Symmetry Properties of Nested Canalyzing Functions.

Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call r-symmetric functions (where r is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF f can be easily computed given a standard representation of f. We also present an algorithm for testing whether a given r-symmetric function is an NCF. Further, we show that for any NCF f with n variables, the notion of strong asymmetry considered in the literature is equivalent to the property that f is n-symmetric. We use this result to derive a closed form expression for the number of n-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.