The expressibility and completeness conditions for sheaves of logic functions (Q1593940)

In this paper it is proved that for any \(k\geq 2\), the completeness problem and the expressibility problem of finite systems in finitely generated functional systems of \(s\)-functions of kind \(k\) are algorithmically solvable. Also, the same property holds for finite systems of \(s\)-functions in \({\mathcal P}_{k,s}\), and for any \(k\geq 3\) and \(n\geq 3\), there exists a Sheffer \(s\)-function in \({\mathcal P}_{k,s}\) that is essentially dependent of \(n\) variables. Here \({\mathcal P}_{k,s}\) denotes a class of functions of \(k\)-valued logic (\(s\)-functions) together with a closure operator related to the set of all \(s\)-functions realized by all possible circuits in a specified class which is defined in the paper. No proofs are given.