Abstract
Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Löf. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to the following result:
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(i)
P = NP if and only if there exists an algorithmically random set that is ≤ Pbtt -hard for NP.
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(ii)
P = PSPACE if and only if there exists an algorithmically random set that is ≤ Pbtt -hard for PSPACE.
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(iii)
The polynomial-time hierarchy collapses if and only if there exists k>0 such that some algorithmically random set is σ Pk -hard for PH.
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(iv)
PH = PSPACE if and only if there exists a algorithmically random set that is PH-hard for PSPACE.
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References
K. Ambos-Spies. Randomness, relativations, and polynomial reducibilities. In Lecture Notes in Computer Sci. 223, pages 23–34. Proc. 1st Conf. Stucture in Complexity Theory, Springer-Verlag, 1986.
J. Balcázar, R. Book, and U. Schöning. The polynomial-time hierarchy and sparse oracles. J. Assoc. Comput. Mach., 33:603–617, 1986.
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer-Verlag, 1988.
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity II. Springer-Verlag, 1990.
C. Bennett. Logical depth and physical complexity. In R. Herken (ed.), The Universal Turing Machine: A Half-Century Survey, pages 227–257. Oxford University Press, 1988.
C. Bennett and J. Gill. Relative to a random oracle PA ≠ NPA ≠ co-NPA with probability 1. SIAM J. Computing, 10:96–113, 1981.
J.-Y. Cai. Probability one separation of the boolean hierarchy. In Lecture Notes in Computer Sci. 38, pages 148–158. STACS 87, Springer Verlag, 1987.
J.-Y. Cai. With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy. J. Comput. Systems Sci., 38:68–85, 1989.
R. Karp and R. Lipton. Turing machines, that take advice. L'Enseignement Mathématique, 28 2nd series:191–209, 1982.
T. Long and A. Selman. Relativizing complexity classes with sparse oracles. J. Assoc. Comput. Mach., 33:618–627, 1986.
P. Martin-Löf. On the definition of random sequences. Info. and Control, 9:602–619, 1966.
P. Martin-Löf. Complexity oscillations in infinite binary sequences. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 19:225–230, 1971.
M. Ogiwara and A. Lozano. On one query self reducible sets. In Proc. 6th IEEE Conference on Structure in Complexity Theory, pages 139–151, 1991.
M. Ogiwara and 0. Watanabe. On polynomial bounded truth table reducibility of NP sets to sparse sets. SIAM J. Computing, 20:471–483, 1991.
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© 1992 Springer-Verlag Berlin Heidelberg
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Book, R.V., Lutz, J.H., Wagner, K.W. (1992). On complexity classes and algorithmically random languages. In: Finkel, A., Jantzen, M. (eds) STACS 92. STACS 1992. Lecture Notes in Computer Science, vol 577. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55210-3_193
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DOI: https://doi.org/10.1007/3-540-55210-3_193
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