Slowly synchronizing automata with fixed alphabet size

TL;DR

This paper investigates the role of alphabet size in the synchronization length of automata, proves Cerny's conjecture for small cases, and constructs automata with slow synchronization for various sizes, revealing new automata and structural properties.

Contribution

It provides new constructions of automata with fixed alphabet sizes that synchronize slowly, and confirms Cerny's conjecture for automata with up to 6 states.

Findings

Confirmed Cerny's conjecture for n ≤ 6.

Constructed automata with fixed alphabet sizes that synchronize in n^2 - 3n + O(1) steps.

Discovered 19 new automata with maximal synchronization length not in Cerny's sequence.

Abstract

It was conjectured by \v{C}ern\'y in 1964 that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. In this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on $n \le 6$ states which synchronize in $(n-1)^2 - e$ steps, for all $e < 2\lceil n/2 \rceil$. Furthermore, we give constructions of automata with any number of states, and $3$, $4$, or $5$ symbols, which synchronize slowly, namely in $n^2 - 3n + O(1)$ steps. In addition, our results prove \v{C}ern\'y's conjecture for $n \le 6$. Our computation has led to $27$ DFAs on $3$, $4$, $5$ or $6$ states, which synchronize in $(n-1)^2$ steps, but do not belong to \v{C}ern\'y's sequence. Of these $27$ DFA's, $19$ are new, and the remaining $8$ which were already known are exactly the \emph{minimal} ones: they will not synchronize any more after removing a symbol. So the $19$ new DFAs are extensions of automata which were already known, including the \v{C}ern\'y automaton on $3$ states. But for $n > 3$, we prove that the \v{C}ern\'y automaton on $n$ states does not admit non-trivial extensions with the same smallest synchronizing word length $(n-1)^2$.