I, EATCS Monographs on Theoretical Computer Science, Vol. II, Springer-Verlag, New York, 1988.
R. Beigel, Bounded Queries to SAT and the Boolean Hierarchy, Johns Hopkins Technical Report 87-8 (1988), to appear inTheoretical Computer Science.
R. Beigel, R. Chang, and M. Ogiwara, A Relationship Between Difference Hierarchies and Relativized Polynomial Hierarchies, to appear inMathematical Systems Theory.
R. Beigel, N. Reingold, and D. Spielman, PP is Closed Under Intersection,Proceedings of the 23rd Annual Symposium on Theory of Computing, ACM Press, New York, 1991, pp. 1–9.
A. Bertoni, D. Brushi, D. Joseph, M. Sitharam, and P. Young, Generalized Boolean Hierarchies and Boolean Hierarchies over RP,Proceedings of the 7th Conference on Fundamentals of Computation Theory, Lecture Notes in Computer Science, Vol. 380, Springer-Verlag, Berlin, 1989.
S. R. Buss and L. Hay, On Truth-Table Reducibility to SAT and the Difference Hierarchy Over NP,Proceedings of the 3rd Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1988, pp. 224–233.
J.-y. Cai, Probability One Separation of the Boolean Hierarchy,Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 247, Springer-Verlag, Berlin, 1987, pp. 148–158.
J.-y. Cai, T. Gunderman, J. Hartmanis, L. A. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung, The Boolean Hierarchy I: Structural Properties,SIAM Journal of Computing,17 (1988), 1232–1252.
R. Chang and J. Kadin, The Boolean Hierarchy and the Polynomial Hierarchy: a Closer Connection,Proceedings of the 5th Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1990, pp. 169–178.
T. Gundermann, N. A. Nasser, and G. Wechsung, A Survey on Counting Classes,Proceedings of the 5th Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1990, pp. 140–153.
L. A. Hemachandra, The Strong Exponential Hierarchy Collapses,Journal of Computer and System Science,39 (1989), 299–322.
J. Kadin, Restricted Turing Reducibilities and the Structure of the Polynomial Time Hierarchy, Ph.D. thesis, Cornell University, February 1988.
C. Lautenmann, BPP and the Polynomial Hierarchy,Information Processing Letters,17 (1983), 215–217.
M. Ogiwara, Generalized Theorems on Relationships Among Reducibility Notions to Certain Complexity Classes, Manuscript, April 1991.
J. Tarui, Randomized Polynomials, Threshold Circuits, and the Polynomial Hierarchy,Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 480, Springer-Verlag, Berlin, 1991, pp. 238–250.
J. Tarui, Degree Complexity of Boolean Functions and Its Applications to Relativized Separations,Proceedings of the 6th Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1991, pp. 382–390.
S. Toda, On the computational power of PP and ⊕ P,Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, 1989, pp. 514–519.
S. Toda, On Polynomial-Time Truth-Table Reducibility to C = P Sets, Colloquium, Department of Computer Science, University of Chicago, October 26, 1990.
S. Toda and M. Ogiwara, Counting Classes Are at Least as Hard as the Polynomial-Time Hierarchy,Proceedings of the 6th Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1991, pp. 2–12.
J. Torán, Structural Properties of the Counting Hierarchy, Ph.D. thesis, Facultat d'lnformatica de Barcelona, 1988.
J. Torán, An Oracle Characterization of the Counting Hierarchy,Proceedings of the 3rd Annual Conference on Structure in Complexity Theory, IEEE Computer Society Press, New York, 1988, pp. 213–223.
K. Wagner, Compact Descriptions and the Counting Polynomial Time Hierarchy,Acta Informatica,23 (1986), 325–356.
K. Wagner, Bounded Query Classes,SIAM Journal of Computing,19 (1990), 833–846.
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This research was supported by a grant from the Directión General de Investigation Cientifica y Técnica (DGICYT), Spanish Ministry of Education, while the author was vising the Facultat d'lnformàtica, Universitat Politècnica de Catalunya, Barcelona.
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Green, F. On the power of deterministic reductions to C=P. Math. Systems Theory 26, 215–233 (1993). https://doi.org/10.1007/BF01202284
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DOI: https://doi.org/10.1007/BF01202284