Description of Generalized Continued Fractions by Finite Automata

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 43)

Abstract

A generalized continued fraction algorithm associates every real number x with a sequence of integers; x is rational iff the sequence is finite. For a fixed algorithm A, call a sequence of integers valid if it is the result of A on some input x 0. We show that, if the algorithm is sufficiently well behaved, then the set of all valid sequences is accepted by a finite automaton.

Keywords

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Notes

Acknowledgements

I am very grateful to the referee for many suggestions that considerably improved the chapter.

Supported by a grant from NSERC.

References

  1. 1.
    J.-P. Allouche, Automates finis en théorie des nombres. Exposition Math. 5, 239–266 (1987)MathSciNetMATHGoogle Scholar
  2. 2.
    W.M. Beynon, A formal account of some elementary continued fraction algorithms. J. Algorithms 4, 221–240 (1983)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    F. Blumer, Über die verschiedenen Kettenbruchentwicklungen beliebiger reeller Zahlen und die periodischen Kettenbruchentwicklungen quadratischer Irrationalitäten. Acta Arith. 3, 3–63 (1939)Google Scholar
  4. 4.
    C. Brezinski, History of Continued Fractions and Padé Approximants (Springer, New York, 1991)MATHCrossRefGoogle Scholar
  5. 5.
    K. Dajani, C. Kraaikamp, The mother of all continued fractions. Colloq. Math. 84/85(Pt 1), 109–123 (2000)Google Scholar
  6. 6.
    J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, Boston, 1979)MATHGoogle Scholar
  7. 7.
    A. Hurwitz, Über die Entwicklung complexer Grössen in Kettenbrüche. Acta Math. 11, 187–200 (1888) [= Werke, II, pp. 72–83]Google Scholar
  8. 8.
    A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen. Acta Math. 12, 367–405 (1889) [ = Werke, II, pp. 84–115]Google Scholar
  9. 9.
    S. Istrail, On formal construction of algebraic numbers of degree two. Rev. Roum. Math. Pures Appl. 22, 1235–1239 (1977)MathSciNetMATHGoogle Scholar
  10. 10.
    D.E. Knuth, The Art of Computer Programming, V. II (Seminumerical Algorithms), 2nd edn (Addison-Wesley, Boston, 1981)Google Scholar
  11. 11.
    J. Lehner, Semiregular continued fractions whose partial denominators are 1 or 2, in The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry, and Special Functions, ed. by W. Abikoff, J.S. Birman, K. Kuiken (Amer. Math. Soc., New York, 1994), pp. 407–410CrossRefGoogle Scholar
  12. 12.
    E.E. McDonnell, Integer functions of complex numbers, with applications. IBM Philadelphia Scientific Center, Technical Report 320-3015, February 1973Google Scholar
  13. 13.
    O. Perron, Die Lehre von den Kettenbrüchen, in Band I: Elementare Kettenbrüche (Teubner, 1977)Google Scholar
  14. 14.
    G.N. Raney, On continued fractions and finite automata. Math. Ann. 206, 265–283 (1973)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    J.O. Shallit, Integer functions and continued fractions. A.B. Thesis, Princeton University, 1979. Available at http://www.cs.uwaterloo.ca/~shallit/papers.html
  16. 16.
    J. Steinig, On the complete quotients of semi-regular continued fractions for quadratic irrationals. Arch. Math. 43, 224–228 (1984)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    S. Tanaka, S. Ito, On a family of continued-fraction transformations and their ergodic properties. Tokyo J. Math. 4, 153–175 (1981)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H. Tietze, Über Kriterien für Konvergenz und Irrationalität unendlichen Kettenbrüche. Math. Annalen 70, 236–265 (1911)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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