Short complete fault detection tests for logic networks with fan-in two

It is established that one can implement almost any Boolean function on $n$ variables by a logic network in the basis $\{x\& y, x\vee y,x\oplus y, 1\}$, allowing a complete fault detection test with length not exceeding $4$ under arbitrary stuck-at faults at outputs of gates. The following assertions are also proved: one can implement any Boolean function on $n$ variables by a logic network in the basis $\{x\& y, x\vee y,x\oplus y, 1\}$ (in the basis $\{x\& y, x\vee y, x\vee\overline y, x\oplus y\}$), containing not more than one dummy variable and allowing a complete fault detection test with length not exceeding $5$ (not exceeding $4$, respectively) under faults of the same type.