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Unavoidable Sets of Partial Words

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Abstract

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of “do not know” symbols or “holes,” appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a k-letter alphabet.

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Authors and Affiliations

  1. Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC, 27402-6170, USA

    F. Blanchet-Sadri

  2. Department of Mathematics, University of Central Florida, P.O. Box 161364, Orlando, FL, 32816-1364, USA

    N. C. Brownstein & T. Weyand

  3. Franklin W. Olin College of Engineering, Olin Way, Needham, MA, 02492-1200, USA

    Andy Kalcic

  4. UCLA Mathematics Department,, Box 951555, Los Angeles, CA, 90095-1555, USA

    Justin Palumbo

Authors
  1. F. Blanchet-Sadri
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  2. N. C. Brownstein
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  3. Andy Kalcic
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  4. Justin Palumbo
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  5. T. Weyand
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Corresponding author

Correspondence to F. Blanchet-Sadri.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0452020. Part of this paper was presented at DLT’07 [4]. We thank the referees as well as Robert Mercaş and Geoffrey Scott for very valuable comments and suggestions. World Wide Web server interfaces have been established at www.uncg.edu/mat/research/unavoidablesets and www.uncg.edu/cmp/research/unavoidablesets2 for automated use of the programs.

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Blanchet-Sadri, F., Brownstein, N.C., Kalcic, A. et al. Unavoidable Sets of Partial Words. Theory Comput Syst 45, 381–406 (2009). https://doi.org/10.1007/s00224-008-9106-1

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  • DOI: https://doi.org/10.1007/s00224-008-9106-1

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