Relaibility of dual circuits in $P_{k}$

Background. Increasing complexity of modern data processing, transferring and soring systems highlights a demand for reliability and control of various controlling and computing systems. The article is devoted to a topical problem of constructing reliable circuits realizing functions from $P_k$ at random gate failures in a complete finite basis. It has been proved earlier at $k=2$ that reliability of a circuit, which realizes Boolean function f, equals unreliability of a dual circuit, built from gates of dual basi s$B^*$ and realizing a function that is dual to function f. This property makes it possible to transfer unreliability results of a circuit that realizes Boolean function f in basis $B$ with given gate failures into another dual basis $B^*$ for a dual circuit that realizes dual function $f^*$ with given failures. For example, unreliability results, proved for a circuit that trealizes Boolean function f in basis $B$ with similar constant failures of type 0 at gate outputs are fair for a dual circuit that realizes function $f^*$ in basis $B^*$ with similar constant failures of type 1 at gate outputs. The goal of the work is to find answers to the following questions: «Does this property occur in $P_k$ ($k=3$ )?», «If “yes”, for what bases, functions and failures?». Materials and methods. The study employed well-known methods of synthesis of circuits containing unreliable gates. Results. It has been proved that unreliabilities of dual (in relation to a permutation, set by the function known as the Likasiewicz's negation) circuits are equal for functions of k-valued logic. The results obtained may be used in technical systems design to improve their relaibility.