In this paper we generate upper and lower bounds for the sensitivity to noise of a Boolean function using relaxed assumptions on input choices and noise. The robustness of a Boolean network to noisy inputs is related to the average sensitivity of that function. The average sensitivity measures how sensitive to changes in the inputs the output of the function is. The average sensitivity of Boolean functions can indicate whether a specific random Boolean network constructed from those functions is ordered, chaotic, or in critical phase. We give an exact formula relating the sensitivity to noise and the average sensitivity of a Boolean function. The analytic approach is supplemented by numerical results that illustrate the overall behavior of the sensitivities as various Boolean functions are considered. It is observed that, for certain parameter combinations, the upper estimates in this paper are sharper than other estimates in the literature and that the lower estimates are very close to the actual values of the sensitivity to noise of the selected Boolean functions.
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October 2009
Research Article|
October 02 2009
On the sensitivity to noise of a Boolean function
Mihaela T. Matache;
Mihaela T. Matache
a)
Department of Mathematics,
University of Nebraska at Omaha
, Omaha, Nebraska 68182-0243, USA
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Valentin Matache
Valentin Matache
b)
Department of Mathematics,
University of Nebraska at Omaha
, Omaha, Nebraska 68182-0243, USA
Search for other works by this author on:
a)
Electronic mail: dmatache@mail.unomaha.edu.
b)
Electronic mail: vmatache@mail.unomaha.edu.
J. Math. Phys. 50, 103512 (2009)
Article history
Received:
February 18 2009
Accepted:
August 20 2009
Citation
Mihaela T. Matache, Valentin Matache; On the sensitivity to noise of a Boolean function. J. Math. Phys. 1 October 2009; 50 (10): 103512. https://doi.org/10.1063/1.3225563
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