-
-
-
In what follows, we consider the Boolean algebra on the set
$\Bool=\{0,1\}$ with the classical operators of conjunction '.',
of disjunction '+', of negation '$\overline{~}$', and of
disjunctive union $\oplus$.
Let $n$ be a positive integer. A {\emph{Boolean map} $f$ is
-a function from the Boolean domain
- to itself
+a function from $\Bool^n$
+to itself
such that
$x=(x_1,\dots,x_n)$ maps to $f(x)=(f_1(x),\dots,f_n(x))$.
Functions are iterated as follows.
Let us consider for instance $n=3$.
Let
$f^*: \Bool^3 \rightarrow \Bool^3$ be defined by
-
$f^*(x_1,x_2,x_3) =
(x_2 \oplus x_3, \overline{x_1}\overline{x_3} + x_1\overline{x_2},
-\overline{x_1}\overline{x_3} + x_1x_2)$
+\overline{x_1}\overline{x_3} + x_1x_2)$.
The iteration graph $\Gamma(f^*)$ of this function is given in
Figure~\ref{fig:iteration:f*}.
\vspace{-1em}
\begin{figure}[ht]
\begin{center}
-\includegraphics[scale=0.5]{images/iter_f0b}
+\includegraphics[scale=0.5]{images/iter_f0c}
\end{center}
\vspace{-0.5em}
\caption{Iteration Graph $\Gamma(f^*)$ of the function $f^*$}\label{fig:iteration:f*}
\end{figure}
\end{xpl}
-\vspace{-0.5em}
-It is easy to associate a Markov Matrix $M$ to such a graph $G(f)$
-as follows:
-
-$M_{ij} = \frac{1}{n}$ if there is an edge from $i$ to $j$ in $\Gamma(f)$ and $i \neq j$; $M_{ii} = 1 - \sum\limits_{j=1, j\neq i}^n M_{ij}$; and $M_{ij} = 0$ otherwise.
-
-\begin{xpl}
-The Markov matrix associated to the function $f^*$ is
-
-\[
-M=\dfrac{1}{3} \left(
-\begin{array}{llllllll}
-1&1&1&0&0&0&0&0 \\
-1&1&0&0&0&1&0&0 \\
-0&0&1&1&0&0&1&0 \\
-0&1&1&1&0&0&0&0 \\
-1&0&0&0&1&0&1&0 \\
-0&0&0&0&1&1&0&1 \\
-0&0&0&0&1&0&1&1 \\
-0&0&0&1&0&1&0&1
-\end{array}
-\right)
-\]
-
-
-
-
-
-\end{xpl}
+% \vspace{-0.5em}
+% It is easy to associate a Markov Matrix $M$ to such a graph $G(f)$
+% as follows:
+
+% $M_{ij} = \frac{1}{n}$ if there is an edge from $i$ to $j$ in $\Gamma(f)$ and $i \neq j$; $M_{ii} = 1 - \sum\limits_{j=1, j\neq i}^n M_{ij}$; and $M_{ij} = 0$ otherwise.
+
+% \begin{xpl}
+% The Markov matrix associated to the function $f^*$ is
+
+% \[
+% M=\dfrac{1}{3} \left(
+% \begin{array}{llllllll}
+% 1&1&1&0&0&0&0&0 \\
+% 1&1&0&0&0&1&0&0 \\
+% 0&0&1&1&0&0&1&0 \\
+% 0&1&1&1&0&0&0&0 \\
+% 1&0&0&0&1&0&1&0 \\
+% 0&0&0&0&1&1&0&1 \\
+% 0&0&0&0&1&0&1&1 \\
+% 0&0&0&1&0&1&0&1
+% \end{array}
+% \right)
+% \]
+%\end{xpl}
It is usual to check whether rows of such kind of matrices
