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Symbolic Dynamics and Finite Automata

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Handbook of Formal Languages

Abstract

Symbolic dynamics is a field which was born with the work in topology of Marston Morse at the beginning of the 1920s [44]. It is, according to Morse, an “algebra and geometry of recurrence”. The idea is the following. Divide a surface into regions named by certain symbols. We then study the sequences of symbols obtained by scanning the successive regions while following a trajectory starting from a given point. A further paper by Morse and Hedlund [45] gave the basic results of this theory. Later, the theory was developed by many authors as a branch of ergodic theory (see for example the collected works in [59] or [12]). One of the main directions of research has been the problem of the isomorphism of shifts of finite type (see below the definition of these terms). This problem is not yet completely solved although the latest results of Kim and Roush [35] indicate a counterexample to a long-standing conjecture formulated by F. Williams [61].

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References

  1. Roy Adler, I. Goodwin, and Benjamin Weiss. Equivalence of topological Markov shifts. Israel J. Math., 27: 49–63, 1977.

    Article  MathSciNet  Google Scholar 

  2. Roy L. Adler, D. Coppersmith, and M. Hassner. Algorithms for sliding block codes. IEEE Trans. Inform. Theory, IT-29: 5–22, 1983.

    Google Scholar 

  3. Alfred V. Aho, John E. Hoperoft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison Wesley, 1974.

    Google Scholar 

  4. Jonathan Ashley, Brian Marcus, Dominique Perrin, and Selim Tuncel. Surjective extensions of sliding block codes. SIAM J. Discrete Math., 6: 582–611, 1993.

    Article  MathSciNet  Google Scholar 

  5. Frédérique Bassino. Non-negative companion matrices and star-height of N-rational series. Theoret. Comput. Sci.,1996. (to appear).

    Google Scholar 

  6. Frédérique Bassino. Star-height of an N-rational series. In C. Puech and R. Reischuk, editors, STACS 96, Lecture Notes in Computer Science, volume 1046, pages 125–135. Springer-Verlag, Berlin, 1996.

    Google Scholar 

  7. Marie-Pierre Béal. Codage Symbolique. Masson, 1993.

    Google Scholar 

  8. Marie-Pierre Béal. Puissance extérieure d’un automate déterministe, application au calcul de la fonction zeta d’un système sofique. R.A.I.R.OInformatique Théorique et Applications, 29: 85–103, 1996.

    Google Scholar 

  9. Marie-Pierre Beal, Oliver Carton, and Christophe Reutenauer. Cyclic languages and strongly cyclic languages. In C. Puech and R. Reischuk, editors, STACS 96, Lecture Notes in Computer Science, volume 1046, pages 49–59. Springer-Verlag, Berlin, 1996.

    Google Scholar 

  10. Marie-Pierre Beal, Filippo Mignosi, and Antonio Restivo. Minimal forbidden words and symbolic dynamics. In C. Puech and R. Reischuk, editors, STACS 96, Lecture Notes in Computer Science, volume 1046, pages 555–566. Springer-Verlag, Berlin, 1996.

    Google Scholar 

  11. Danièle Beauquier. Minimal automaton for a factorial, transitive rational language. Theoret. Comput. Sci., 67: 65–73, 1989.

    Article  MathSciNet  Google Scholar 

  12. Tim Bedford, Michael Keane, and Caroline Series, editors. Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Oxford University Press, 1991.

    Google Scholar 

  13. Jean Berstel and Dominique Perrin. Theory of codes. Academic Press, 1985.

    Google Scholar 

  14. Jean Berstel and Christophe Reutenauer. Zeta functions of formal languages. Trans. Amer. Math. Soc., 321: 533–546, 1990.

    Article  Google Scholar 

  15. Garett D. Birkhoff. Quelques théorèmes sur le mouvement des systèmes dynamiques. Bull. Soc. Math. France, 40: 305–323, 1912.

    MathSciNet  Google Scholar 

  16. François Blanchard and Alejandro Maass. On dynamical properties of generalized cellular automata. In Ricardo Baeza-Yates and Eric Goles, editors. LATIN 95: Theoretical Informatics, Lecture Notes in Computer Science, volume 911, pages 84–98, Springer-Verlag, Berlin, 1995.

    Chapter  Google Scholar 

  17. Rufus Bowen and O. Lanford. Zeta functions of restrictions of the shift transformation. Proc. Symp. Pure Math., 14: 43–50, 1970.

