Nested canalyzing, unate cascade, and polynomial functions

Abstract

This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.

Introduction

Canalyzing functions were introduced by Kauffman [10] as appropriate rules in Boolean network models of gene regulatory networks. The definition is reminiscent of the concept of “canalisation” introduced by the geneticist Waddington [22] to represent the ability of a genotype to produce the same phenotype regardless of environmental variability. Canalyzing functions are known to have other important applications in physics, engineering and biology. They have been used to study the convergence behavior of a class of nonlinear digital filters, called stack filters, which have applications in image and video processing [5], [23], [24]. Canalyzing functions also play an important role in the study of random Boolean networks [10], [13], [19], [20], and have been used extensively as models for dynamical systems as varied as gene regulatory networks [10], evolution, [20] and chaos [13]. One important characteristic of canalyzing functions is that they exhibit a stabilizing effect on the dynamics of a system. For example, in [5], it is shown that stack filters which are defined by canalyzing functions converge to a fixed point called a root signal after a finite number of passes. Moreira and Amaral [15] showed that the dynamics of a Boolean network which operates according to canalyzing rules is robust with regard to small perturbations.

A special type of canalyzing function, so-called nested canalyzing functions (NCFs), were introduced recently in [8], and it was shown in [9] that Boolean networks made from such functions show stable dynamic behavior and might be a good class of functions to express regulatory relationships in biochemical networks. Little is known about this class of functions, however. For instance, there is no known formula for the number of nested canalyzing functions in a given number of variables.

Another field in which special families of Boolean functions have been studied extensively is the theory of computing, in particular the design of efficient logical switching circuits. Since the 1970s, several families of Boolean functions have been investigated for use in circuit design. For instance, the family of fanout-free functions has been studied extensively, as well as the family of cascade functions. A subclass of these are the unate cascade functions, see, e.g., [14], [16], which we focus on here. It turns out that this class of functions has some very useful properties. For instance, it was shown recently [3] that the class of unate cascade functions is precisely the class of Boolean functions that have good properties as binary decision diagrams. In particular, the unate cascade functions (on $n$ variables) are precisely those functions whose binary decision diagrams have the smallest average path length $(2-\frac{1}{2^{n-1}})$ among all Boolean functions of $n$ variables.

The notion of average path length is one cost measure for binary decision trees, which measures the average number of steps to evaluate the function on which the tree is based. One way of assessing the relative efficacy of classes of Boolean function for logic circuit or binary decision tree design is to look at the number of different circuits or trees that can be realized with a particular class. That is, one would like to count the number of functions in a given class. This has led to a formula for the number of unate cascade functions [2]. One of the results in this paper shows that the classes of unate cascade functions and nested canalyzing functions are identical (as classes of functions rather than as classes of logical expressions). As a result of the equivalence we will establish, this formula then also counts the number of nested canalyzing functions.

A third framework for studying Boolean functions, in the context of models for biochemical networks, was introduced in [11]. There, a new method to reverse engineer gene regulatory networks from experimental data was proposed. The proposed modeling framework is that of time-discrete deterministic dynamical systems with a finite set of states for each of the variables. The number of states is chosen so as to support the structure of a finite field. One consequence is that each of the state transition functions can be represented by a polynomial function with coefficients in the finite field, thereby making available powerful computational tools from polynomial algebra. This class of dynamical systems in particular includes Boolean networks, when network nodes take on two states. It is straightforward to translate Boolean functions into polynomial form, with multiplication corresponding to AND, addition to XOR, and addition of the constant 1 to negation. In this paper we provide a characterization of those polynomial functions over the field with two elements that correspond to nested canalyzing (and, therefore, unate cascade) functions. Using a parametrized  polynomial representation, one can characterize the parameter set in terms of a well-understood mathematical object, a common method in mathematics. This is done using the concepts and language from algebraic geometry. To be precise, we describe the parameter set as an algebraic variety, that is a set of points in an affine space that represents the set of solutions of a system of polynomial equations. This algebraic variety turns out to have special structure that can be used to study the class of nested canalyzing functions as a rich mathematical object.