Abstract
We show that, for any set A which cannot be computed in polynomial time, the class of sets p-many-one incomparable with A has measure 1, whereas in case of p-Turing reducibility the class of sets incomparable with A has measure 1 if and only if A is not in the class BPP, the class of problems which can be probabilisticly solved with uniformly bounded error probability in polynomial time. Consequences for the reducibility relation between a randomly chosen pair of problems are discussed. Moreover, it is shown that any class, which in the relativized case collapses to P with probability one, is actually contained in BPP.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
K.Ambos-Spies, Three theorems on polynomial degrees of NP-sets, Proc. FOCS 1985.
T. Baker, J. Gill and R. Solovay, Relativizations of the P=?NP question, SIAM J. Comput. 4(1975) 431–442.
C.H. Bennett and J. Gill, Relative to a random oracle A, PA≠NPA≠co-NPA with probability 1, SIAM J. Comput. 10 (1981) 96–113.
S.A.Cook, The complexity of theorem proving procedures, Proc. 3rd STOC 1971, 151–158.
W. Feller, An Introduction to Probability Theory and its Applications, John Wiley, New York, 1957.
J. Gill, Computational complexity of probabilistic Turing machines, SIAM J. Comput. 6 (1977) 675–695.
J. Hartmanis, Solvable problems with conflicting relativizations, Bull. EATCS 27, October 1985.
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Reading, MA., 1979.
R.M. Karp, Reducibility among combinatorial problems, in “Complexity of Computer Computations” (R.E. Miller and J.W. Thatcher, Eds.), Plenum, New York, 1972, 85–103.
S.A.Kurtz, On the random oracle hypothesis, Proc. 14th STOC 1982.
R.E. Ladner, On the structure of polynomial time reducibility, J. ACM 22 (1975) 155–171.
L.H. Landweber, R.J. Lipton and E.L. Robertson, On the structure of sets in NP and other complexity classes, Theor. Comp. Sci. 15 (1981) 181–200.
K.Mehlhorn, The “almost all” theory of subrecursive degrees is decidable, Tech. Rep. TR-73-170, 1973, Cornell University.
K.Mehlhorn, The “almost all” theory of subrecursive degrees is decidable, Proc. 2nd ICALP, Lecture Notes in Comp. Sci. 15 (1974), Springer Verlag.
K. Mehlhorn, Polynomial and abstract subrecursive classes, J. Comput. Sytem Sci. 12 (1976) 147–178.
P.Orponen, Complexity of alternating machines with oracles, Proc. 10th ICALP, Lecture Notes in Comp. Sci. 154 (1983), Springer Verlag.
H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw Hill, New York, 1967.
G.E. Sacks, Degrees of Unsolvability, Second edition,Annals of Mathematics Studies Number 55, Princeton University Press, 1966.
U. Schöning, Minimal pairs for P, Theor. Comp. Sci. 31 (1984) 41–48.
R.I.Soare, Recursively Enumerable Sets and Degrees: The Study of Computable Functions and Computably Generated Sets, Springer Verlag (to appear).
J.Stillwell, Decidability of the “almost all” theory of degrees, J. Symbolic Logic, 1972.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ambos-Spies, K. (1986). Randomness, relativizations, and polynomial reducibilities. In: Selman, A.L. (eds) Structure in Complexity Theory. Lecture Notes in Computer Science, vol 223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16486-3_87
Download citation
DOI: https://doi.org/10.1007/3-540-16486-3_87
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16486-9
Online ISBN: 978-3-540-39825-7
eBook Packages: Springer Book Archive