Mathematical systems theory

, Volume 7, Issue 4, pp 353–359 | Cite as

Propriétés booléennes des langages stochastiques

  • Michel Fliess
Article

Abstract

In this paper we use results and techniques from the theory of rational power series to show that the complement of a one-letter stochastic language is stochastic, but that the family of stochastic languages is closed neither under union and intersection nor under product and homomorphism. We also give a condition on the poles of a rational one-variable power seriesr to ensure that the stochastic language defined byr and any cut-point is rational.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. [1]
    B. Benzaghou, Algèbres de Hadamard,Bull. Soc. Math. France 98 (1970), 209–252.Google Scholar
  2. [2]
    J. Berstel, Sur les pôles et le quotient de Hadamard de séries N-rationnelles,C.R. Acad. Sci. Paris 272 (1971), Série A, 1079–1081 (voir aussi: Contribution à l'étude des propriétés arithmétiques des langages formels, Thèse Sc. Math., Univ. Paris VII, Paris, 1972).Google Scholar
  3. [3]
    S. Eilenberg,Automata, Languages and Machines, Academic Press, New York, à paraître.Google Scholar
  4. [4]
    M. Fliess, Formal languages and formal power series, Séminaire IRIA Logique et Automates, Le Chesnay, 1971, pp. 77–85.Google Scholar
  5. [5]
    M. Fliess, Sur certaines familles de séries formelles, Thèse Sc. Math., Univ. Paris VII, Paris, 1972.Google Scholar
  6. [5a]
    M. Fliess, Séries rationnelles positives et processus stochastiques,Ann. Inst. H. Poincaré, Sect. B, à paraître. (Voir aussi: Automates stochastiques et séries rationnelles non commutatives,Automata, Languages and Programming (M. Nivat, ed.), North Holland, Amsterdam, 1973, pp. 397–411.)Google Scholar
  7. [6]
    G. H. Hardy etE. M. Wright,An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1938.Google Scholar
  8. [7]
    C. Lech, A note on recurring series,Arkiv Mat. 2 (1953), 417–421.Google Scholar
  9. [8]
    K. Mahler, On the Taylor coefficients of rational functions,Proc. Cambridge Philos. Soc. 52 (1956), 39–48.Google Scholar
  10. [9]
    Z. A. Melzak, Power series representing certain rational functions,Canadian J. Math. 12 (1960), 20–26.Google Scholar
  11. [10]
    M. Nasu etN. Honda, A context-free language which is not acceptable by a probabilistic automaton,Information and Control 18 (1971), 233–236.Google Scholar
  12. [11]
    A. Paz, Some aspects of probabilistic automata,Information and Control 9 (1966), 26–60.Google Scholar
  13. [12]
    M. P. Schützenberger, On the definition of a family of automata,Information and Control 4 (1961), 245–270.Google Scholar
  14. [13]
    M. P. Schützenberger, On a theorem of R. Jungen,Proc. Amer. Math. Soc. 13 (1962), 885–890.Google Scholar
  15. [14]
    M. P. Schützenberger, Parties rationnelles d'un monoïde libre, Actes Congrès Internat. Math. Nice (1970), vol. 3, pp. 281–282, Gauthier-Villars, Paris, 1971.Google Scholar
  16. [15]
    T. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, Comptes Rendus 8e Congrès Math. Scandin. Stockholm (1934), pp. 163–188, Lund, 1935.Google Scholar
  17. [16]
    P. Turakainen, On probabilistic automata and their generalizations,Annales Acad. Sci. Fennicae, Ser. A.I., no.429, 1968.Google Scholar
  18. [17]
    P. Turakainen, Generalized automata and stochastic languages,Proc. Amer. Math. Soc. 21 (1969), 303–309.Google Scholar
  19. [18]
    P. Turakainen, On languages represented in rational probabilistic automata,Annales Acad. Sci. Fennicae, Ser. A.I., no.439, 1969.Google Scholar
  20. [19]
    P. Turakainen, The family of stochastic languages is closed neither under catenation nor under homomorphism,Annal. Univ. Turku, Ser. A.I. no.133, 1970.Google Scholar
  21. [20]
    P. Turakainen, Some closure properties of the family of stochastic languages,Information and Control 18 (1971), 253–256.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1973

Authors and Affiliations

  • Michel Fliess
    • 1
  1. 1.C.N.R.S.ParisFrance

Personalised recommendations