Preset Distinguishing Sequences and Diameter of Transformation Semigroups

TL;DR

This paper analyzes the minimal length of preset distinguishing sequences in automata and establishes the asymptotic behavior of the maximal subsemigroup diameter in transformation semigroups, revealing exponential growth rates.

Contribution

It proves the asymptotic formula for the diameter of the full transformation semigroup and derives the growth rate of the shortest preset distinguishing sequences.

Findings

The diameter of the full transformation semigroup grows as $2^n imes ext{exp}igrace{ ext{sqrt}(rac{n}{2} ext{ln} n)}$.

Asymptotic behavior of $ ext{log}_2 ext{ell}(n,k)$ derived for large $n,k$ with fixed ratio.

Provides new bounds and formulas for preset distinguishing sequences in automata theory.

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 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 $k/n\to a\in (0,1)$.