Abstract:
The random Boolean expressions are considered that are obtained by the random and independent substitution with the probabilities $p$ and $1-p$ of the constantly one function and constantly zero function for variables of repetition-free formulas over a given basis. The probability is studied that the expressions are equal to one. It is shown that, for each finite basis and $p\in(0,1)$, this probability tends to some finite limit $P_1(p)$ as the length of an expression grows. Explicit representation of the probability function $P_1(p)$ is found for all finite bases, the analytic properties of this function are studied, and its behavior is investigated in dependence on the properties of the basis.
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