Lower bounds for unreliability of circuits in the Rosser–Tourkett basis

We consider the implementation of four-valued logic functions by circuits consisting of unreliable functional gates in the Rosser–Tourkett basis. It is assumed that all elements of the circuit independently with probability $p$ are subject to inverse failures on the outputs, i.e. each basic element can give each of incorrect values with the probability $p$ and correct value with the probability $1-3p$. In this paper, a class $K$ of four-valued functions is introduced in such a way that almost all four-valued functions are contained in $K$ and any circuit realizing a function from $K$ operates with an unreliability that is asymptotically (for small $p$) not less than $9p$.