Abstract
We define a code in a sofic shift as a set of blocks of symbols of the shift such that any block of the shift has at most one decomposition in code words. It is maximal if it is not strictly included in another one. Such a code is complete in the sofic shift if any block of the shift occurs within some concatenation of code words. We prove that a maximal code in an irreducible sofic shift is complete in this shift. We give an explicit construction of a regular completion of a regular code in a sofic shift. This extends the well known result of Ehrenfeucht and Rozenberg to the case of codes in sofic systems. We also give a combinatorial proof of a result concerning the polynomial of a code in a sofic shift.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Béal, M.P., Perrin, D.: Codes and sofic constraints. Theoret. Comput. Sci. 340(2), 381–393 (2005)
Reutenauer, C.: Ensembles libres de chemins dans un graphe. Bull. Soc. Math. France 114(2), 135–152 (1986)
Restivo, A.: Codes and local constraints. Theoret. Comput. Sci. 72(1), 55–64 (1990)
Ashley, J., Marcus, B., Perrin, D., Tuncel, S.: Surjective extensions of sliding-block codes. SIAM J. Discrete Math. 6(4), 582–611 (1993)
Berstel, J., Perrin, D.: Theory of codes. Pure and Applied Mathematics, vol. 117. Academic Press Inc., Orlando (1985), http://www-igm.univmlv.fr/~berstel/LivreCodes/Codes.html
Ehrenfeucht, A., Rozenberg, G.: Each regular code is included in a maximal regular code. RAIRO Inform. Théor. Appl. 20(1), 89–96 (1986)
Williams, S.: Lattice invariants for sofic shifts. Ergodic Theory and Dynamical Systems 11, 787–801 (1991)
Nasu, M.: An invariant for bounded-to-one factor maps between transitive sofic subshifts. Ergodic Theory Dynam. Systems 5(1), 89–105 (1985)
Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)
Lothaire, M.: Combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 17. Addison-Wesley Publishing Co., Reading (1983)
Restivo, A.: Codes with constraints. In: Mots. Lang. Raison. Calc., Hermès, Paris, pp. 358–366 (1990)
Kitchens, B.P.: Symbolic dynamics. Universitext. Springer, Berlin (1998); One-sided, two-sided and countable state Markov shifts
Berstel, J., Reutenauer, C.: Rational series and their languages. EATCS Monographs on Theoretical Computer Science, vol. 12. Springer, Berlin (1988)
Sakarovitch, J.: Éléments de Théorie des Automates, Vuibert, Paris. Cambridge University Press, Cambridge (2003) (english translation to appear)
Hansel, G., Perrin, D.: Mesures de probabilités rationnelles. In: Mots. Lang. Raison. Calc., Hermès, Paris, pp. 335–357 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Béal, MP., Perrin, D. (2006). Complete Codes in a Sofic Shift. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_9
Download citation
DOI: https://doi.org/10.1007/11672142_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32301-3
Online ISBN: 978-3-540-32288-7
eBook Packages: Computer ScienceComputer Science (R0)