Abstract
The Billaud Conjecture, which has been open since 1993, is a fundamental problem on finite words w and their heirs, i.e., the words obtained by deleting every occurrence of a given letter from w. It posits that every morphically primitive word, i.e. a word which is a fixed point of the identity morphism only, has at least one morphically primitive heir. In this paper, we introduce and investigate the related class of so-called Billaud words, i.e. words whose all heirs are morphically imprimitive. We provide a characterisation of morphically imprimitive Billaud words, using a new concept. We show that there are two phenomena through which words can have morphically imprimitive heirs, and we highlight that only one of those occurs in morphically primitive words. Finally, we examine our concept further, use it to rephrase the Billaud Conjecture and study its difficulty.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Billaud, M.: A problem with words. Newsgroup âcomp.theoryâ (1993). https://groups.google.com/d/msg/comp.theory/V_xDDtoR9a4/zgcM4We0CisJ
Filè, G.: The relation of two patterns with comparable languages patterns. RAIRO - Theor. Inf. Appl. (Informatique ThĂŠorique et Applications) 23(1), 45â57 (1989)
Geser, A.: Your âProblem with Wordsâ (1993). Private communication to M. Billaud
Hamm, D., Shallit, J.: Characterization of finite and one-sided infinite fixed points of morphisms on free monoids. Technical report CS-99-17, University of Waterloo, Ontario, Canada (July 1999)
Head, T.: Fixed languages and the adult languages of OL schemes. Int. J. Comput. Math. 10(2), 103â107 (1981)
Holub, Ĺ : Polynomial-time algorithm for fixed points of nontrivial morphisms. Discret. Math. 309(16), 5069â5076 (2009)
Jiang, T., Salomaa, A., Salomaa, K., Yu, S.: Decision Problems for Patterns. J. Comput. Syst. Sci. 50(1), 53â63 (1995)
Kociumaka, T., Radoszewski, J., Rytter, W., WaleĹ, T.: Linear-time version of Holubâs algorithm for morphic imprimitivity testing. Theoret. Comput. Sci. 602, 7â21 (2015)
LevĂŠ, F., Richomme, G.: On a conjecture about finite fixed points of morphisms. Theoret. Comput. Sci. 339(1), 103â128 (2005)
Matocha, V., Holub, Ĺ : Complexity of testing morphic primitivity. Kybernetika 49(2), 216â223 (2013)
Nevisi, H., Reidenbach, D.: Morphic primitivity and alphabet reductions. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 440â451. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31653-1_39
Reidenbach, D.: Discontinuities in pattern inference. Theoret. Comput. Sci. 397(1), 166â193 (2008)
Reidenbach, D., Schneider, J.C.: Morphically primitive words. Theoret. Comput. Sci. 410(21), 2148â2161 (2009)
Walter, T.: Ăber die Billaudsche Vermutung. Diplomarbeit, Universität Stuttgart, Fakultät Informatik, Elektrotechnik und Informationstechnik, Stuttgart, Germany (August 2011)
Zimmermann, P.: A problem with words from Michel Billaud (1993). Private communication to M. Billaud
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Š 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Ĺopaciuk, S., Reidenbach, D. (2021). On Billaud Words and Their Companions. In: Lecroq, T., Puzynina, S. (eds) Combinatorics on Words. WORDS 2021. Lecture Notes in Computer Science(), vol 12847. Springer, Cham. https://doi.org/10.1007/978-3-030-85088-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-85088-3_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-85087-6
Online ISBN: 978-3-030-85088-3
eBook Packages: Computer ScienceComputer Science (R0)