Estimations of unreliability of circuits in Rosser–Turkett basis (in $P_3$) with faults of type $0$ at the outputs of gates

This work belongs to one of the most important branches of mathematical cybernetics such as the theory of the reliability of control systems. The synthesis problem for reliable control systems is one of the main problems in discrete mathematics and mathematical cybernetics. The topicality of the research in this field is due to the importance of numerous applications arising in various sections of science and technology. We consider the realization of ternary logic functions by circuits consisting of unreliable functional elements in Rosser–Turkett basis. We assume that all the circuit elements are exposed to faults of type 0 at their outputs and pass to fault states independently with the probability $\varepsilon$ ($\varepsilon<1/2$). We have obtained the following results: 1) any function of ternary logic can be realized by a circuit with unreliability that is asymptotically not more than $\varepsilon$ for small $\varepsilon$; 2) for any function except the constant $0$ and the variable $x_i$ ($i\in\mathbb N$), such a circuit has the asymptotically optimal reliability and operates with the unreliability asymptotically equal to $\varepsilon$ for small $\varepsilon$; 3) the functions $0$ and $x_i $ can be realized absolutely reliably.