Recognizing 3-collapsing words over a binary alphabet

A finite deterministic automaton A=(Q,Σ,δ)A=(Q,Σ,δ) is k  -compressible if there is a word w∈Σ+w∈Σ+ such that the image of the state set Q under the natural action of w is reduced by at least k states. In such case w is called a k  -compressing word for AA. It is known that, for any alphabet Σ and any k≥2k≥2, there exist words that are k-compressing for each k-compressible automaton with the input alphabet Σ. Such words are called k  -collapsing. It has been proved that recognizing 2-collapsing words over a 2-element alphabet may be done in polynomial time, while recognizing 2-collapsing words over an alphabet of size n≥3n≥3 is co-NP-complete. The computational complexity of recognizing k  -collapsing words with k≥3k≥3 over an n  -element alphabet has remained open. In this paper we prove that this remaining problem is co-NP-complete in the simplest case with k=3k=3 and n=2n=2.