this is for holding javascript data
Michela Ceria edited bits4.tex
about 6 years ago
Commit id: 7ddcdfc2d3a8bce5ee47406ec48aa5d0642c5bfb
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We can now conclude this section with the most general formulation of the Absolute Normal Form Theorem.
\begin{Theorem}[Absolute Normal Form Theorem]
Let $n\in \NN$.
Any Boolean function
$$f: (\Fb)^n \rightarrow (\Fb)$$
with $n\in \NN$, can be written as a polynomial in
$\Fb[x1,..,x_n]$. $\Fb[x_1,..,x_n]$. More precisely, $f$ can be written as a sum of all the \emph{squarefree monomials} of degree from $0$ to $n$ (with coefficients in $\Fb$), i.e.
$$f(x_1,...,x_n)=a_0+a_1x_1+...+a_n x_n+a_{n+1}x_1x_2+...+a_dx_1x_2...x_n $$f=a_0+a_1x_1+...+a_n x_n+a_{1,2}x_1x_2+...+a_{1,2,...,n}x_1x_2...x_n \, ,$$
where
$a_0,...,a_n $a_0,...,a_{1,2,...,n} \in \Fb .$
\end{Theorem}
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\begin{Example}\label{ExMorevar}