Advertisement

Rational and Recognisable Power Series

  • Jacques Sakarovitch
Chapter
Part of the Monographs in Theoretical Computer Science. An EATCS Series book series (EATCS)

Abstract

This chapter presents the theory of weighted automata over graded monoids and with weights taken in arbitrary semirings. The first benefit of broadening the scope beyond free monoids is that it makes clearer the distinction between the rational and the recognisable series. As the topological machinery is set anyway, the star of series is defined in a slightly more general setting than cycle-free series. The main subjects covered in the chapter are then: the notion of covering of automata (also called bisimulation by some authors) and its relationship with the conjugacy of automata; the closure of recognisable series by Hadamard and shuffle products; the derivation of weighted rational expressions over a free monoid; the reduction theory of series over a free monoid and with weights in a (skew) field, that leads to a procedure for the decidability of equivalence (with a cubic complexity); and the basics for a theory of weighted rational relations. As a result, this chapter, among other things, lays the bases for the proof of the decidability of the equivalence of deterministic k-tape transducers which is one of the most striking examples of the application of algebra to ‘machine theory’.

Keywords

Rational Series Formal Power Series Division Ring Proper Part Word Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Antimirov. Partial derivatives of regular expressions and finite automaton constructions. Theoretical Computer Science, 155:291–319, 1996. zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Arnold. Systèmes de transitions finis et sémantique des processus communiquants. Masson, Paris, 1992. Translation: Finite Transitions Systems. Prentice–Hall, New York, 1994. Google Scholar
  3. 3.
    M.-P. Béal, S. Lombardy, and J. Sakarovitch. On the equivalence of ℤ-automata. In ICALP 2005, volume 3580 of Lecture Notes in Computer Science, pages 397–409. Springer, Berlin, 2005. Google Scholar
  4. 4.
    M.-P. Béal, S. Lombardy, and J. Sakarovitch. Conjugacy and equivalence of weighted automata and functional transducers. In CSR 2006, volume 3967 of Lecture Notes in Computer Science, pages 58–69. Springer, Berlin, 2006. Google Scholar
  5. 5.
    J. Berstel and C. Reutenauer. Les séries rationnelles et leurs langages. Masson, Paris, 1984. Translation: Rational Series and Their Languages. Springer, Berlin, 1988. New revised English edition available from http://www-igm.univ-mlv.fr/~berstel/. zbMATHGoogle Scholar
  6. 6.
    M. Bird. The equivalence problem for deterministic two-tape automata. Journal of Computer and System Sciences, 7:218–236, 1973. zbMATHMathSciNetGoogle Scholar
  7. 7.
    S.L. Bloom and Z. Ésik. Iteration Theories. Springer, Berlin, 1993. zbMATHGoogle Scholar
  8. 8.
    A. Cardon and M. Crochemore. Détermination de la représentation standard d’une série reconnaissable. Theoretical Informatics and Applications, RAIRO, 14:371–379, 1980. zbMATHGoogle Scholar
  9. 9.
    P. Caron and M. Flouret. Glushkov construction for multiplicities. In A. Paun and S. Yu, editors, CIAA 2000, volume 2088 of Lecture Notes in Computer Science, pages 67–79. Springer, Berlin, 2001. Google Scholar
  10. 10.
    J.-M. Champarnaud and D. Ziadi. Canonical derivatives, partial derivatives and finite automaton constructions. Theoretical Computer Science, 289:137–163, 2002. zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P.M. Cohn. Algebra. Wiley, New York, 1974. 2nd edition: volume I, 1982; volume II, 1989; volume III, 1991. zbMATHGoogle Scholar
  12. 12.
    J.H. Conway. Regular Algebra and Finite Machines. Chapman & Hall, London, 1971. zbMATHGoogle Scholar
  13. 13.
    A. Ehrenfeucht, R. Parikh, and G. Rozenberg. Pumping lemmas for regular sets. SIAM Journal on Computing, 10:536–541, 1981. zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Eilenberg. Automata, Languages and Machines, volume A. Academic Press, San Diego, 1974. Google Scholar
  15. 15.
    S. Eilenberg and M.P. Schützenberger. Rational sets in commutative monoids. Journal of Algebra, 13:173–191, 1969. zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    C.C. Elgot and J.E. Mezei. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9:47–68, 1965. zbMATHMathSciNetGoogle Scholar
  17. 17.
    M. Fliess. Formal languages and formal power series. In Séminaire Logique et Automates, IRIA, 1971, pages 77–85. Google Scholar
  18. 18.
    M. Fliess. Matrices de Hankel. Journal de Mathématiques Pures et Appliquées, 53:197–222, 1974. Erratum in: Journal de Mathématiques Pures et Appliquées, 54, 1975. zbMATHMathSciNetGoogle Scholar
  19. 19.
    M. Fliess. Sur divers produit de séries formelles. Bulletin de la Société Mathématique de France, 102:181–191, 1974. zbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Flouret and E. Laugerotte. Noncommutative minimization algorithms. Information Processing Letters, 64:123–126, 1997. CrossRefMathSciNetGoogle Scholar
  21. 21.
    W. Gröbner. Matrizenrechnung. Oldenburg, München, 1956. zbMATHGoogle Scholar
  22. 22.