    Article  MathSciNet  Google Scholar 

  18. Michael Boyle. Symbolic dynamics and matrices. In S. Friedland, V. Brualdi, and V. Klee, editors, Combinatorial and Graph-Theoretic Problems in Linear Algebra, IMA Volumes in Mathematics and its Applications, volume 50, Springer-Verlag, Berlin, 1993.

    Google Scholar 

  19. Michael Boyle, Douglas Lind, and D. Rudolph. The automorphism group of a shift of finite type. Trans. of Amer. Math. Soc., 306: 71–114, 1988.

    Article  MathSciNet  Google Scholar 

  20. Véronique Bruyère and Michel Latteux. Variable-length maximal codes. In ICALP 96, Lecture Notes in Computer Science,Springer-Verlag, Berlin, 1996. (to appear).

    Google Scholar 

  21. Véronique Bruyère, L. Wang, and Liang Zhang. On completion of codes with finite decriphering delay. European J. Combin., 11: 513–521, 1990.

    Article  MathSciNet  Google Scholar 

  22. Christian Choffrut. Conjugacy in free inverse monoids. Int. J. Alg. Comput., 3: 169–188, 1993.

    Article  MathSciNet  Google Scholar 

  23. Aldo De Luca and Stefano Varricchio. Combinatorics on Words and Regularity Conditions. Springer-Verlag, Berlin, 1996. (to appear).

    Google Scholar 

  24. Samuel Eilenberg. Automata, Languages and Machines,volume A. Academic Press, 1974.

    Google Scholar 

  25. R. Fischer. Sofic systems and graphs. Monatshefte Math., 80: 179–186, 1975.

    Article  MathSciNet  Google Scholar 

  26. David Fried. Finitely presented dynamical systems. Ergod. Th. Dynam. Sys., 7: 489–507, 1987.

    MathSciNet  Google Scholar 

  27. Joel Friedman. On the road coloring problem. Proc. Amer. Math. Soc., 110: 1133–35, 1990.

    Article  MathSciNet  Google Scholar 

  28. Christiane Frougny and Boris Solomyak. Finite beta-expansions. Ergod. Th. and Dynam. Syst., 12: 45–82, 1992.

    Google Scholar 

  29. F. R. Gantmacher. The Theory of Matrices. Chelsea, 1960.

    Google Scholar 

  30. Andrew Gleason. Semigroups of shift register matrices. Mathematical Systems Theory, 25:253–267, 1992. (notes of a course given at Princeton in 1960 ).

    Google Scholar 

  31. David Handelman. Positive matrices and dimension groups affiliated to C.-algebras and topological Markov chains. J. Operator Theory, 6: 55–74, 1981.

    MathSciNet  Google Scholar 

  32. David Handelman. Spectral radii of primitive integral companion matrices and log-concave polynomials. In Peter Walters, editor, Symbolic Dynamics and its Applications, Contemporary Mathematics, volume 135, pages 231–238. Amer. Math. Soc., 1992.

    Google Scholar 

  33. George Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory, 3: 320–375, 1969.

    Article  MathSciNet  Google Scholar 

  34. K. H. Kim and F. W. Roush. Decidability of shift equivalence. In Symbolic Dynamics, Lecture Notes in Mathematics, volume 1342, pages 374–424. Springer-Verlag; Berlin, 1988.

    Google Scholar 

  35. K. H. Kim and F. W. Roush. Williams’s conjecture is false for reducible matrices. J. Amer. Math. Soc., 5: 213–215, 1992.

    MathSciNet  MATH  Google Scholar 

  36. Wolfgang Krieger. On the subsystems of topological Markov chains. Ergod. Th. E1 Dynam. Syst., 2: 195–202, 1982.

    MathSciNet  Google Scholar 

  37. Joseph Kruskal. The theory of well-quasi-ordering: a frequently rediscovered concept. J. Comb. Theory (ser. A), 13: 297–305, 1972.

    Article  Google Scholar 

  38. Douglas Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1996.

    Google Scholar 

  39. M. Lothaire. Combinatorics on Words. Cambridge University Press, 1983.

    Google Scholar 

  40. Anthony Manning. Axiom a diffeomorphisms have rational zeta function. Bull. London Math. Soc., 3: 215–220, 1971.

    Article  MathSciNet  Google Scholar 

  41. Brian Marcus. Factors and extensions of full shifts. Monats. Math., 88: 239247, 1979.

    MathSciNet  MATH  Google Scholar 

  42. Brian Marcus. Sofic systems and encoding data. IEEE Trans. Inf. Theory, IT-31: 366–377, 1985.

    Article  MathSciNet  Google Scholar 

  43. J. C. Martin. Substitution minimal sets. Amer. J. Math., 93: 503–526, 1971.

    Article  MathSciNet  Google Scholar 

  44. Marston Morse. Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc., 22: 84–110, 1921.

    Article  MathSciNet  Google Scholar 

  45. Marston Morse and George Hedlund. Symbolic dynamics. Amer. J. Math., 3: 286–303, 1936.

    Google Scholar 

  46. Masakazu Nasu. Textile Systems for Endomorphisms and Automorphisms of the Shift, volume 546 of Memoirs of Amer. Math. Soc. Amer. Math. Soc., 1995.

