Fife’s Theorem Revisited

  • Jeffrey Shallit
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)


We give another proof of a theorem of Fife — understood broadly as providing a finite automaton that gives a complete description of all infinite binary overlap-free words. Our proof is significantly simpler than those in the literature. As an application we give a complete characterization of the overlap-free words that are 2-automatic.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allouche, J.-P., Currie, J., Shallit, J.: Extremal infinite overlap-free binary words. Electronic J. Combinatorics 5(1), R27 (1998) (electronic),
  2. 2.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berstel, J.: A rewriting of Fife’s theorem about overlap-free words. In: Karhumäki, J., Rozenberg, G., Maurer, H. (eds.) Results and Trends in Theoretical Computer Science. LNCS, vol. 812, pp. 19–29. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Blondel, V.D., Cassaigne, J., Jungers, R.M.: On the number of α-power-free binary words for 2 < α ≤ 7/3. Theoret. Comput. Sci. 410, 2823–2833 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carpi, A.: Overlap-free words and finite automata. Theoret. Comput. Sci. 115, 243–260 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cassaigne, J.: Counting overlap-free binary words. In: Enjalbert, P., Wagner, K.W., Finkel, A. (eds.) STACS 1993. LNCS, vol. 665, pp. 216–225. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  7. 7.
    Fife, E.D.: Binary sequences which contain no BBb. Trans. Amer. Math. Soc. 261, 115–136 (1980)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105, 335–347 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rampersad, N., Shallit, J., Shur, A.: Fife’s theorem for \(7\over 3\)-powers (2011) (preprint) Google Scholar
  10. 10.
    Restivo, A., Salemi, S.: Overlap free words on two symbols. In: Perrin, D., Nivat, M. (eds.) Automata on Infinite Words. LNCS, vol. 192, pp. 198–206. Springer, Heidelberg (1985)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jeffrey Shallit
    • 1
  1. 1.University of WaterlooWaterlooCanada

Personalised recommendations