Asymptotic reliability-optimal circuits in the Кosser-Еurkett basis in $P_4$

Background. The multivalued logic offers ample opportunities for various algorithms development in multiple fields. It allows to decrease both the computational complexity and the magnitude, a number of connections in various arithmetic and logic units, to increase the density of gate placement on circuits, to find alternative methods of problem solving. Already nowadays the multivalued logic is successfully applied for solution of multiple problems and in many technological developments. The latter include various arithmetical devices, systems of artificial intelligence and data processing, complex digital signal processing etc. The research of reliability of circuit functioning in the complete finite basis from $k$-valued functions ($k \geq 3$) is of certain interest. The problem of reliable circuit building in a random complete basis from three-valued functions (i.e. $k=3$) has been solved in the thesis work by O.Yu. Barsukova. The aim of the work is to build asymptotically reliability-optimal circuits in the Rosser-Turkett basis at $k=4$. Results. The authors found a circuit that may be used to increase the reliability of initial circuits, obtained a recurrent correlation for unreliabilities of the initial circuit and the estimated circuit. The researchers described the method of reliable circuits synthesis, obtained the upper estimate of circuit unreliability. The article describes $K$ class functions, containing almost all four-valued functions, proves the lower estimate of circuit unreliability, realizing function of the said class. For the $K$ class functions the authors built a circuit, the lower and upper estimates of which are asymptotically equal. Cocnlusions. Almost any function of four-valued logic may be realized by an asymptotically reliability-optimal circuit.