    Google Scholar 

  47. G. L. O’Brien. The road coloring problem. Israel J. Math, 39: 145–154, 1981.

    Article  MathSciNet  Google Scholar 

  48. William Parry and R. F. Williams. Block coding and a zeta function for finite markov chains. Proc. London Math. Soc., 35: 483–495, 1977.

    Article  MathSciNet  Google Scholar 

  49. Dominique Perrin. Arbres et séries rationnelles. C. R. Acad. Sci. Paris, 309: 713–716, 1989.

    MathSciNet  Google Scholar 

  50. Dominique Perrin. On positive matrices. Theoretical Computer Science, 94: 357–366, 1992.

    Article  MathSciNet  Google Scholar 

  51. Dominique Perrin. Symbolic dynamics and finite automata. In Jiri Wiedermann and Petr Hajek, editors, Mathematical Foundations of Computer Science 1995, Lecture Notes in Computer Science, volume 969, pages 94–104. Springer-Verlag, Berlin, 1995.

    Google Scholar 

  52. Dominique Perrin and Marcel-Paul Schützenberger. Synchronizing prefix codes and automata and the road coloring problem. In Peter Walters, editor, Symbolic Dynamics and its Applications, Contemporary Mathematics, volume 135, pages 295–318. Amer. Math. Soc. 1992.

    Chapter  Google Scholar 

  53. Antonio Restivo. Codes and local constraints. Theoret. Comput. Sci., 72: 55–64, 1990.

    Article  MathSciNet  Google Scholar 

  54. Christophe Reutenauer. Non commutative factorization of variable length codes. J. Pure and Applied Algebra, 36: 157–186, 1985.

    Google Scholar 

  55. Christophe Reutenauer. N-rationality of zeta functions. Advances in Mathematics, 1996. (to appear).

    Google Scholar 

  56. Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag, New York, 1978.

    Book  Google Scholar 

  57. Marcel-Paul Schützenberger. On synchronizing prefix codes. Inform and Control, 11: 396–401, 1967.

    Article  Google Scholar 

  58. Wolfgang Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. volume vol. B, Formal models and semantics, pages 135–191. Elsevier, 1990.

    Google Scholar 

  59. Peter Walters, editor. Symbolic Dynamics and its Applications, Contemporary Mathematics, volume 135, Amer. Math. Soc., 1992.

    Google Scholar 

  60. Benjamin Weiss. Subshifts of finite type and sofic systems. Monats. Math.. 77: 462–474. 1973.

    Article  MathSciNet  Google Scholar 

  61. Frank Williams. Classification of subshifts of finite type. Ann. of Math.. 98:120–153, 1973. (Errata ibid. 99: 380–381, 1974 ).

    Google Scholar 

  62. Liang Zhang and Zhonghui Shen. Completion of recognizable bifix codes. Theoret. Comput. Sci., 145: 345–355, 1995.

    Article  MathSciNet  Google Scholar 

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Authors
  1. Marie-Pierre Béal
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  2. Dominique Perrin
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Leiden University, P.O. Box 9512, NL-2300 RA, Leiden, The Netherlands

    Grzegorz Rozenberg

  2. Department of Computer Science, University of Colorado at Boulder, Campus 430, 80309, Boulder, CO, USA

    Grzegorz Rozenberg

  3. Turku Centre for Computer Science Data City, FIN-20520, Turku, Finland

    Arto Salomaa

  4. Academy of Finland and Turku Centre for Computer Science (TUCS), Lemninkäisenkatu 14 A, FIN-20520, Turku, Finland

    Arto Salomaa

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© 1997 Springer-Verlag Berlin Heidelberg

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Béal, MP., Perrin, D. (1997). Symbolic Dynamics and Finite Automata. In: Rozenberg, G., Salomaa, A. (eds) Handbook of Formal Languages. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07675-0_10

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  • DOI: https://doi.org/10.1007/978-3-662-07675-0_10

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