Parallel computation for well-endowed rings and space-bounded probabilistic machines
Information and Control
Relativized circuit complexity
Journal of Computer and System Sciences
The expressive power of voting polynomials
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
PP is closed under intersection
Selected papers of the 23rd annual ACM symposium on Theory of computing
PP is closed under truth-table reductions
Information and Computation
When do extra majority gates help?: polylog(N) majority gates are equivalent to one
Computational Complexity - Special issue on circuit complexity
On Probabilistic Time and Space
Proceedings of the 12th Colloquium on Automata, Languages and Programming
On some central problems in computational complexity.
On some central problems in computational complexity.
Structural properties of complexity classes (immunity, core)
Structural properties of complexity classes (immunity, core)
NP trees and Carnap's modal logic
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
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Wilson's model of oracle gates provides a framework for considering reductions whose strength is intermediate between truth-table and Turing. Improving on a stream of results by Beigel, Reingold, Spielman, Fortnow, and Ogihara (STOC '91, Structures '91, FOCS '96), we prove that PL and PP are closed under NC_1 reductions. This answers an open problem of Ogihara (FOCS '96). More generally, we show that NC_{k+1}^PP = AC_k^PP and NC_{k+1}^PL = AC_k^PL for all k = 0. On the other hand, we construct an oracle A such that NC_k^{PP^A} NC_{k+1}^{PP^A} for all integers k = 1. Slightly weaker than NC_1 reductions are Boolean formula reductions. We ask whether PL and PP are closed under Boolean formula reductions. This is a nontrivial question despite NC_1 = BF, because that equality is easily seen not to relativize. We prove that P^PP_{log^2{n}/loglogn-T} is a subset of BF^PP is a subset of PrTIME(n^{O(logn)}). Because P^PP_{log^2{n}/loglogn-T} is not a subset of PP relative to an oracle, we think it is unlikely that PP is closed under Boolean formula reductions. We also show it is unlikely that PL is closed under BF reductions.