Recurrent relationships for unreliabilities of circuits with single-type constant malfunctions of types $0$ and ($k-1$) in the basis consisting from a web function, in $P_{k}$

Background. The increasing complexity of modern systems for processing, transmitting and storing information highlights the requirement for the reliability of control and computing systems. This article is devoted to the actual problem of building reliable circuits that implement functions of $k$-valued logic ($k\geq 3$), with one-type constant failures of type $0$ only or of type ($k-1$) on outputs of gates. The goal of the work is to find a method for synthesizing reliable circuits and to obtain recurrent relations for the unreliability of the original and proposed circuits with the above-mentioned failures. Materials and methods. The wor applied the methods of the theory of synthesis of reliable circuits and combinatorics. Results. In each of the two cases of failures of functional gates, recurrent relations were found for the unreliability of the original and proposed circuits. The obtained results can be used in the synthesis of reliable circuits, obtaining the upper and lower estimates of the reliability of circuits.