Composing short 3-compressing words on a 2-letter alphabet

A finite deterministic (semi) automaton A = (Q, Sigma, delta) is k-compressible if there is some word w is an element of Sigma(+) such that the image of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is called a k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing for each k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata with k + 1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabet is 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and, with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we show that the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 <= c(3, 2) <= 53 for the length c (3, 2) of the shortest 3-collapsing word on a two-letter alphabet.