Amplification by Read-Once Formulas

@article{Dubiner1997AmplificationBR,
  title={Amplification by Read-Once Formulas},
  author={Moshe Dubiner and Uri Zwick},
  journal={SIAM J. Comput.},
  year={1997},
  volume={26},
  pages={15-38}
}
Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulas of size $O(n^{\alpha+2})$ (where $\alpha=\log_{\sqrt{5}-1}2\simeq 3.27$) that amplify $(\psi-\frac{1}{n},\psi+\frac{1}{n})$ (where $\psi=(\sqrt{5}-1)/2\simeq0.62$) to $(2^{-n},1-2^{-n})$ and deduced as a consequence the existence of monotone formulas of the same size that compute the majority… 

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References

SHOWING 1-10 OF 20 REFERENCES

Amplification of Bounded Depth Monotone Read-Once Boolean Formulae

The size of monotone read-once formulae in $\Sigma _d \cup \Pi _d $ that amplify $(p,p + 1 / m)$ to $(p',p' + 1/ c)$ is $\exp (\theta ((d - m)(m / c)^{1 / (d - 1)} ))$ under certain conditions.

Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1

Let A be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to A, and${\bf NP}^A is shown, with probability 1, to contain a-immune set, i.e., a set having no infinite subset in ${\bf P]^A $.

Complexity of the realization of a linear function in the class of II-circuits

AbstractIt is proved that the linear function gn(x1,..., xn) = x1 + ... + xnmod 2 is realized in the class of II-circuits with complexity Lπ(gn) ≥n2. Combination of this result with S. V.

A theorem on probabilistic constant depth Computations

Stockmeyer [St] showed that probabilistic bounded depth circuits can approximate the exact number of ones in the input with very low probability of error.

Faster circuits and shorter formulae for multiple addition, multiplication and symmetric Boolean functions

The shallowest known multiplication circuits are constructed on the basis of the optimal constructions of multiple carry-save adders using any given basic addition unit.

Two theorems on random polynomial time

  • L. Adleman
  • Computer Science
    19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
  • 1978
Where the traditional method of polynomial reduction has been inapplicable, randomness has been used in demonstrating intractibility by Adleman and Manders, and in showing problems equivalent by Rabin, a new examination of randomness is in order.

Optimal carry save networks

In this paper simple basic carry save adders are described using which multiplication circuits of depth 3.71 log n and majority formulae of size O (n3.13) are constructed and the shallowest known multiplication circuits and the shortest formULae for the majority function are obtained.

Amplification of probabilistic boolean formulas

  • R. Boppana
  • Computer Science
    26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
  • 1985
This paper shows that the amount of amplification that Valiant obtained is optimal, and gives an O(k4.3 n log n) upper bound for the size of monotone formulas computing the kth threshold function of n variables.

On the Shrinkage Exponent for Read-Once Formulae

How Do Read-Once Formulae Shrink?

It is shown that f e depends, on the average, on only O (e α n + e n 1/α ) variables, where .