Binary Completely Reachable Automata

Abstract

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. It was conjectured that in each completely reachable automaton with n states, every set of k > 0 states is the image of a word of length at most n(n − k). We confirm the conjecture for completely reachable automata with two input letters satisfying certain restrictions on the action of the letters.