3x_1^4+11\,x_1^4x_3^4x_2^4\\ 
&& +x_2^4x_3^2x_1+x_2^5x_3^3x_1^2-3\,x_2^5x_3^3x_1^3-3\,x_2^4x_3^3x_1 - 2\,x_2^5x_3^4x_1^2+6\,x_2^5x_3^4x_1^3\\ 
&& +3\,x_2^5x_3^3x_1^4-6\,x_1^4x_3^4x_2^5-x_2^3x_3^3x_1^5+2\,x_1^5x_3^4x_2^3+2\,x_2^4x_3^3x_1^5- \\ 
&& 4\,x_1^5x_3^4x_2^4- x_1^5x_2^5x_3^3+2\,x_1^5x_2^5x_3^4+x_2^2x_3^4x_1+x_2^3x_3^5x_1^2-3\,x_2^3x_3^5x_1^3\\ 
&& +3\,x_1^4x_3^5x_2^3- 3\,x_2^3x_3^4x_1-2\,x_2^4x_3^5x_1^2+6\,x_2^4x_3^5x_1^3-6\,x_1^4x_3^5x_2^4+2\,x_2^4x_3^4x_1\\ 
&& +x_2^5x_3^5x_1^2- 3\,x_2^5x_3^5x_1^3 +3\,x_1^4x_3^5x_2^5-x_1^5x_3^5x_2^3+2\,x_1^5x_3^5x_2^4-x_1^5x_2^5x_3^5.
\end{eqnarray*}
Now the {\it look up table} of a rule $g$ on $[0,1]^3$ is the image of the 8 boolean triples $(0,0,0), (0,0,1), \ldots, (1,1,1)$ under $g$. Given the nature of probabilistic ECA 110 (PCA 110) as seen above, we observe that the look up tables of probabilistic and FCA 110 are identical. In other words, the values of the corresponding fuzzification $g(x,y,z)$ all agree on the 8 basic triples. Although more difficult to write down explicitly, it can be verified directly without too much effort that this is also the case for Product ECA 110 (see Table~\ref{ta1} below), G\"{o}del ECA 110 and even the Lukasiewicz ECA 110 (for any $p \geq 1$). In addition, these continuous ECA also agree with Boolean ECA 110 when restricted to the Boolean space $\{0,1\}$. In fact, the reason for this equivalence of look-up tables, for any rule number, is that all the multi-valued logics mentioned here agree with standard propositional calculus when restricted to the two-point set $\{0,1\}$.

\begin{table}
\begin{center}
\begin{tabular}{||c|c||c|c|c||}\hline
Boolean triples\, &\, Boolean 110\, &\,  Fuzzy 110\, &\,  Probabilistic 110\,&\, Product 110\, \\ \hline
$000$&$	0	 $&$ 0			$&$ 0			$&$ 0			$ \\
$001$&$	1	 $&$ 1			$&$ 1			$&$ 1			$ \\
$010$&$	1	 $&$ 1			$&$ 1			$&$ 1			$ \\
$011$&$	1	 $&$ 1			$&$ 1			$&$ 1			$ \\
$100$&$	0	 $&$ 0			$&$ 0			$&$ 0			$ \\
$101$&$	1	 $&$ 1			$&$ 1			$&$ 1			$ \\
$110$&$	1	 $&$ 1			$&$ 1			$&$ 1			$ \\
$111$&$	0	 $&$ 0			$&$ 0			$&$ 0			$ \\
\hline
\end{tabular}
\end{center}
\caption{The look-up tables of the elementary cellular automaton, ECA 110, under diverse multi-valued logics}
\label{ta1}
\end{table}



\section{The long term dynamics of ECA under CFMS logic}
\begin{Exa}
\label{ex184}
Since $184 = 2^3+2^4+2^5+2^7$ we see that the rule number $184 = \sum_{i=0}^{7} r_i\,2^i $
forces $r_i = 1$ only for $i=3, 4, 5, 7$. Use of the disjunctive normal form expression above gives us
\begin{eqnarray}
g_{184}(x_1,x_2, x_3) & = &\vee_{i|r_i=1} \wedge_{j=1}^{3} x_j^{d_{ij}},\nonumber\\
&=&  (x_1^{d_{31}} \wedge x_2^{d_{32}} \wedge x_3^{d_{33}}) \vee (x_1^{d_{41}} \wedge x_2^{d_{42}} \wedge x_3^{d_{43}}) \vee  (x_1^{d_{51}} \wedge x_2^{d_{52}} \wedge x_3^{d_{53}})\nonumber \\
&& \vee (x_1^{d_{71}} \wedge x_2^{d_{72}} \wedge x_j^{d_{73}}), \nonumber \\
& = & (x_1^{{0}} \wedge x_2^{{1}} \wedge x_3^{{1}}) \vee (x_1^{{1}} \wedge x_2^{{0}} \wedge x_3^{{0}}) \vee (x_1^{{1}} \wedge x_2^{{0}} \wedge x_3^{{1}}) \vee (x_1^{{1}} \wedge x_2^{{1}} \wedge x_3^{{1}}), \nonumber \\
& = & (\neg x_1 \wedge  x_2 \wedge x_3) \vee (  x_1 \wedge  \neg x_2 \wedge  \neg x_3) 
\vee ( x_1 \wedge \neg x_2 \wedge  x_3) \vee \nonumber \\
 && \vee (  x_1 \wedge  x_2 \wedge  x_3) \label{b11}\\
&=& (1-x_1)x_2x_3 + x_1(1-x_2)(1-x_3) + x_1(1-x_2)x_3 + x_1x_2x_3,\label{f10}\\
& = & x_1-x_1x_2+x_2x_3.\label{f1bis}
\end{eqnarray}
\end{Exa}
Note that in ``fuzzifying" the DNF (\ref{b1}),  we replaced $\neg x$ by $1-x$, $ x \vee y$ by $x+y$, and $x \wedge y$ in (\ref{b1}) by their product, $xy$, so as to find the fuzzy form (or rule) given by (\ref{f10}) or equivalently (\ref{f1bis}), in accordance with the fuzzification process defined above. In this case, the local rule maps the triples of zeros and ones as follows:
$$000, 001, 010, 011, 100, 101, 110, 111 \rightarrow
0,0,0,1,1,1,0,1.$$ Thus, (boolean) rule 184 is given by (\ref{b11}) above while FCA 184, given by (\ref{f1bis}), may be written as
$$g_{184}(x,y,z) = x - xy + yz,$$ for any value of $(x,y,z) \in [0,1]^3,$  (the unit cube in $\mathbf{R}^3$.) From now on, ``rule $n$" refers simply to the local ECA correponding to (classical) boolean cellular automata defined above (or in \cite{nks}) while ``F