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Covering Problems from a Formal Language Point of View

  • Marcella Anselmo
  • Maria Madonia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2710)

Abstract

We consider the formal language of all words that are ‘covered’ by words in a given language. This language is said cov-free when any word has at most one minimal covering over it. We study the notion of cov-freeness in relation with its counterpart in classical monoids and in monoids of zig-zag factorizations. In particular cov-freeness is characterized by the here introduced notion of cov-stability. Some more properties are obtained using this characterization. We also show that the series counting the minimal coverings of a word over a regular language is rational.

Keywords

Covering Problem Regular Language Minimal Covering Classical Stability Covering Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Anselmo: The zig-zag power-series: a two-way version of the star operator. Theor. Comp. Sc. 79 no 1 (1991) 3–24zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Anselmo: Two-way Automata with Multiplicity. Procs. ICALP 90, LNCS 443 Springer-Verlag (1990) 88–102Google Scholar
  3. 3.
    A. Apostolico and A. Ehrenfeucht: Efficient Detection of quasiperiodicities in strings. Theor. Comp. Sc. 119 (1993) 247–265.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Berstel and D. Perrin: Theory of codes. Academic Press (1985).Google Scholar
  5. 5.
    G. S. Brodal and C. N. S. Pedersen: Finding Maximal Quasiperiodities in Strings. Procs. 11 Annual Symp. on Comb. Pattern Matching LNCS 1848 (2000) 347–411Google Scholar
  6. 6.
    G. Cohen, I. Honkala, S. Litsyn and A. Lobstein: Covering Codes. Elsevier, North-Holland Mathematical Library 54 (1997)Google Scholar
  7. 7.
    Do Long Van, B. Le Saëc and I. Litovsky: Stability for the zigzag submonoids. Theor. Comp. Sc. 108 (1993) 237–249.zbMATHCrossRefGoogle Scholar
  8. 8.
    D. Gusfield: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge Univ. Press (1997)Google Scholar
  9. 9.
    T. Head: Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors. Bull. Math. Biol. 49 (1987) 737–759.zbMATHMathSciNetGoogle Scholar
  10. 10.
    J. E. Hopcroft and J. D. Ullman: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Reading, MA, (1995).Google Scholar
  11. 11.
    C. S. Iliopoulos, D. W. G. Moore and K. Park: Covering a string. Proc. 4th Symp. Combinatorial Pattern Matching LNCS 684 (1993) 54–62.CrossRefGoogle Scholar
  12. 12.
    C. S. Iliopoulos and W. F. Smyth: On-line Algorithms for k-Covering. Procs. ninth AWOCA (1998) 97–106.Google Scholar
  13. 13.
    B. Le Saec, I. Litovsky and B. Patrou: A more efficient notion of zigzag stability. RAIRO Informatique Théorique 30 n.3 (1996) 181–194.zbMATHGoogle Scholar
  14. 14.
    I. Litovsky and B. Patrou: On a binary zigzag operation. Proc. 3rd Int. Conf. Developments in Languages Theory, Thessaloniki, July 1997, Bozapalidis Ed., Aristotle University of Thessaloniki, 273–289.Google Scholar
  15. 15.
    M. Madonia, S. Salemi and T. Sportelli: Covering submonoids and covering codes. Journal of Automata, Languages and Combinatorics 4 n.4 (1999) 333–350.zbMATHMathSciNetGoogle Scholar
  16. 16.
    M. Madonia, S. Salemi and T. Sportelli: On z-submonoids and z-codes. RAIRO Informatique théorique 25 n.4 (1991) 305–322.zbMATHMathSciNetGoogle Scholar
  17. 17.
    D. Moore and W. F. Smyth: An optimal algorithm to compute all the covers of a string. Inf. Proc. Letters 50(5) (1994) 239–246 and further corrections in Inf. Proc. Letters 50 (2) (1995) 101–103.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. S. Sim, C. S. Iliopoulos, K. Park and W. F. Smyth: Approximate Periods of Strings. Theor. Comp. Sc. 262 (2001) 557–568.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Marcella Anselmo
    • 1
  • Maria Madonia
    • 2
  1. 1.Dip. di Informatica ed ApplicazioniUniversità di SalernoBaronissi (SA)Italy
  2. 2.Dip. di Matematica ed InformaticaUniversità di CataniaCataniaItaly

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