Sofic and Almost of Finite Type Tree-Shifts

  • Nathalie Aubrun
  • Marie-Pierre Béal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)

Abstract

We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique minimal deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Shannon cover of the tree-shift. We define the notion of almost finite type tree-shift which is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible sofic tree-shifts. We characterize the Shannon cover of an almost finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubrun, N., Béal, M.-P.: Decidability of conjugacy of tree-shifts of finite type. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 132–143. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bates, T., Eilers, S., Pask, D.: Reducibility of covers of AFT shifts. Israel Journal of Mathematics (to appear, 2010)Google Scholar
  3. 3.
    Béal, M.-P., Fiorenzi, F., Perrin, D.: A hierarchy of shift equivalent sofic shifts. Theor. Comput. Sci. 345(2-3), 190–205 (2005)MATHCrossRefGoogle Scholar
  4. 4.
    Boyle, M., Kitchens, B., Marcus, B.: A note on minimal covers for sofic systems. Proc. Amer. Math. Soc. 95(3), 403–411 (1985)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007), http://www.grappa.univ-lille3.fr/tata (release October 12, 2007)
  6. 6.
    Coven, E., Johnson, A., Jonoska, N., Madden, K.: The symbolic dynamics of multidimensional tiling systems. Ergodic Theory and Dynamical Systems 23(02), 447–460 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fujiwara, M., Osikawa, M.: Sofic systems and flow equivalence. Math. Rep. Kyushu Univ. 16(1), 17–27 (1987)MathSciNetGoogle Scholar
  8. 8.
    Johnson, A.S.A., Madden, K.M.: The decomposition theorem for two-dimensional shifts of finite type. Proc. Amer. Math. Soc. 127(5), 1533–1543 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jonoska, N., Marcus, B.: Minimal presentations for irreducible sofic shifts. IEEE Trans. Inform. Theory 40(6), 1818–1825 (1994)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kitchens, B.P.: Symbolic dynamics. In: Universitext. Springer, Berlin (1998); One-sided, two-sided and countable state Markov shiftsGoogle Scholar
  11. 11.
    Lind, D., Schmidt, K.: Symbolic and algebraic dynamical systems. In: Handbook of dynamical systems, vol. 1A, pp. 765–812. North-Holland, Amsterdam (2002)CrossRefGoogle Scholar
  12. 12.
    Lind, D.A., Marcus, B.H.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)MATHCrossRefGoogle Scholar
  13. 13.
    Marcus, B.H.: Sofic systems and encoding data. IEEE Transactions on Information Theory 31(3), 366–377 (1985)MATHCrossRefGoogle Scholar
  14. 14.
    Nasu, M.: Textile Systems for Endomorphisms and Automorphisms of the Shift. American Mathematical Society, Providence (1995)Google Scholar
  15. 15.
    Perrin, D., Pin, J.: Infinite words. Elsevier, Boston (2004)MATHGoogle Scholar
  16. 16.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  17. 17.
    Seidl, H.: On the finite degree of ambiguity of finite tree automata. In: Csirik, J.A., Demetrovics, J., Gecseg, F. (eds.) FCT 1989. LNCS, vol. 380, pp. 395–404. Springer, Heidelberg (1989)Google Scholar
  18. 18.
    Thomas, W.: Automata on infinite objects. In: Handbook of theoretical computer science, vol. B, pp. 133–191. Elsevier, Amsterdam (1990)Google Scholar
  19. 19.
    Williams, R.F.: Classification of subshifts of finite type. In: Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn.). Lecture Notes in Math., vol. 318, pp. 281–285. Springer, Berlin (1973)Google Scholar
  20. 20.
    Williams, S.: Covers of non-almost-finite type sofic systems. Proc. Amer. Math. Soc. 104(1), 245–252 (1988)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nathalie Aubrun
    • 1
  • Marie-Pierre Béal
    • 1
  1. 1.Laboratoire d’informatique Gaspard-Monge, CNRSUniversité Paris-Est 

Personalised recommendations