Abstract
It is known that for almost every oracle set A, P(A) ≠ NP(A) and PH(A) ≠ PSPACE(A); but there are no known results that relate these facts to the P =?NP or PH =?PSPACE problems. Here the following result is shown (Theorem 6):
If for almost every oracle set A, the polynomial-time hierarchy relative to A collapses (to some finite level), then the unrelativized polynomial-time hierarchy collapses.
Also, a new characterization (Theorem 2) of classes of the form ALMOST-R = {A ¦ for almost every B, A ∃ R(B)}, where 72. is any appropriate reducibility, is established. This new characterization, The Random Oracle Characterization, is based on algorithmically random oracle sets, and shows that for any appropriate reducibility R, ALMOST-R is precisely the recursive part of R(B) for every algorithmically random language B. The paper itself is part of a development of the properties of classes of the form ALMOST-R, extending the results of Book, Lutz, and Wagner [4].
This work was supported in part by the National Science Foundation under Grants CCR-8913584 and CCR-9302057.
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© 1993 Springer-Verlag Berlin Heidelberg
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Book, R.V. (1993). Relativizing complexity classes with Random Oracles. 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_255
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DOI: https://doi.org/10.1007/3-540-57568-5_255
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