On Non-complete Sets and Restivo’s Conjecture

  • Vladimir V. Gusev
  • Elena V. Pribavkina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)


A finite set S of words over the alphabet Σ is called non-complete if \(\textit{Fact}(S^*)\ne\Sigma^*\). A word \(w\in\Sigma^*\setminus\textit{Fact}(S^*)\) 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Vladimir V. Gusev
    • 1
  • Elena V. Pribavkina
    • 1
  1. 1.Ural State UniversityEkaterinburgRussia

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