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International Computer Science Symposium in Russia

CSR 2010: Computer Science – Theory and Applications pp 12–24Cite as

Sofic and Almost of Finite Type Tree-Shifts

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

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References

  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)

    Chapter  Google Scholar 

  2. Bates, T., Eilers, S., Pask, D.: Reducibility of covers of AFT shifts. Israel Journal of Mathematics (to appear, 2010)

    Google Scholar 

  3. Béal, M.-P., Fiorenzi, F., Perrin, D.: A hierarchy of shift equivalent sofic shifts. Theor. Comput. Sci. 345(2-3), 190–205 (2005)

    Article  MATH  Google Scholar 

  4. Boyle, M., Kitchens, B., Marcus, B.: A note on minimal covers for sofic systems. Proc. Amer. Math. Soc. 95(3), 403–411 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fujiwara, M., Osikawa, M.: Sofic systems and flow equivalence. Math. Rep. Kyushu Univ. 16(1), 17–27 (1987)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jonoska, N., Marcus, B.: Minimal presentations for irreducible sofic shifts. IEEE Trans. Inform. Theory 40(6), 1818–1825 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kitchens, B.P.: Symbolic dynamics. In: Universitext. Springer, Berlin (1998); One-sided, two-sided and countable state Markov shifts

    Google Scholar 

  11. Lind, D., Schmidt, K.: Symbolic and algebraic dynamical systems. In: Handbook of dynamical systems, vol. 1A, pp. 765–812. North-Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  12. Lind, D.A., Marcus, B.H.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  13. Marcus, B.H.: Sofic systems and encoding data. IEEE Transactions on Information Theory 31(3), 366–377 (1985)

    Article  MATH  Google Scholar 

  14. Nasu, M.: Textile Systems for Endomorphisms and Automorphisms of the Shift. American Mathematical Society, Providence (1995)

    Google Scholar 

  15. Perrin, D., Pin, J.: Infinite words. Elsevier, Boston (2004)

    MATH  Google Scholar 

  16. Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  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. Thomas, W.: Automata on infinite objects. In: Handbook of theoretical computer science, vol. B, pp. 133–191. Elsevier, Amsterdam (1990)

    Google Scholar 

  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. Williams, S.: Covers of non-almost-finite type sofic systems. Proc. Amer. Math. Soc. 104(1), 245–252 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’informatique Gaspard-Monge, CNRS, Université Paris-Est,  

    Nathalie Aubrun & Marie-Pierre Béal

Authors
  1. Nathalie Aubrun
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  2. Marie-Pierre Béal
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Editor information

Editors and Affiliations

  1. Dept. of Theoretical Cybernetics, Kazan State University, 420008, Kazan, Russia

    Farid Ablayev

  2. Institut für Informatik, Technische Universität München, 85748, Garching, Germany

    Ernst W. Mayr

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Aubrun, N., Béal, MP. (2010). Sofic and Almost of Finite Type Tree-Shifts. In: Ablayev, F., Mayr, E.W. (eds) Computer Science – Theory and Applications. CSR 2010. Lecture Notes in Computer Science, vol 6072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13182-0_2

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  • DOI: https://doi.org/10.1007/978-3-642-13182-0_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13181-3

  • Online ISBN: 978-3-642-13182-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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