Synthesis of circuits containing unreliable gates in $P_k$

Background. Multivalued logic gives ample opportunities for creation of various algorithms in many fields and can be successfully applied in solving problems and in multiple technical developments. This explains the interest to the problem of building reliable circuits in the complete finite basis of $k$-valued functions ($k \geq 3$), which is solved with $k$ equaling to $3$ and $4$. The aim of the work is to reveal features of $k$-valued functions ($k \geq 5$), the circuits of which may be used for improvement of initial circuits' reliability, and to describe the corresponding synthesis method. Materials and methods. The study included well-known methods of discrete mathematics and mathematical cybernetics for obtaining circuit unreliability values and for estimating a number of functions of a special type. Besides, the author suggests a new method of synthesizing circuits containing unreliable functional elements. Results. The author revealed $k$-valued functions ($k \geq 5$), the circuits that can be used for improvement of initial circuits' reliability, and described the corresponding synthesis method. The researcher obtained the upper and the lower values for a number of such functions. Conclusions. Features of three-valued and four-valued functions, revealed previously, circuits that can be used for improvement of initial circuits' reliability may be generalized in case of $k$-valued functions at any natural $k \geq 5$.