Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words

Abstract

We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a $k$-reachable permutation group of degree $n$ with $6\le k\le n-6$ is either the alternating or the symmetric group.

shortened the 1st version by leaving out the part about binary automata; corrected some typos; 2nd version accepted at the 15th International Conference on Language and Automata Theory and Applications (LATA 2021), see https://irdta.eu/lata2020-2021//acceptedpapers/