On reliability of non-branching programs in a basis containing the Sheffer stroke

Background. In mathematical cybernetics, one of the main areas of research is the study of control systems. Control systems are models of real computing devices. Such models include, for example, circuits of functional elements, branching and non-branching programs, etc. The relevance of these studies is connected with numerous applications arising in various fields of science and technology. In this article the reliability of non-branching programs with conditional stop operator is searched. Studies show that the use of conditional stop operators can significantly raise the reliability of non-branching programs. In this paper one particular case is considered, namely, the implementation of Boolean functions by non-branching programs in a complete finite basis containing the Scheffer stroke function. It is assumed that both computational and conditional stop operators can independently switch to the fault conditions: of an arbitrary type (computational operators) and of the first and second kind (stop operators). Materials and methods. Methods of discrete mathematics, mathematical cybernetics, mathematical analysis were used. Results. In the considered basis an upper bound of the unreliability of non-branching programs with a conditional stop operator is found, this estimate tends to zero with increasing number of iterations. Conclusions. In a complete finite basis containing the Scheffer stroke any Boolean function can be implemented by an arbitrarily reliable non-branching program with unreliable operators (both computational and stopping), at that failures of computational operators are arbitrary.