On Non-complete Sets and Restivo's Conjecture

Abstract

A finite set S of words over the alphabet ∑ is called non-complete if Fact(S*) ≠ ∑*. A word w ∈ ∑* \ Fact(S*) is said to be uncompletable. We present a series of non-complete sets S k whose minimal uncompletable words have length 5k 2-17k+13, where k ≥ 4 is the maximal length of words in S k . This is an infinite series of counterexamples to Restivo's conjecture, which states that any non-complete set possesses an uncompletable word of length at most 2k 2. © 2011 Springer-Verlag.