Algebraic properties of neuro-function kernel (Q2759396)

Let \(\mathbb{Z}_2=\{0,1\}\), \(A=\{a_1,\ldots,a_{q}\}\subset \mathbb{Z}_2^{n}\), \(A'=\mathbb{Z}_2^{n}\setminus A\). The author proves that \(A\subset \mathbb{Z}_2^{n}\) admits a representation by tolerance matrices if and only if \(\text{conv} A\cap\text{conv} A'=\emptyset\). Let us denote \(\rho(a,b)=\sum_{i=1}^{n}|\alpha_{i}-\beta_{i}|\), where \(a=(\alpha_1,\ldots,\alpha_{n})\in \mathbb{Z}_2^{n}\), \(b=(\beta_1,\ldots,\beta_{n})\in \mathbb{Z}_2^{n}\), and let \(O(a,b)\) be the set of unit vectors \(e_{i_1},\ldots,e_{i_{s}}\in \mathbb{Z}_2^{n}\) such that \(a\oplus b=e_{i_1}+\ldots+e_{i_{s}}\), where \(a,b\in A\subset \mathbb{Z}_2^{n}, a\neq b\). We denote by \(H(a,b)=\langle e_{i_1},\ldots,e_{i_{s}}\mid e_{i_{j}}\in O(a,b) \rangle\) the subgroup of the group \(\mathbb{Z}_2^{n}\) generated by elements from \(O(a,b)\). Let \(a \& b\) be the coordinate-wise conjunction of Boolean vectors \(a\) and \(b\), and let \(H(a\& b)\) be the residue class of group \(\mathbb{Z}_2^{n}\) with respect to the subgroup \(H(a,b)\). The following result is proved.NEWLINENEWLINENEWLINEIf \(A\subset \mathbb{Z}_2^{n} (|A|\geq 2)\) admits a representation by tolerance matrices, then for any \(a,b \in A\) for which \(|H(a\& b)\cap A'|\geq 2\) and for any \(g,h\in H(a\& b)\cap A'\) we have \(\rho(g,h)<\rho(a,b)\). On the basis of these results a necessary condition for the realizability of a Boolean function on the single neuron element is obtained.