Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Allouche, J.-P., Currie, J., Shallit, J.: Extremal infinite overlap-free binary words. Electronic J. Combinatorics 5(1), R27 (1998) (electronic), http://www.combinatorics.org/Volume_5/Abstracts/v5i1r27.html
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
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)
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)
Carpi, A.: Overlap-free words and finite automata. Theoret. Comput. Sci. 115, 243–260 (1993)
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)
Fife, E.D.: Binary sequences which contain no BBb. Trans. Amer. Math. Soc. 261, 115–136 (1980)
Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105, 335–347 (2004)
Rampersad, N., Shallit, J., Shur, A.: Fife’s theorem for \(7\over 3\)-powers (2011) (preprint)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shallit, J. (2011). Fife’s Theorem Revisited. In: Mauri, G., Leporati, A. (eds) Developments in Language Theory. DLT 2011. Lecture Notes in Computer Science, vol 6795. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22321-1_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-22321-1_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22320-4
Online ISBN: 978-3-642-22321-1
eBook Packages: Computer ScienceComputer Science (R0)