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

Evans, William S.; Schulman, Leonard J.

doi:10.1109/tit.2003.818405

Cited by 51 publications

(82 citation statements)

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“…As expected, the value of h is larger for larger k, for the same value p, and the critical noise is also larger. The critical values p c match exactly the upper bounds for Boolean formulas using noisy majority functions of k inputs found by Evans and Schulman [15], given by,…”

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Redundancy and Error Resilience in Boolean Networks

Peixoto

2010

Phys. Rev. Lett.

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We consider the effect of noise in sparse Boolean Networks with redundant functions. We show that they always exhibit a non-zero error level, and the dynamics undergoes a phase transition from non-ergodicity to ergodicity, as a function of noise, after which the system is no longer capable of preserving a memory of its initial state. We obtain upper-bounds on the critical value of noise for networks of different sparsity. PACS numbers: 89.75.Da,05.65.+b,91.30.Dk,91.30.Px Introduction-Biological systems are unavoidably noisy in their nature, but often need to function in a predictable fashion [1]. In such a situation, strategies to diminish the harmful effect of noise will significantly impact the fitness of a given organism. The most fundamental protection mechanism a system can adopt is the redundancy of its underlying components, since the resulting coincidences necessary to impact the proper function of the system can drastically diminish the probability of error. In this letter we are concerned with the effect of redundancy in gene regulation; in particular in a simple Boolean Network (BN) model. We assume that each component in the system is arbitrarily redundant, with the only restriction that the number of inputs per component is fixed and finite. In a general manner, we are able to show that redundancy can always guarantee reliable dynamics, up to a given critical value of noise, above which the system is incapable of maintaining any memory of its past states. From simple considerations, we are able to obtain upper bounds on the maximum resilience attainable. This provides an important frame of reference to determine the reliability of a system with a given sparsity.We begin by defining the model, and how noise is introduced. A Boolean Network (BN) [2] is a directed graph with N nodes, representing the genes, which have an associated Boolean state σ i ∈ {0, 1}, corresponding to the transcription state, and a function f i ({σ j } i ), which determines the state of node i given the states of its input nodes {σ j } i . The number of inputs of a given node is k i , or simply k if its the same for all nodes. This system is usually updated in parallel, such that at each time step t, we have σ i (t + 1) = f i ({σ j (t)} i ). Starting from an initial configuration, the system will evolve, and eventually settle on an attractor. In a real system, the expression level of a particular gene can fluctuate, despite the stability of its input states [3]. This characteristic can be incorporated qualitatively in the BN model as uniform noise [4][5][6][7][8][9][10], defined as a probability p that, at each time step, the value of a given input σ j ∈ {σ j } i of a node i is flipped, prior to the evaluation of the function f i . The value of p plays the role of a temperature in the system. If p = 0 the original deterministic model is recovered, and if p = 1/2 the system becomes effectively decoupled, with entirely stochastic dynamics.In the model above, it is known that error resilience does not spontaneously emerge, sinc...

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confidence: 79%

Redundancy and Error Resilience in Boolean Networks

Peixoto

2010

Phys. Rev. Lett.

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“…In comparison to its noiseless counterpart, a noisy formula that computes reliably has greater depth due to the presence of restitution-gates, implying longer computation times [3]. These results have been refined and extended for circuits [4], for k-ary Boolean formulae [5,6] and different gates [7,8].…”

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On the robustness of random Boolean formulae

Mozeika

Saad

Raymond

2010

J. Phys.: Conf. Ser.

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Abstract. 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. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

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“…Von Neumann studied a model of random errors, where each gate has an (arbitrary) error independently with small fixed probability, and his goal was to obtain correctness (as opposed to privacy). There have been numerous follow up papers to this seminal work, including [16,53,52,27,23,36,28,22], who considered the same noise model, ultimately showing that any circuit of size σ can be encoded into a circuit of size O(σ log σ) that tolerates a fixed constant noise rate, and that any such encoding must have size Ω(σ log σ).…”

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confidence: 99%

Securing Circuits against Constant-Rate Tampering

Dachman-Soled

Kalai

2012

Lecture Notes in Computer Science

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Abstract. We present a compiler that converts any circuit into one that remains secure even if a constant fraction of its wires are tampered with. Following the seminal work of Ishai et al. (Eurocrypt 2006), we consider adversaries who may choose an arbitrary set of wires to corrupt, and may set each such wire to 0 or to 1, or may toggle with the wire. We prove that such adversaries, who continuously tamper with the circuit, can learn at most logarithmically many bits of secret information (in addition to black-box access to the circuit). Our results are information theoretic.

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

Abstract: Abstract-We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.

Cited by 51 publications

References 6 publications

Redundancy and Error Resilience in Boolean Networks

Redundancy and Error Resilience in Boolean Networks

On the robustness of random Boolean formulae

Securing Circuits against Constant-Rate Tampering

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