On the robustness of random Boolean formulae

Abstract

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.

Dynamics of Boolean Networks: An Exact Solution

“…For lim t→∞ m(t) = m, q = 0 is a fixed point of (9) iff { S α(S)} 2 α = 0 which occurs only for balanced Boolean functions, with an equal number of ±1 in the output. By similar argument to the one used in the previous paragraph we show [17] that for m = 0 the point q = 0 is a unique stable solution of (9) when tanh 2 β < b(k). The α-averages in equations ( 8)-( 9) can be computed for a uniform distribution over all balanced Boolean functions to obtain m(t) = 0 for all t > 0, which implies q = tanh 2 (β)…”