Abstract
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 \(k/n\rightarrow a\in (0,1)\).
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Panteleev, P. (2015). Preset Distinguishing Sequences and Diameter of Transformation Semigroups. In: Dediu, AH., Formenti, E., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2015. Lecture Notes in Computer Science(), vol 8977. Springer, Cham. https://doi.org/10.1007/978-3-319-15579-1_27
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DOI: https://doi.org/10.1007/978-3-319-15579-1_27
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