$\mathbb {Z}_2$ Z 2 -algebras in the Boolean function irreducible decomposition

Abstract

We develop further the consequences of the irreducible-Boolean classification established by Zertuche, [“On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions,” J. Math. Phys. 50, 043513 (2009)10.1063/1.3116166] which have the advantage of allowing strong statistical calculations in disordered Boolean function models, such as the NK-Kauffman networks. We construct a ring-isomorphism \documentclass[12pt]{minimal}\begin{document}$\mathfrak {R}_K \left\lbrace i_1, \dots , i_\lambda \right\rbrace \cong \mathcal {P}^2 \left[ K \right]$\end{document}RKi1,⋯,iλ≅P2K of the set of reducible K-Boolean functions that are reducible in the Boolean arguments with indexes {i1, …, iλ}, and the double power set \documentclass[12pt]{minimal}\begin{document}$\mathcal {P}^2 \left[ K \right]$\end{document}P2K of the first K natural numbers. This allows us, among other things, to calculate the number ϱK(λ, ω) of K-Boolean functions which are λ-irreducible with weight ω. ϱK(λ, ω) is a fundamental quantity in the study of the stability of 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.

Canalization and the stability of NK-Kauffman networks

“…That is, canalizing functions constitute a minority with respect the total K -Boolean functions. In contrast, for K c ≫ 1 the ratio of the totally irreducible functions with respect to # Ξ K goes like 1 − O  K 2 2 K −1  [12,13]. The only way in which canalization could play a role would be if canalizing functions have better chances to be extracted in the regions p c ∼ 0 and p c ∼ 1.…”

“…Let us now go to the formal definition of canalization [13,14]. A K -Boolean function β is canalizing iff ∃j = 1, .…”

“…I want to stress, that canalization and irreducibility are not related concepts [13]. The reader may find two examples in Table 3.…”

“…For that scope, I use the calculation done in [13] for the number of β ∈ Ξ K which have irreducible degree λ and weight ω, denoted by ϱ K (λ, ω). The result was:…”

“…In [12] the irreducible classification was used to give a best approach to the phase transition curve (4) obtaining curve (5), which is going to be used for the calculations along this work. In [13] an algebraic approach was used to obtain the all important expression ϱ K (λ, ω) for the number of K Boolean which have a degree of irreducibility λ and weight ω (see Section 5). By means of ϱ K (λ, ω) (5) was calculated in [12] and it Table 1 The general form of a truth table of a K -Boolean function.…”