S. Toda. Restricted relativizations of probabilistic polynomial time. Theoretical Computer Science, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is at least as large as an odd number of the elements of X . Since x is the number of accepting paths of a nondeterministic, polynomial time bounded Turing machine, this is a polynomial time parity reduction to a PP language. Related Work Corollary 3 can be obtained as a corollary of Toda s [18] independent work on restricted relativization. Corollary 4 is due to Gundermann, Nasser, and Wechsung [6] Both of those corollaries also follow from Beigel, Reingold, and Spielman s (personal communication, 1990) very recent discovery that P PP[log ] PP. Corollary 5 is related to a result of ....

S. Toda. Restricted relativizations of probabilistic polynomial time. Theoretical Computer Science, 1991.

....but equivalent to, the usual definition; see Section 2) Gill noted that PP is closed under complementation, but stated that it was not known if PP is closed under intersection and union. Since Gill s paper, PP and related counting classes have been studied extensively by numerous researchers [2, 8, 16, 19, 25, 28, 29, 30, 31], though few closure properties have been shown for the class. In 1985 Russo [25] showed that the symmetric difference of two sets in PP is also in PP, and in 1991 Beigel, Hemachandra, The authors may be reached by writing to Department of Computer Science, P.O. Box 2158, New Haven, CT ....

....reductions. That is, P O(log n) T = PP. Proof: Every polynomial time O (log n) Turing reduction can be converted to a polynomial time depth 2 Boolean formula reduction (write the reduction as a CNF or DNF formula over the query answers) It was previously known that P O(logn) T PP [8, 29], and that P C=P O(log n) T PP [16] A language L is in C=P if there is a nondeterministic Turing machine N such that for all inputs X, X 2 L if and only if Gap(N;X) 0. Definition 18. A polynomial time threshold reduction is a polynomial time truthtable reduction in which the ....

Seinosuke Toda. Restricted relativizations of probabilistic polynomial time. Theoretical Computer Science, 1991. To appear.

....is at least as large as an odd number of the elements of X 0 . Since x is the number of accepting paths of a nondeterministic, polynomial time bounded Turing machine, this is a polynomial time parity reduction to a PP language. Related Work Corollary 3 can be obtained as a corollary of Toda s [18] independent work on restricted relativization. Corollary 4 is due to Gundermann, Nasser, and Wechsung [6] Both of those corollaries also follow from Beigel, Reingold, and Spielman s (personal communication, 1990) very recent discovery that P PP[log ] PP. Corollary 5 is related to a result of ....

S. Toda. Restricted relativizations of probabilistic polynomial time. Theoretical Computer Science, 1991.

....but equivalent to, the usual definition; see Section 2) Gill noted that PP is closed under complementation, but stated that it was not known if PP is closed under intersection and union. Since Gill s paper, PP and related counting classes have been studied extensively by numerous researchers [2, 8, 16, 19, 25, 28, 29, 30, 31], though few closure properties have been shown for the class. In 1985 Russo [25] showed that the symmetric difference of two sets in PP is also in PP, and in 1991 Beigel, Hemachandra, The authors may be reached by writing to Department of Computer Science, P.O. Box 2158, New Haven, CT ....

....reductions. That is, P PP O(log n) T = PP. Proof: Every polynomial time O (log n) Turing reduction can be converted to a polynomial time depth 2 Boolean formula reduction (write the reduction as a CNF or DNF formula over the query answers) It was previously known that P NP O(logn) T PP [8, 29], and that P C=P O(log n) T PP [16] A language L is in C=P if there is a nondeterministic Turing machine N such that for all inputs X, X 2 L if and only if Gap(N;X) 0. Definition 18. A polynomial time threshold reduction is a polynomial time truthtable reduction in which the truth table ....

Seinosuke Toda. Restricted relativizations of probabilistic polynomial time. Theoretical Computer Science, 1991. To appear.

....recent result in [13] that Sigma Sigma k SPARSE k P Sigma k [O(logn) Remark 3.10. The advice (even depending on the input) provided by an OptP[O(log n) function does not increase the power of the probabilistic class PP: PP= OptP[O(log n) PP. This follows from the result by Toda [40] that PP NP R = PP, since PP= OptP[O(log n) coincides with the class PP= FP NP[O(logn) see Lemma 3.4) that is clearly contained in PP NP R . Next, we consider uniform subclasses of P=log and P=poly. Whereas the proof of Corollary 3.5 ii) also yields the inclusion of P=OptP[O(log n) in ....

S. Toda. Restricted relativizations of probabilistic polynomial time. In Theoretical Computer Science, 93(2):265-277, 1992.

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