On non-complete sets and Restivo's conjecture

Authors:
Vladimir V. Gusev;Elena V. Pribavkina
Affiliations:
Ural State University, Ekaterinburg, Russia;Ural State University, Ekaterinburg, Russia
Venue:
DLT'11 Proceedings of the 15th international conference on Developments in language theory
Year:
2011

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Abstract

A finite set S of words over the alphabet Σ is called noncomplete if Fact(S*) ≠ Σ*. A word w ∈ Σ* \ Fact(S*) is said to be uncompletable. We present a series of non-complete sets Sk whose minimal uncompletable words have length 5k2 - 17k + 13, where k ≥ 4 is the maximal length of words in Sk. 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 2k2.