    T. Harju and J. Karhumäki. The equivalence problem of multitape finite automata. Theoretical Computer Science, 78:347–355, 1991. zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    G. Higman. Ordering by divisibility in abstract algebra. Proceedings of the London Mathematical Society. Second Series, 2:326–336, 1952. zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    G. Jacob. Représentations et substitutions matricielles dans la théorie algébrique des transductions. Thèse Sci. Math. Univ. Paris VII, 1975. Google Scholar
  25. 25.
    G. Jacob. Sur un théorème de Shamir. Information and Control, 27:218–261, 1975. zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    S.C. Kleene. Representation of events in nerve nets and finite automata. In C. Shannon and J. McCarthy, editors, Automata Studies, pages 3–41. Princeton University Press, Princeton, 1956. Google Scholar
  27. 27.
    D. Krob. Complete systems of B-rational identities. Theoretical Computer Science, 89:207–343, 1991. zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    D. Krob. The equality problem for rational series with multiplicities in the tropical semiring is undecidable. In W. Kuich, editor, ICALP’92, volume 623 of Lecture Notes in Computer Science, pages 101–112. Springer, Berlin, 1992. Google Scholar
  29. 29.
    W. Kuich and A. Salomaa. Semirings, Automata, Languages. Springer, Berlin, 1986. zbMATHGoogle Scholar
  30. 30.
    F.W. Levi. On semigroups. Bulletin of the Calcutta Mathematical Society, 36:141–146, 1944 and 38:123–124, 1946. zbMATHMathSciNetGoogle Scholar
  31. 31.
    D. Lind and B. Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995. zbMATHGoogle Scholar
  32. 32.
    S. Lombardy and J. Sakarovitch. Derivation of rational expressions with multiplicity. In MFCS’02, volume 2420 of Lecture Notes in Computer Science, pages 471–482. Springer, Berlin, 2002. Google Scholar
  33. 33.
    S. Lombardy and J. Sakarovitch. Derivation of rational expressions with multiplicity. Theoretical Computer Science, 332:141–177, 2005. zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    A.I. Malcev. On the embedding of group algebras in division algebras. Doklady Akademii Nauk SSSR (N.S.), 60:1409–1501, 1948 (in Russian). MathSciNetGoogle Scholar
  35. 35.
    J. McKnight. Kleene’s quotient theorems. Pacific Journal of Mathematics, 14:43–52, 1964. MathSciNetGoogle Scholar
  36. 36.
    B.H. Neumann. On ordered division ring. Transactions of the American Mathematical Society, 66:202–252, 1949. zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    M.O. Rabin and D. Scott. Finite automata and their decision problems. IBM Journal of Research and Development, 3:125–144, 1959. Reprinted in: E. Moore, editor, Sequential Machines: Selected Papers, Addison–Wesley, Reading, 1965. MathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Restivo and C. Reutenauer. On cancellation properties of languages which are support of rational series. Journal of Computer and System Sciences, 29:153–159, 1984. zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    A.R. Richardson. Simultaneous linear equations over a division algebra. Proceedings of the London Mathematical Society, 28:395–420, 1928. CrossRefGoogle Scholar
  40. 40.
    J.M. Rutten. Automata, power series, and coinduction: Taking input derivatives seriously. In J. Wiedermann, P. van Emde Boas, and M. Nielsen, editors, ICALP’99, volume 1644 of Lecture Notes in Computer Science, pages 645–654. Springer, Berlin, 1999. Google Scholar
  41. 41.
    J.M. Rutten. Behavioural differential equations: A coinductive calculus of streams, automata, and power series. Theoretical Computer Science, 308:1–53, 2003. zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    J. Sakarovitch. Kleene’s theorem revisited. In A. Kelemenova and K. Kelemen, editors, Trends, Techniques and Problems in Theoretical Computer Science, volume 281 of Lecture Notes in Computer Science, pages 39–50. Springer, Berlin, 1987. Google Scholar
  43. 43.
    J. Sakarovitch. Éléments de théorie des automates. Vuibert, Paris, 2003. Corrected English edition: Elements of Automata Theory, Cambridge University Press, Cambridge, 2009. zbMATHGoogle Scholar
  44. 44.
    J. Sakarovitch. The language, the expression and the (small) automaton. In CIAA 2005, volume 3845 of Lecture Notes in Computer Science, pages 15–30. Springer, Berlin, 2005. Google Scholar
  45. 45.
    A. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer, Berlin, 1977. Google Scholar
  46. 46.
    M.P. Schützenberger. On the definition of a family of automata. Information and Control, 4:245–270, 1961. zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    M.P. Schützenberger. Certain elementary families of automata. In Symposium on Mathematical Theory of Automata, 1962, pages 139–153. Google Scholar
  48. 48.
    M.P. Schützenberger. On a theorem of R. Jungen. Proceedings of the American Mathematical Society, 13:885–889, 1962. zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    I. Simon. Limited subsets of a free monoid. In FOCS’78, 1978, pages 143–150. Google Scholar
  50. 50.
    E.D. Sontag. On some questions of rationality and decidability. Journal of Computer and System Sciences, 11:375–385, 1975. zbMATHMathSciNetGoogle Scholar
  51. 51.
    A. Tarski. Cardinal Algebras. Oxford University Press, London, 1949. zbMATHGoogle Scholar
  52. 52.
    L.G. Valiant. The equivalence problem for deterministic finite-turn pushdown automata. Information and Control, 25:123–133, 1974. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.LTCI, ENST/CNRSParis Cedex 13France

Personalised recommendations