Synthesis of two-cascade nets of multivalued neuron elements over a finite Galois field (Q2730533)

The author proposes a method of synthesizing multivalued neuron elements and two-cascade nets of such elements over a finite Galois field \(L\). In particular, the following result is proved. Let \(H_{k}=\langle \sigma\mid \sigma^{k}=1\rangle\) be a cyclic group, \(G_{n}=H_{k}\times\ldots\times H_{k}\) the direct product of \(H_{k}\), \(\sigma=\varepsilon^{(l-1)/k}\), where \(\varepsilon\) is the primitive element of \(L\), and the integer \(k\) divide \(l-1\). The \(k\)-valued logic function \(f: G_{k}\to H_{k}\) \((k\geq 2)\) is realized on the single neuron element with the structure vector \(w=(w_0,w_1,\ldots,w_{n})\) over the field \(L\) if and only if there exists a function \(r(x)\) defined on \(G_{n}\) with nonzero value in the field \(L\) such that \(r(x)\otimes f(x)=w(x)\), \(0\leq \deg r(x)<(l-1)/k\). Here \(w(a)=w_0\oplus\sum_{i=1}^{n}w_{i}\otimes\alpha_{i}\), \(w_{i}\in L\), \(a=(\alpha_{1},\ldots, \alpha_{n})\in G_{n}\). A necessary condition for functional completeness of a two-cascade net of multivalued neuron elements is obtained.