Some Researches On Synchronizing Automata And (?)ern(?) Conjecture

Synchronizing automaton is a special finite automaton. In robotics, industrial automationand biocomputing, it has an important application. Detection of synchronization as well as thelength of the shortest synchronizing word（especially （?）ern（?） conjecture） are the two coreproblems on the synchronizing automaton.To research the synchronizing automaton, a common method is to classify thesynchronizing automata according to their structural feature fristly, then analysis thesynchronization of the different synchronizing automata, or compute the length of theirshortest synchronizing word, therefore, it can be verified whether or not they meet the ernyconjecture.In this article, we research three special synchronizing automaton: quasi-trappedsynchronizing automaton,C_n,i synchronizing automaton and LC synchronizing automaton. Toquasi-trapped synchronizing automaton, by using the number of the states of the stronglyconnected subautomaton, a upper bound of the length of the shortest synchronizing word isgiven, and then a sufficient condition that the automaton satisfy （?）ern（?） Conjecture is obtained.To C_n,i synchronizing automaton, by using the mode complete residue system, the necessaryand sufficient condition that the automaton is synchronizing is given, and then the onlyshortest synchronizing word is obtained. To LC synchronizing automaton, by analysising thesynchronization of its isomorphism automaton LCA, the conclusion that it conforms the（?）ern（?） conjecture is obtained.