Slowly synchronizing automata with fixed alphabet size
Abstract
It was conjectured by Černý in 1964 that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n1)^2$, 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 $(n1)^2  e$ 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 Černý's conjecture for $n \le 6$. Our computation has led to $27$ DFAs on $3$, $4$, $5$ or $6$ states, which synchronize in $(n1)^2$ steps, but do not belong to Černý'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 Černý automaton on $3$ states. But for $n > 3$, we prove that the Černý automaton on $n$ states does not admit nontrivial extensions with the same smallest synchronizing word length $(n1)^2$.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.06853
 Bibcode:
 2016arXiv160906853D
 Keywords:

 Computer Science  Formal Languages and Automata Theory;
 Mathematics  Combinatorics
 EPrint:
 Replacing and extending the paper titled 'Finding DFAs with maximal shortest synchronizing word length'. Source code included