Binary Patterns in Infinite Binary Words

  • Antonio Restivo
  • Sergio Salemi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2300)


In this paper we study the set P(ω) of binary patterns that can occur in one infinite binary word ω, comparing it with the set F(ω) of factors of the word. Since the set P(ω) can be considered as an extension of the set F(ω), we first investigate how large is such extension, by introducing the parameter △(ω) that corresponds to the cardinality of the difference set P(ω) / F(ω). Some non trivial results about such parameter are obtained in the case of the Thue-Morse and the Fibonacci words. Since, in most cases, the parameter △(ω) is infinite, we introduce the pattern complexity of ω, which corresponds to the complexity of the language P(ω). As a main result, we prove that there exist infinite words that have pattern complexity that grows more quickly than their complexity. We finally propose some problems and new research directions.


Pattern Complexity Injective Morphism Binary Word Binary Alphabet Standard Word 
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  1. 1.
    Allouche, J.P.: Sur la complexité des suites infinies. Bull. Belg. Math. Soc. 1 (1994) 133–143zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bean, D.R., Ehrenfeucht, A., McNulty, G.F.: Avoidable Patterns in Strings of Symbols. Pacific J. Math. 85 (1984) 261–294MathSciNetGoogle Scholar
  3. 3.
    Berstel, J., Séébold, P.: Sturmian Words. In: Lothaire, M. (Ed.): Algebraic Combinatorics on Words. Chap. 2. Cambridge University Press (2001)Google Scholar
  4. 4.
    Cassaigne, J.: Motifs evitables et regularites dans les mots. These de Doctotat, Universite Paris VI, Report LITP TH 94.04 (1994)Google Scholar
  5. 5.
    Cassaigne, J.: Unavoidable Pattern. In: Lothaire, M. (Ed.): Algebraic Combinatorics on Words. Chap. 3. Cambridge University Press (2001)Google Scholar
  6. 6.
    Choffrut, C., Karhumaki, J.: Combinatorics onWords. In: Rozenberg, G., Salomaa, A. (Eds.) The Handbook of Formal Languages. Springer, Berlin (1997)Google Scholar
  7. 7.
    Ehrenfeucht, A., Lee, K.P., Rozenberg, G.: Subword Complexities of Various Classes of Deterministic Developmental Languages without Interactions. Theoret. Comput. Sci. 1 (1975) 59–75zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Guaiana, D.: On the Binary Patterns of the Thue-Morse Infinite Word. Internal Report, University of Palermo (1996)Google Scholar
  9. 9.
    Kolpakov, R., Kucherov, G.: On Maximal Repetitions in Words. In: Proc. 12-th International Symposium on Fundamentals of Computer Science Lecture Notes in Comput. Sci., Vol. 1684. Springer-Verlag (1999) 374–385Google Scholar
  10. 10.
    Mignosi, F.: On the Number of Factors of Sturmian Words. Theor. Comp. Sci. 82 (1991) 71–84zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mignosi, F., Pirillo, G.: Repetitions in the Fibonacci Infinite Word. RAIRO Theoretical Informatics and Applications 26(3) (1992) 199–204zbMATHMathSciNetGoogle Scholar
  12. 12.
    Mignosi, F., Restivo, A.: Periodicity. In: Lothaire, M. (Ed.): Algebraic Combinatorics on Words. Chap. 8. Cambridge University Press (2001)Google Scholar
  13. 13.
    Mignosi, F., Séébold, P.: If a DOL Language Is k-power-free Then It Is Circular. In: Proc. ICALP’93, Lecture Notes in Comput. Sci., Vol. 700. Springer-Verlag (1993)Google Scholar
  14. 14.
    Mignosi, F., Restivo, A., Sciortino, M.: Words and Forbidden Factors. Theoret. Comput. Sci., to appearGoogle Scholar
  15. 15.
    Restivo, A., Salemi, S.: Patterns andWords. In: Proc. 5th International Conference DLT 2001, Wien, Austria, July 16–21, Lecture Notes in Comput. Sci., to appearGoogle Scholar
  16. 16.
    Thue, A.: Über unendliche Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl., Christiana 7 (1906)Google Scholar
  17. 17.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl., Christiana 12 (1912)Google Scholar
  18. 18.
    Zimin, A.I.: Blocking Sets of Terms. Math. USSRSb 47 (1979) 353–364CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Antonio Restivo
    • 1
  • Sergio Salemi
    • 1
  1. 1.Dipartimento di Matematica ed ApplicazioniUniversity of PalermoPalermoItaly

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