Noisy random Boolean formulae: A statistical physics perspective

Abstract

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

On the robustness of random Boolean formulae

“…While in this paper we concentrated on the majority gates, we have looked at other gate types [16], hard noise (fabrication errors) and other properties of noisy circuits [18]. We believe that much can be explored about the properties of noisy circuits using the methodology developed here, for instance, the type of functions generated depending on the gate types and the level of gate noise.…”

Characterizing short-term stability for Boolean networks over any distribution of transfer functions

“…Specifically, if a single node has its state flipped, does the effect of this perturbation die out (quiescence), exponentially cascade over time (chaos), or is the system right in between (criticality)? There have been numerous empirical and mathematical observations about the characteristics of critical transition points in classes of Boolean networks [4][5][6][7][8][9][11][12][13][14][15], These results require F to have specific properties: for example, each truth table entry is i.i.d. or that functions are balanced (number of +1 and −1 outcomes is the same) on average.…”

Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise

“…Then intuitively, both models are in the ordered state. This intuition turns out to be correct for models with non-uniform topologies [19,20], but whether this is true in a more general case is not clear.…”