Transformation completeness properties of SVPC transformation sets

AbstractA set T of permutations of a finite set D is said to be transformation complete if the orbits of 〈T〉, the group generated by T, acting on F(D), the power set of D, are exactly the set of subsets of D having the same cardinality, where the orbit of xϵF(D) is {α(x)|αϵ〈T〉}. This paper studies the transformation completeness properties of suppressed variable permutation and complementation (SVPC) transformations which act on Boolean variables with domain being D = {0, 1}n. An SVPC transformation with r control variables is an identity on the n-cube except on an (n − r)-subcube where the acting is like a variable permutation and complementation (VPC) transformation on n − r variables, 0≤r<n. Let Pnr be the set of all SVPC transformations on n variables with r control variables. It is shown that Pnr is transformation complete for n>r≥1. In particular, it is shown that S2n = 〈Pnn−1〉 = 〈Pnn−2〉 ⊃ 〈Pnn−3〉 = 〈Pnn−4〉 = ⋯ = 〈Pn1〉 = A2n ⊃ 〈Pn0〉, where S2n and A2n are the symmetric group and alternating group of degree 2n, respectively. Pn0, i.e., the VPC transformation group on n variables, is not transformation complete, however. Thus, one control variable is necessary and sufficient to make Pnr transformation complete