The number and probability of canalizing functions

Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of canalizing functions to other classes of functions with respect to their evolutionary plausibility as emergent control rules in genetic regulatory systems, it is informative to know the number of canalizing functions with a given number of input variables. This is also important in the context of using the class of canalizing functions as a constraint during the inference of genetic networks from gene expression data. To this end, we derive an exact formula for the number of canalizing Boolean functions of n variables. We also derive a formula for the probability that a random Boolean function is canalizing for any given bias p of taking the value 1. In addition, we consider the number and probability of Boolean functions that are canalizing for exactly k variables. Finally, we provide an algorithm for randomly generating canalizing functions with a given bias p and any number of variables, which is needed for Monte Carlo simulations of Boolean networks

On the Existence and Design of the Best Stack Filter Based Associative Memory

The associative memory of a stack filter is defined to be the set of root signals of that filter. If the root sets of two stack filters both contain a desired set of patterns, but one filter’s root set is smaller than the other, then the filter with the smaller root set is said to be better for that set of patterns. Any filter which has the smallest number of roots containing the specified set of patterns is said to be a best filter. The configuration of the family of best filters is described via a graphical approach which specifies an upper and lower bound for the subset of possible best filters which are furthest from the sets of type-1 and type-2 stack filters. Knowledge of this configuration leads to an algorithm which can produce a near-best filter. This new method of constructing associative memories does not require the desired set of patterns to be independent and it can construct a much better filter than the methods in [I]