Preset Distinguishing Sequences and Diameter of Transformation Semigroups

We investigate the length \(\ell (n,k)\) 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 \(\ell (\mathbf {T}_n)\) for the full transformation semigroup \(\mathbf {T}_n\) of an \(n\)-element set. We prove that \(\ell (\mathbf {T}_n)=2^n\exp \{\sqrt{\frac{n}{2}\ln n}(1+ o(1))\}\) as \(n\rightarrow \infty \) and, using approach of Sokolovskii, find the asymptotics of \(\log _2 \ell (n,k)\) as \(n,k\rightarrow \infty \) and \(k/n\rightarrow a\in (0,1)\).