Preset Distinguishing Sequences and Diameter of Transformation Semigroups

We investigate the length `(n, k) of a shortest preset distinguishing sequence (PDS) in the worst case for a k-element subset of an n-state Mealy automaton. It was mentioned by Sokolovskii [18] that this problem is closely related to the problem of finding the maximal subsemigroup diameter `(Tn) for the full transformation semigroup Tn of an n-element set. We prove that `(Tn) = 2 n exp{ √ n 2 lnn(1 + o(1))} as n → ∞ and, using approach of Sokolovskii, find the asymptotics of log2 `(n, k) as n, k →∞ and k/n→ a ∈ (0, 1).