Abstract
Threshold machines [21] are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines [11] are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. Simon [21] proved that for unbounded-error polynomial-time machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored.
In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class, BPPpath, and, more generally, we study the structural properties of BPPpath. We prove that BPPpath contains both NPBPP and PNP[log], and that BPPpath is contained in \(P^{\Sigma _2^p [\log ]}\), BPPNP, and PP. We conclude that, unless the polynomial hierarchy collapses, bounded-error threshold computation is strictly more powerful than bounded-error probabilistic computation.
We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length [5]. We show, for both BPP and BPPpath, that if there is any database for which this formalization of security differs from the security given by oblivious [9] database access, then BPP≠PP. It follows that if any set lacking small circuits can be securely accepted, then BPP≠PP.
Research supported in part by grants NSF-CCR-8957604, NSF-CCR-9057486, and NSF-INT-9116781/JSPS-ENG-207, and by DFG Postdoctoral Stipend Th 472/1-1.
Work done while visiting the University of Rochester and Princeton University.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
M. Abadi, J. Feigenbaum, and J. Kilian. On hiding information from an oracle. Journal of Computer and System Sciences, 39:21–50, 1989.
L. Babai. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on Theory of Computing, pages 421–429, April 1985.
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.
J. Balcázar, R. Book, and U. Schöning. The polynomial-time hierarchy and sparse oracles. Journal of the ACM, 33(3):603–617, 1986.
D. Beaver and J. Feigenbaum. Hiding instances in multioracle queries. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, pages 37–48. Springer-Verlag Lecture Notes in Computer Science #415, 1990.
R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 14–19. IEEE Computer Society Press, June 1992.
R. Book. Restricted relativizations of complexity classes. In J. Hartmanis, editor, Computational Complexity Theory, Proceedings of Symposia in Applied Mathematics #38, pages 47–74. American Mathematical Society, 1989.
R. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25:127–132, 1987.
J. Feigenbaum, L. Fortnow, C. Lund, and D. Spielman. The power of adaptiveness and additional queries in random-self-reductions. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 338–346. IEEE Computer Society Press, June 1992.
S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 30–42. IEEE Computer Society Press, June/July 1991.
J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, 1977.
Y. Han, L. Hemachandra, and T. Thierauf. Threshold computation and cryptographic security. Technical Report TR-461, University of Rochester, Department of Computer Science, Rochester, NY, 1993.
J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.
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, April 1980.
K. Ko. Some observations on the probabilistic algorithms and NP-hard problems. Information Processing Letters, 14(1):39–43, 1982.
J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272–286, 1992.
R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities. Theoretical Computer Science, 1(2):103–124, 1975.
T. Long and A. Selman. Relativizing complexity classes with sparse oracles. Journal of the ACM, 33(3):618–627, 1986.
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.
M. Ogiwara and L. Hemachandra. A complexity theory for feasible closure properties. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 16–29. IEEE Computer Society Press, June/July 1991.
J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell Univeristy, Ithaca, N.Y., January 1975. Available as Cornell Department of Computer Science Technical Report TR75-224.
M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330–335, 1983.
L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.
N. Vereshchagin. On the power of PP. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 138–143. IEEE Computer Society Press, June 1992.
S. Zachos. Robustness of probabilistic complexity classes under definitional perturbations. Information and Control, 54:143–154, 1982.
S. Zachos. Probabilistic quantifiers and games. Journal of Computer and System Sciences, 36:433–451, 1988.
S. Zachos and H. Heller. A decisive characterization of BPP. Information and Control, 69:125–135, 1986.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Han, Y., Hemaspaandra, L.A., Thierauf, T. (1993). Threshold computation and cryptographic security. In: Ng, K.W., Raghavan, P., Balasubramanian, N.V., Chin, F.Y.L. (eds) Algorithms and Computation. ISAAC 1993. Lecture Notes in Computer Science, vol 762. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57568-5_253
Download citation
DOI: https://doi.org/10.1007/3-540-57568-5_253
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57568-9
Online ISBN: 978-3-540-48233-8
eBook Packages: Springer Book Archive