Z_2-Algebras in the Boolean Function Irreducible Decomposition

TL;DR

This paper advances the classification of Boolean functions by their reducibility and irreducibility, enabling precise statistical analysis of disordered Boolean models like NK-Kauffman networks and their stability.

Contribution

It constructs a ring-isomorphism linking reducible Boolean functions to double power sets, facilitating calculation of function counts based on irreducibility and weight.

Findings

Derived a formula for counting $ ho_K (\lambda, \omega)$ functions

Enabled statistical analysis of NK-Kauffman network stability

Provided tools for mean field dynamical studies

Abstract

We develop further the consequences of the irreducible-Boolean classification established in Ref. [9]; which have the advantage of allowing strong statistical calculations in disordered Boolean function models, such as the \textit{NK}-Kauffman networks. We construct a ring-isomorphism $ mathfrak{R}_K {i_1, ..., i_\lambda} \cong \mathcal{P}^2 -[K] $ of the set of reducible $K$-Boolean functions that are reducible in the Boolean arguments with indexes ${i_1, ..., i_\lambda}$; and the double power set $\mathcal{P}^2 [K]$, of the first $K$ natural numbers. This allows us, among other things, to calculate the number $\varrho_K (\lambda, \omega)$ of $K$-Boolean functions which are $\lambda $-irreducible with weight $\omega$. $\varrho_K (\lambda, \omega)$ is a fundamental quantity in the study of the stability of \textit{NK}-Kauffman networks against changes in their connections between their Boolean functions; as well as in the mean field study of their dynamics when Boolean irreducibility is taken into account.