Abstract
We address the characterization of finite test-sets for cubefreeness of morphisms between free monoids, that is, the finite sets T such that a morphism f is cube-free if and only if f(T) is cube-free. We first prove that such a finite test-set does not exist for morphisms defined on an alphabet containing at least three letters. Then we prove that for binary morphisms, a set T of cube-free words is a test-set if and only if it contains twelve particular factors. Consequently, a morphism f on {a; b} is cube-free if and only if f(aabbababbabbaabaababaabb) is cube-free (length 24 is optimal). Another consequence is an unpublished result of Leconte: A binary morphism is cube-free if and only if the images of all cube-free words of length 7 are cube-free.
We also prove that, given an alphabet A containing at least two letters, the monoid of cube-free endomorphisms on A is not finitely generated.
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References
D.R. Bean, A. Ehrenfeucht and G. McNulty, Avoidable patterns in string of symbols, Pacific J. Math. 95, p261–294, 1979.
J. Berstel, Mots sans carré et morphismes itérés, Discrete Mathematics 29, p235–244, 1980.
J. Berstel, Axel Thue’s Work on repetitions in words, (4th Conf. on Formal Power Series and Algebraic Combinatorics, Montréal 1992) LITP Technical Report 92.70, 1992.
J. Berstel, Axel Thue’s papers on repetitions in words: a translation, Publications of LaCIM 20, University of Québec at Montréal.
J. Berstel and P. Séébold, A characterization of overlap-free morphisms, Discrete Applied Mathematics 46, p275–281, 1993.
F.-J. Brandenburg, Uniformly Growing k-th Power-Free Homomorphisms, Theoretical Computer Science 23, p69–82, 1983.
C. Choffrut and J. Karhumäki, chapter: Combinatorics of Words in Handbook of Formal Languages vol.1 (G. Rozenberg and A. Salomaa Eds), Springer, 1997.
M. Crochemore, Sharp Characterization of Squarefree Morphisms, Theoretical Computer Science 18, p221–22, 1982.
J. Karhumäki, On cube-free ω-words generated by binary morphisms, Discrete Applied Mathematics 5, p279–297, 1983.
V. Keränen, On the k-freeness of morphisms on free monoids, Annales Academia Scientiarum Fennicae, 1986.
M. Leconte, Codes sans répétition, Thèse de 3ème cycle, LITP Université P. et M. Curie, 1985.
M. Leconte, A characterization of power-free morphisms, Theoretical Computer Science 38, p117–122, 1985.
M. Lothaire, Combinatorics on words, Encyclopedia of Mathematics, Vol. 17, Addison-Wesley; reprinted in 1997 by Cambridge University Press in the Cambridge Mathematical Library.
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press, to appear.
M. Morse, Recurrent geodesics on a surface of negative curvature, Transactions Amer. Math. Soc. 22, p84–100, 1921.
F. Mignosi and P. Séébold, If a D0L language is k-power free then it is circular, ICALP’93, LNCS 700, Springer-Verlag, 1993, p507–518.
G. Richomme and P. Séébold, Characterization of test-sets for overlap-free morphisms, LaRIA Internal report 9724, 1997, to appear in Discrete Applied Mathematics.
P. Séébold, Sequences generated by infinitely iterated morphisms, Discrete Applied Mathematics 11, p255–264, 1985.
A. Thue, Über unendliche Zeichenreihen, Videnskapsselskapets Skrifter, I. Mat.-naturv. Klasse, Kristiania, p1–22, 1906.
A. Thue, Über die gegenseitige Lage gleigher Teile gewisser Zeichenreihen, Videnskapsselskapets Skrfter, I. Mat.-naturv. Klasse, Kristiania, p1–67, 1912.
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Richomme, G., Wlazinski, F. (2000). About Cube-Free Morphisms. In: Reichel, H., Tison, S. (eds) STACS 2000. STACS 2000. Lecture Notes in Computer Science, vol 1770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46541-3_8
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DOI: https://doi.org/10.1007/3-540-46541-3_8
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