Z_2-Algebras in the Boolean Function Irreducible Decomposition

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.

Published

References in corpus (1)

• The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness

Cited by in corpus (1)

• The Role of Boolean Irreducibility in \textit{NK}-Kauffman Networks