6 problems

Let $Q=[n]$, let the alphabet be $Σ=\{a,b\}$, and choose the transition function $\delta:Q\times Σ\to Q$ uniformly at random, so that each pair in $[n]\times\{a,b\}$ is mapped inde…

Cameron's conjecture. A random automaton is synchronizable with high probability; equivalently, the probability that it is synchronizable tends to $1$ as $n$ tends to infinity.

Consider a random automaton with $n$ states and two random mapping letters, and let its reset threshold be the length of its shortest synchronizing word. Kisielewicz et al.'s conje…

Consider an $n$-state finite automaton, and let its reset threshold be the length of its shortest synchronizing word. Skvortsov–Tipikin conjecture. The expected value of the reset…

Let $\mathcal{A}_n(\mathbf{a},\mathbf{b})$ be a circular automaton of order $n$, where $\mathbf{a}:\mathbb{Z}_n\to\mathbb{Z}_n$ is fixed and $\mathbf{b}\in\mathcal{M}_n$ is chosen…

For each positive integer $n$, let $\mathcal{M}_n$ be the set of maps used to define the random circular automaton in the paper, and let $D$ denote the associated random variable c…