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### A Generalization of the Noisy-Or Model [chapter]

Sampath Srinivas
1993 Uncertainty in Artificial Intelligence
Here we generalize the model to nary input and output variables and to ar bitrary functions other than the Boolean OR function.  ...  The Noisy-Or model is convenient for de scribing a class of uncertain relationships in Bayesian networks [Pearl 1988 ]. Pearl describes the Noisy-Or model for Boolean variables.  ...  We now suggest a particular form of F that is 'compat ible' with the Boolean Noisy-Or, i.e., F degenerates to the Boolean OR function when the inputs and outputs are Boolean23: In essence, this function  ...

### Reliable computation from contextual correlations

André L. Oestereich, Ernesto F. Galvão
2017 Physical Review A
Here we investigate the requirements for reliable computation in this setting, that is, the evaluation of any Boolean function with success probability bounded away from 1/2.  ...  functions.  ...  An ǫ-noisy, k-input gate computes a k-bit Boolean function, outputting the correct answer with probability 1−ǫ, and its negation with probability ǫ, independently of the input.  ...

### On networks of noisy gates

Nicholas Pippenger
1985 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
This may be contrasted with results of von Neumann, Dobrushin and Ortyukov to the effect that (1) for every Boolean function, the number of noisy gates needed is larger by at most a logarithmic factor,  ...  We show that many Boolean functions (including, in a certain sense, "almost all" Boolean functions) have the . property that the number of noisy gates needed to compute them differs from the number of  ...  We shall take e=I/512, 8=1/128 and show that the noisy network strongly (E, B)-computes the same function as the noiseless one. Let A be a set of xm inputs with xs 1/64.  ...

### On the maximum tolerable noise of k-input gates for reliable computation by formulas

W.S. Evans, L.J. Schulman
2003 IEEE Transactions on Information Theory
We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.  ...  An -noisy gate computes a Boolean function of its inputs that is then complemented with probability to become the gate's output.  ...  An -noisy, k-input gate is designed to compute a Boolean function of its k Boolean inputs; however, it has the property that for any assignment to the inputs, there is probability that the output of the  ...

### Average-Case Lower Bounds for Noisy Boolean Decision Trees

William Evans, Nicholas Pippenger
1998 SIAM journal on computing (Print)
We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions.  ...  The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds.  ...  Suppose that the noisy dynamic decision tree T (ε, δ)-computes the Boolean function f with expected cost C averaged over both inputs and noise. Let δ be such that δ < δ < 1/2.  ...

### Computing with Noise: Phase Transitions in Boolean Formulas

Alexander Mozeika, David Saad, Jack Raymond
2009 Physical Review Letters
Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting.  ...  This framework paves the way for obtaining new results on error-rates, function-depth and sensitivity, and their dependence on the gate-type and noise model used.  ...  A noisy circuit with > 0 represents a given deterministic function with a maximum error probability δ over all possible circuit inputs determining its reliability.  ...

### On the maximum tolerable noise for reliable computation by formulas

B. Hajek, T. Weller
1991 IEEE Transactions on Information Theory
Boolean function, the network required for the proof is described as follows. Start with a formula for computing the function using only noiseless 2-input NAND gates.  ...  Consider a Boolean function F of at least 1 + 3'--l essential arguments, and consider any formula constructed of e-noisy 3-input gates for computing F.  ...

### On the robustness of random Boolean formulae

Alexander Mozeika, David Saad, Jack Raymond
2010 Journal of Physics, Conference Series
Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics.  ...  We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used.  ...  (iii) The -noisy gate is designed to compute a Boolean function α : {−1, 1} k → {−1, 1}, but for each input S ∈ {−1, 1} k there is an error probability such that α(S) → −α(S).  ...

### On reliable computation by noisy random Boolean formulas [article]

Alexander Mozeika, David Saad
2014 arXiv   pre-print
We show that these gates can be used to compute any Boolean function reliably below the noise bound.  ...  We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable.  ...  The ǫ-noisy gate is designed to compute a Boolean function α(σ), but for each input σ ∈ {−1, 1} k there is an error probability ǫ such that α(σ) → −α(σ).  ...

### Do Distributed Differentially-Private Protocols Require Oblivious Transfer?

Vipul Goyal, Dakshita Khurana, Ilya Mironov, Omkant Pandey, Amit Sahai, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming
optimally accurate, distributed differentially private protocol for any functionality with a boolean XOR embedded on adjacent inputs. • While the previous result holds for optimally accurate protocols  ...  This is because the output of the functionality combined with the input of any individual party reveals completely, the input of the other party.  ...  Alice and Bob, with inputs x and y respectively, execute a protocol to compute a Boolean function f (x, y).  ...

### The Complexity of Combinatorial Computations: An Introduction [chapter]

L. G. Valiant
1978 Informatik-Fachberichte
Noisy Decision Trees An -noisy decision tree on N boolean input variables is a decision tree where each internal node is labelled by one of the N input variables and each leaf is labelled by a boolean  ...  One of them is to find interesting boolean functions that need superlinear number of broadcasts in the noisy broadcast model with random noise.  ...

### On Reliable Computation by Noisy Random Boolean Formulas

Alexander Mozeika, David Saad
2015 IEEE Transactions on Information Theory
We show that these gates can be used to compute any Boolean function reliably below the noise bound. Index Terms-Random Boolean formulas, ǫ-noise, reliable computation.  ...  We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable.  ...  Thus in this regime any Boolean function can be computed with any desired accuracy. III.  ...

### ATEN: And/Or tree ensemble for inferring accurate Boolean network topology and dynamics

2019 Bioinformatics
Using these important features we then infer the Boolean function of the target gene. Finally, the Boolean functions of all target genes are combined as a Boolean network.  ...  To address the problem, we propose a Boolean network inference algorithm which is able to infer accurate Boolean network topology and dynamics from short and noisy time series data.  ...  The directed edges, or the interactions between the target gene and input genes are associated with the regulatory rule, i.e., the Boolean function.  ...

### Noisy random Boolean formulae: A statistical physics perspective

Alexander Mozeika, David Saad, Jack Raymond
2010 Physical Review E
A growth model that gives rise to typical random Boolean functions is mapped onto a layered Ising spin system, which facilitates the study of their ability to represent arbitrary formulae with a given  ...  level of error, the tolerable level of gate-noise, and its dependence on the formulae depth and complexity, the gates used and properties of the function inputs.  ...  However, this strategy fails for inputs with m͑0͒ = 0 and the circuit computes more than one Boolean function.  ...

### On a lower bound for the redundancy of reliable networks with noisy gates

N. Pippenger, G.D. Stamoulis, J.N. Tsitsiklis
1991 IEEE Transactions on Information Theory
We provide a proof that a logarithmic redundancy factor is necessary for the reliable computation of the parity function by means of a network with noisy gates.  ...  We have established the result by following the same steps as Dobrushin and Ortyukov and by replacing the questionable part of their analysis with entirely new argumnents.  ...  INTRODUCTION Computation of Boolean functions by means of noisy gates is a topic that started attracting the attention of researchers in the early '50s.  ...
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