Modifications of threshold criterion for Boolean functions on the basis of properties of characteristic vectors (Q2759398)

Let \(f(x_1,\ldots,x_{n})\) be an \(n\)-placed Boolean function, \(f^{-1}(0)=\{X\in \mathbb{Z}_2^{n}\mid f(X)=0\}\). The neuron element with structure vector \([w;T]\) (where \(w\) is a weight vector, \(T\) is a threshold) realizes the Boolean function \(f(x)\) if for all \(X\in \mathbb{Z}_2^{n}\) we have \(X\in f^{-1}(0)\Leftrightarrow (w,X)<T\). The Boolean function \(f(X)\) in the alphabet \(\{0,1\}\) is assigned to the Boolean function \(g_{f}(Y)\) in the alphabet \(\{-1,1\}\), where \(Y=2X-1\), \(g_{f}(Y)=2f(X)-1\). For \(a=(\alpha_1,\ldots,\alpha_{n})\in G_{n}=\{-1,1\}^{n}\), let \(y_{i}(a)=\alpha_{i}\) if \(i\in\{1,\ldots,n\}\) and \(y_{i}(a)=1\), if \(i=0\). The vector \(b_{f}=(b_1,\ldots,b_{n};b_0)\) is called the characteristic vector of the Boolean function \(f\) if \(b_{i}=\sum_{a\in G_{n}}y_{i}(a)g_{f}(a)\), \(i=0,\ldots,n\). The following result is proved.NEWLINENEWLINENEWLINEAssume that \(b_{n-k+1}=\dots= b_n\) for the characteristic vector \(b_{f}\) of the Boolean function \(f(x_1,\ldots,x_{n})\). Then \(f(x_{1},\ldots,x_{n})\) is realizable on the single neuron element if and only if: 1) for all \(a,c\in \mathbb{Z}_2^{k}:\|a\|=\|c\|\Rightarrow f_{a}(x_1,\ldots,x_{n-k})=f_{c}(x_1,\ldots,x_{n-k})\), where \(\|a\|=\sum_{i=1}^{n}\alpha_{i}\), \(\alpha_{i}\in \mathbb{Z}_2\), \(f_{a}(x_1,\ldots,x_{n-k})=f(x_1,\ldots,x_{n-k},\alpha_1,\ldots,\alpha_{k})\); 2) there exist real numbers \(T\) and \(\omega\), and a real vector \((w_1,\ldots,w_{n-k})\) such that the Boolean functions NEWLINE\[NEWLINEf_{s}=f(x_1,\ldots,x_{n-k},0,\ldots,0,\mathop{\underbrace{1,\ldots,1}}\limits_{s})\qquad s=0,\ldots,kNEWLINE\]NEWLINE are realizable on the neuron elements with the same weight vector \((w_1,\ldots,w_{n-k})\) and corresponding threshold \(T-s\omega\).