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 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 $n\to\infty$ and, using approach of Sokolovskii, find the asymptotics of $\log_2 \ell(n,k)$ as $n,k\to\infty$ and $k/n\to a\in (0,1)$.