# 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…

## 12 Citations

### Tight Space Complexity of the Coin Problem

- Computer Science, Mathematics2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

A tight threshold is shown between the values of $\beta=n^{-c}$ where $O(\log n) $ width suffices and the regime where $poly (n)$ width is needed, with a transition at $c=1/3$.

### Monotone Circuits for the Majority Function

- Computer ScienceAPPROX-RANDOM
- 2006

It is shown that the size can be further reduced at the expense of increased depth, and the circuit obtained is for the majority of size and depth about $n^{1+\sqrt{2}}$ and 9.9logn.

### Growth processes on formulas and reversible circuits

- Computer Science, Mathematics
- 2003

This thesis parameterizes a growth process by its initial conditions and characterize it by the existence and shape of a limiting probability distribution that describes the likelihood of it realizing a particular Boolean function, and proposes and characterize growth processes on reversible circuits.

### THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS

- Computer Science, Mathematics20th Annual IEEE Conference on Computational Complexity (CCC'05)
- 2005

Two new complexity measures for Boolean functions are introduced, which are named sumPI and maxPI, and the main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares ofthe spectral norms of its submatrices.

### On the Size of Probabilistic Formulae

- Mathematics, Computer ScienceISAAC
- 1997

A lower bound method for probabilistic formulae based on the VC-dimension and the Neciporuk lower bound is devised and lower bounds on the size of 2-sided bounded error, Monte Carlo, and Las Vegas formULae over an arbitrary basis are given.

### The Boolean functions computed by random Boolean formulas or how to grow the right function

- Computer ScienceArXiv
- 2003

Growth processes are characterized by their initial conditions to derive conditions under which results such as Valiant's J Algorithms 5 (1984), 363–366 hold and explicit bounds on the convergence rates of several growth processes are obtained.

### On linear rewriting systems for Boolean logic and some applications to proof theory

- Computer ScienceLog. Methods Comput. Sci.
- 2016

This paper studies properties of systems consisting only of linear inferences and finds that the length of any 'nontrivial' derivation in such a system is bound by a polynomial, unless coNP=NP.

### Balanced And/Or Trees and Linear Threshold Functions

- Mathematics, Computer ScienceANALCO
- 2009

It is proved that the probability distribution induced on Boolean functions is shown to have a limit when letting the depth of these formulas grow to infinity, and that its support is made of linear threshold functions.

### Analysis of random processes via And-Or tree evaluation

- Computer ScienceSODA '98
- 1998

These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including random loss-resilient codes, solving random k-SAT formula using the pure literal rule, and the greedy algorithm for matchings in random graphs.

### No complete linear term rewriting system for propositional logic

- Computer ScienceRTA
- 2015

It is shown that, as long as reduction steps are polynomial-time decidable, such a rewriting system does not exist unless coNP=NP, i.e. that every Boolean tautology can be written as a (left- and right-) linear rewrite rule.

## References

SHOWING 1-10 OF 20 REFERENCES

### Amplification of Bounded Depth Monotone Read-Once Boolean Formulae

- MathematicsSIAM J. Comput.
- 1991

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

- Computer Science, MathematicsSIAM J. Comput.
- 1981

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

- Mathematics
- 1971

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

- Computer ScienceSTOC '84
- 1984

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

- Computer ScienceProceedings [1990] 31st Annual Symposium on Foundations of Computer Science
- 1990

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

- Computer Science19th 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

- Computer Science
- 1992

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

- Computer Science26th 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.

### How Do Read-Once Formulae Shrink?

- MathematicsCombinatorics, Probability and Computing
- 1994

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