Composition of Post classes and normal forms of Boolean functions

The class composition C ◦ K of Boolean clones, being the set of composite functions f(g1, . . . , gn) with f ∈ C, g1, . . . , gn ∈ K, is investigated. This composition C ◦K is either the join C ∨K in the Post Lattice or it is not a clone, and all pairs of clones C,K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation.