article

Hausdorff dimension and oracle constructions

Author:

John M. HitchcockAuthors Info & Claims

Theoretical Computer Science, Volume 355, Issue 3

Pages 382 - 388

https://doi.org/10.1016/j.tcs.2006.01.025

Published: 14 April 2006 Publication History

Abstract

Bennett and Gill [Relative to a random oracle A, P^A ≠ NP^A ≠ co-NP^A with probability 1, SIAM J. Comput. 10 (1981) 96-113] proved that P^A ≠ NP^A relative to a random oracle A, or in other words, that the set O_[P=NP] = {A | P^A = NP^A} has Lebesgue measure 0. In contrast, we show that O_[P=NP] has Hausdorff dimension 1.This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Φ, then the set of oracles relative to which Φ holds has Hausdorff dimension 1.We give several other applications including proofs that the polynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that P^A ≠ NP^A ∩ coNP^A relative to a Hausdorff dimension 1 set of oracles.

References

[1]

[1] C.H. Bennett, J. Gill, Relative to a random oracle A, P^A ≠ NP^A ≠ co-NP^A with probability 1, SIAM J. Comput. 10 (1981) 96-113.

Digital Library

[2]

[2] R.V. Book, On collapsing the polynomial-time hierarchy, Inform. Process. Lett. 52 (5) (1994) 235-237.

Digital Library

[3]

[3] R. Chang, B. Chor, O. Goldreich, J. Hartmanis, J. Håstad, D. Ranjan, R. Rohatgi, The random oracle hypothesis is false, J. Comput. System Sci. 49 (1) (1994) 24-39.

Digital Library

[4]

[4] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990.

[5]

[5] L. Fortnow, Relativized worlds with an infinite hierarchy, Inform. Process. Lett. 69 (6) (1999) 309-313.

Digital Library

[6]

[6] J. Håstad, Computational Limitations for Small-Depth Circuits, The MIT Press, Cambridge, MA, 1986.

Digital Library

[7]

[7] F. Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1919) 157-179.

[8]

[8] H. Heller, On relativized polynomial and exponential computations, SIAM J. Comput. 13 (4) (1984) 717-725.

Digital Library

[9]

[9] H. Heller, On relativized exponential and probabilistic complexity classes, Inform. and Control 71 (1986) 231-243.

Digital Library

[10]

[10] J.M. Hitchcock, Fractal dimension and logarithmic loss unpredictability, Theoret. Comput. Sci. 304 (1-3) (2003) 431-441.

Digital Library

[11]

[11] J.M. Hitchcock, J.H. Lutz, E. Mayordomo, Scaled dimension and nonuniform complexity, J. Comput. System Sci. 69 (2) (2004) 97-122.

Digital Library

[12]

[12] C. Lund, L. Fortnow, H.J. Karloff, N. Nisan, Algebraic methods for interactive proof systems, J. ACM 39 (4) (1992) 859-868.

Digital Library

[13]

[13] J.H. Lutz, Dimension in complexity classes, SIAM J. Comput. 32(5)(2003) 1236-1259.

Digital Library

[14]

[14] A. Shamir, IP = PSPACE, J. ACM 39 (4) (1992) 869-877.

Digital Library

[15]

[15] A. Yao, Separating the polynomial-time hierarchy by oracles, in: 26th IEEE Symp. on Foundations of Computer Science, IEEE Computer Society Press, Silver Spring, MD, 1985, pp. 1-10.

Digital Library

Cited By

Hitchcock JSekoni AShafei H(2021)Polynomial-Time Random Oracles and Separating Complexity ClassesACM Transactions on Computation Theory10.1145/343438913:1(11-16)Online publication date: 21-Jan-2021
https://dl.acm.org/doi/10.1145/3434389
Gu XLutz J(2006)Dimension characterizations of complexity classesProceedings of the 31st international conference on Mathematical Foundations of Computer Science10.1007/11821069_41(471-479)Online publication date: 28-Aug-2006
https://dl.acm.org/doi/10.1007/11821069_41

Index Terms

Hausdorff dimension and oracle constructions
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Design and analysis of algorithms

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cover image Theoretical Computer Science

Theoretical Computer Science Volume 355, Issue 3

14 April 2006

135 pages

ISSN:0304-3975

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Elsevier Science Publishers Ltd.

United Kingdom

Publication History

Published: 14 April 2006

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Cited By

Hitchcock JSekoni AShafei H(2021)Polynomial-Time Random Oracles and Separating Complexity ClassesACM Transactions on Computation Theory10.1145/343438913:1(11-16)Online publication date: 21-Jan-2021
https://dl.acm.org/doi/10.1145/3434389
Gu XLutz J(2006)Dimension characterizations of complexity classesProceedings of the 31st international conference on Mathematical Foundations of Computer Science10.1007/11821069_41(471-479)Online publication date: 28-Aug-2006
https://dl.acm.org/doi/10.1007/11821069_41

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