Basis sets for synthesis of switching functions

The synthesis of switching function f(x₁ , x₂, . . ., x_n) from a given family of functions g_i(x₁, x₂, . . ., x_n), 1⩽i⩽k, using a complete set of logic primitives is considered. Necessary and sufficient conditions for the synthesis of f from the g_i´s are derived using the concept of a basis set. The independence between the basis property and the completeness of a set of logic primitives is shown, the conditions for extending a set {g₁, g₂ , . . ., g_j}, j<n, to a basis set are found. Thus, the selection of a basis set and the logic primitives can be treated as separate problems. Finally, it is shown that there is a unique generalized Reed-Muller expansion for any f in terms of the basis functions {g_i}