Michela Ceria edited bits4.tex  about 6 years ago

Commit id: 7ddcdfc2d3a8bce5ee47406ec48aa5d0642c5bfb

deletions | additions      

       

%  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}  %  \begin{Example}\label{ExMorevar}