X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/586298b57c6e1cec5566bbe692ac21c4c701fd51..f18e5003b9da99d6565d366a1a9d56f8d893e5d9:/preliminaries.tex?ds=inline

diff --git a/preliminaries.tex b/preliminaries.tex
index 6d837a1..8f7774a 100644
--- a/preliminaries.tex
+++ b/preliminaries.tex
@@ -1,14 +1,11 @@
-
-
-
 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. 
@@ -37,52 +34,46 @@ the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$.
 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