An oracle separating ΘP from PPPH

doi:10.1016/0020-0190(91)90035-G

Information Processing Letters

Volume 37, Issue 3, 18 February 1991, Pages 149-153

https://doi.org/10.1016/0020-0190(91)90035-G Get rights and content

Abstract

We prove the existence of an oracle A such that ΘP^A is not contained in PP^{PH^A}. This separation follows in a straightforward manner from a circuit complexity result, which is also proved here: To compute the parity of n inputs, any constant depth circuit consisting of a single threshold gate on top of and's and or's requires exponential size in n.

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Relating polynomial time to constant depth
1998, Theoretical Computer Science
Going back to the seminal paper of Furst, Saxe, and Sipser (1984), analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes. In this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization. Our technical tool is the leaf language approach to the definition of complexity classes.
A Lower Bound for Perceptrons and an Oracle Separation of the PP<sup>PH</sup> Hierarchy
1998, Journal of Computer and System Sciences
We show that there are functions computable by linear size boolean circuits of depthkthat require superpolynomial size perceptrons of depthk−1, fork<log n/(6 log log n). This result implies the existence of an oracleAsuch thatΣ^p, A_k⊈PP^{Σ^p, A_k−2}and, in particular, this oracle separates the levels in the PP^PHhierarchy. Using the same ideas, we show a lower bound for another function, which makes it possible to strengthen the oracle separation toΔ^p, A_k⊈PP^{Σ^p, A_k−2}.
A complex-number Fourier technique for lower bounds on the mod-m degree
2000, Computational Complexity
Exponential sums and circuits with a single threshold gate and mod-gates
1999, Theory of Computing Systems
Immunity and simplicity for exact counting and other counting classes
1999, Theoretical Informatics and Applications
A lower bound for perceptrons and an oracle separation of the PP<sup>PH</sup> hierarchy
1997, Proceedings of the Annual IEEE Conference on Computational Complexity

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^☆: A preliminary version of this paper appeared in the Fifth Annual Conference on Structure in Complexity Theory. IEEE Computer Society Press, 1990.

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An oracle separating ΘP from PPPH☆

Abstract

Proc. 28th Annual IEEE Symposium on Foundations of Computer Science

Proc. 30th IEEE Symposium on Foundations of Computer Science

Relativized counting classes: relations among thresholds, parity, and mods

Johns Hopkins Preprint

Relative to a random oracle A PA ≠ NPA ≠ co-NPA with probability 1

SIAM J. Comput.

With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy

J. Comput. System Sci.

An oracle separating ΘP from PP^PH☆

Relative to a random oracle A P^A ≠ NP^A ≠ co-NP^A with probability 1