Zachos, S., \Probabilistic Quanti ers, Adversaries, and Complexity Classes: An Overview," Proc. 1 st Structure in Complexity Theory Conference, vol. 223, Lecture Notes in Computer Science, Springer-Verlag, 1986. 23  Home/Search   Document Not in Database   Summary   Related Articles   Check

....BPP. 14 Recall also that when a BPP algorithm is augmented by an NP oracle, and the number of oracle destined bits is always polynomial in the input size, one obtains the class BPP NP . Finally, when just one oracle call is allowed in a BPP NP algorithm, one obtains the ArthurMerlin class AM [Zac86, BM88]. Theorem 3. Koi96] Assuming the truth of GRH, HN2AM. While probabilistic algorithms for HN (and more general problems) have certainly existed at least since the early 1980 s, the above theorem is the rst and only example of a randomized algorithm for HN requiring a number of bit operations ....

....or the algorithm from section 2, is that it could happen that P 6=NP but the higher complexity classes we have been alluding to all collapse to the same level. For example, while it is known that NP[BPP AM P NP NP NP NP NP PSPACE, the properness of each inclusion is still unknown [Zac86, BM88, BF91, Pap95]. The algorithm for theorem 13 is almost as simple as the algorithm for theorem 4 given earlier, and can be outlined as follows: Step 0 Let NF (x) denote the weighted version of F (x) where we instead sum the total 28 number of roots in Z=pZ of the mod p reductions of F over all primes p x. ....

Zachos, S., \Probabilistic Quantiers, Adversaries, and Complexity Classes: An Overview," Proc. 1 st Structure in Complexity Theory Conference, vol. 223, Lecture Notes in Computer Science, Springer-Verlag,

....to the classes in the counting hierarchy in [Tor88a] and [Tor88b] Intuitively, a set A is low for a complexity class K if A does not increase the computational power of K when used as oracle; K A = K. We prove that Few is low for the complexity classes PP, C = P, and PhiP (parity P, [PaZa83]) showing PP Few =PP, C = P Few = C = P and PhiP Few = PhiP. In Section 5 we consider some other interesting sets that are low for the class PP. We prove that all sparse sets in NP, as well as the sets in the probabilistic class BPP are PP low. The proofs of these results relativize, ....

....Next we define the complexity classes PP, C = P and PhiP that are also defined considering the number of computation paths of a nondeterministic machine, but in this case the number of paths is not necessarily polynomially bounded. These classes were first introduced in [Gi77] Wa86] and [PaZa83], respectively. Definition 2.4: A language L is in the class PP if there is a nondeterministic polynomial time machine M and a function f 2 FP such that for every x 2 Sigma , x 2 L ( accM (x) f(x) PP is called CP in the notation of [Wa86] This notation can be generalized to other ....

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C.H. Papadimitriou and S.K. Zachos. Two remarks on the power of counting. 6th GI Conf. on Theor. Comput. Sci., Lecture Notes in Computer Science 145, 269--276, Springer-Verlag, 1983. 15

....the MODm degree of the function MOD n is O(1) by a folklore theorem (Beigel and Gill 1992, Hertrampf 1990, Beigel and Tarui 1991, Barrington 1992a, Smolensky 1987) 4. An oracle for the conjectured relations among MODmP classes The class MODmP is a generalization of the counting class PhiP (Papadimitriou and Zachos 1983, Goldschlager and Parberry 1986) First developed by Cai and Hemachandra (1990) these classes have since been studied by many others (Beigel 1991, Beigel and Gill 1992, Hertrampf 1990, Babai and Fortnow 1990, Toda and Ogiwara 1992, Tarui 1993) It is known that MODmP = MODm 0 P where m 0 is ....

C. Papadimitriou and S. Zachos, Two remarks on the power of counting. Proc. Sixth GI Conf. Theoret. Comp. Sci., Lecture Notes in Computer Science 145, Springer-Verlag, Berlin, 1983, 269--276.

....C 6. Phi Delta C Phi DeltaP C 7. NP BP Delta Phi DeltaP. This follows from the proof of [VV 86] showing that SAT is reducible via probabilistic reductions to the unique satisfiability problem . 8. Phi Delta Phi Delta C = Phi Delta C. 9. Phi DeltaP Phi DeltaP = Phi DeltaP. [PZ 83] (Thus Phi DeltaP is closed under p T . 10. PP BPP = PP. KSTT 89] If the underlying class C is closed under positive reducibility (see [Sc 87] then the usual techniques of amplification can be used to exponentially reduce the error probability for sets in the class BP Delta C. Thus ....

C. Papadimitriou and S. Zachos, Two remarks on the power of counting, Proc. 6th GI Conference, Lecture Notes in Computer Science 145, pp. 269--275.

.... x 2 L ( #M (x) f(x) ffl (Beigel, Gill, Hertrampf [5] For k 2, define Mod k P to be the class of all languages L such that there exists M such that, for all x, x 2 L ( #M (x) 6j 0 mod k: The class Mod 2 P is also called PhiP ( Parity P ) This class was defined by Papadimitriou Zachos [20] and by Goldschlager Parberry [10] see [4] for details) The following two classes will also be of interest to us: Definition 2.4 ffl (Allender [1] For any language L, L 2 FewP if and only if there exist a CM M and a polynomial p such that for all x 2 Sigma , #M (x) p(jxj) and x 2 L ....

....same gap as M L but without an oracle. Thus L 2 SPP as witnessed by N . Conversely, we show that if M is an OCM and L is a language in SPP , there is a CM N (without an oracle) such that gap N = gap M L : This part of the proof has the same flavor as the proof that PhiP PhiP = PhiP in [20]. Let M 1 be an SPP machine recognizing L. We may assume without loss of generality that for any oracle A and input x of length n, M A (x) makes exactly k(1 n ) oracle queries on each path, where k 2 FP . Fix n and let k df = k(1 n ) The CM N does the following in sequence on input ....

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C. H. Papadimitriou and S. K. Zachos. Two Remarks on the Power of Counting, pages 269--276. Lecture Notes in Computer Science 145. Springer-Verlag, 1983.

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Zachos, S., \Probabilistic Quanti ers, Adversaries, and Complexity Classes: An Overview," Proc. 1 st Structure in Complexity Theory Conference, vol. 223, Lecture Notes in Computer Science, Springer-Verlag, 1986. 23

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S. Zachos and M. Furer, Probabilistic quantifiers vs. distrustful adversaries, Proc. 7th Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science 287, pp. 443--455.

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