Theory of Computing Systems

, Volume 30, Issue 3, pp 285–331 | Cite as

Automatic maps in exotic numeration systems

  • J. P. Allouche
  • E. Cateland
  • W. J. Gilbert
  • H. O. Peitgen
  • J. O. Shallit
  • G. Skordev


We generalize the classical notion of ab-automatic sequence for a sequence indexed by the natural numbers. We replace the integers by a semiring and use a numeration system consisting of the powers of a baseb and an appropriate set of digits. For example, we define (−3)-automatic sequences (indexed by the ordinary integers or by the rational integers) and (−1 +i)-automatic sequences (indexed by the Gaussian integers). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties that permit the replacement of one numeration system by another.


Automatic Sequence Finite Subset Finite Automaton Numeration System Double Sequence 
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.


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  1. [1]
    J.-P. Allouche, Somme des chiffres et transcendance,Bull. Soc. Math. France 110 (1982), 279–285.MathSciNetzbMATHGoogle Scholar
  2. [2]
    J.-P. Allouche, Automates finis en théorie des nombres,Exposition. Math. 5 (1987), 239–266.MathSciNetzbMATHGoogle Scholar
  3. [3]
    J.-P. Allouche, Finite automata in 1-dimensional and 2-dimensional physics, inNumber Theory and Physics, J.-M. Lucket al., eds., Proceedings in Physics, vol. 47, Springer-Verlag, Berlin, 1990, pp. 177–184.CrossRefGoogle Scholar
  4. [4]
    J.-P. Allouche, E. Cateland, H.-O. Peitgen, J. Shallit, G. Skordev, Automatic maps on a semiring with digits,Fractals 3 (1995), 663–677.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    J.-P. Allouche, F. von Haeseler, H.-O. Peitgen, A. Petersen, G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata,Theoret. Comput. Sci., to appear.Google Scholar
  6. [6]
    J.-P. Allouche, F. von Haeseler, H.-O. Peitgen, G. Skordev, Linear cellular automata, finite automata and Pascal’s triangle,Discrete Appl. Math. 66 (1996), 1–22.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    J.-P. Allouche, J. Shallit, The ring ofk-regular sequences,Theoret. Comput. Sci. 98 (1992), 163–197.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    A. Barbé, F. von Haeseler, H.-O. Peitgen, G. Skordev, Coarse-graining invariant patterns of one-dimensional two-state linear cellular automata,Internat. J. Bifurcation and Chaos 5 (1995), 1611–1631.zbMATHCrossRefGoogle Scholar
  9. [9]
    V. Bruyère, G. Hansel, Recognizable sets of numbers in nonstandard bases, inLATIN ’95: Theoretical Informatics, R. Baeza-Yates, E. Goles, P. V. Poblete, eds., Lecture Notes in Computer Science, vol. 911, Springer-Verlag, Berlin, 1995, pp. 167–179.CrossRefGoogle Scholar
  10. [10]
    V. Bruyère, G. Hansel, Bertrand numeration systems and recognizability, preprint, 1995. Available from or Scholar
  11. [11]
    A. Cerny, J. Gruska, Modular trellises, inThe Book of L, G. Rozenberg, A. Salomaa, eds., Springer-Verlag, Berlin, 1985, pp. 45–61.Google Scholar
  12. [12]
    L. Childs,A Concrete Introduction to Higher Algebra, Springer-Verlag, New York, 1990.Google Scholar
  13. [13]
    G. Christol, T. Kamae, M. Mendès France, G. Rauzy, Suites algébriques, automates et substitutions,Bull. Soc. Math. France 108 (1980), 401–419.MathSciNetzbMATHGoogle Scholar
  14. [14]
    A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata,Math. Systems Theory 3 (1969), 186–192.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    A. Cobham, Uniform tag sequences,Math. Systems Theory 6 (1972), 164–192.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    C. Davis, D. Knuth, Number representations and dragon curves, I and II,J. Recreational Math. 3 (1970), 61–81; 133–149.Google Scholar
  17. [17]
    M. Dekking, M. Mendès France, A. J. van der Poorten, FOLDS!,Math. Intelligencer 4 (1982), 130–138; 173–181; 190–195.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    S. Eilenberg,Automata, Languages and Machines, vol. A, Academic Press, New York, 1985.Google Scholar
  19. [19]
    C. Frougny, Representation of numbers and finite automata,Math. Systems Theory 25 (1992), 37–60.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    C. Frougny, B. Solomyak, On representation of numbers in linear numeration systems, inErgodic Theory of ℤ d -Actions,Proceedings of the Warwick Symposium,Warwick, UK, 1993–94, M. Pollicottet al., eds., London Mathematical Society Lecture Notes Series, vol. 228, Cambridge University Press, New York, 1996, pp. 345–368.Google Scholar
  21. [21]
    W. Gilbert, Radix representations of quadratic fields,J. Math. Anal. Appl. 83 (1981), 264–274.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    W. Gilbert, Geometry of radix representations, inThe Geometric Vein, C. Davis, B. Grünbaum, F. Sherk, eds., Springer-Verlag, New York, 1982, pp. 129–139.Google Scholar
  23. [23]
    W. Gilbert, R. Green, Negative based number systems,Math. Mag. 52 (1979), 240–244.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    F. von Haeseler, On algebraic properties of sequences generated by substitutions over a group, Habilitationsschrift, Universität Bremen, 1995.Google Scholar
  25. [25]
    J. Honkala, A decision method for the recognizability of sets derived by number systems,Inform. Théor. Appl. 20 (1986), 395–403.MathSciNetzbMATHGoogle Scholar
  26. [26]
    J. Honkala, On number systems with negative digits,Acad. Sci. Fenn. Ser. A. I. Math. 14 (1989), 149–156.MathSciNetzbMATHGoogle Scholar
  27. [27]
    J. Hopcroft, J. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979.zbMATHGoogle Scholar
  28. [28]
    I. Kátai, J. Szabó, Canonical number systems,Acta Sci. Math. (Szeged) 37 (1975), 255–280.MathSciNetGoogle Scholar
  29. [29]
    D. Knuth,The Art of Computer Programming, vol. 2, 2nd edn., Addison-Wesley, Reading, MA, 1988.Google Scholar
  30. [30]
    P. Kornerup, Digit-set conversions: generalizations and applications,IEEE Trans. Comput. 43 (1994), 622–629.CrossRefGoogle Scholar
  31. [31]
    D. Matula, Radix arithmetic: digital algorithms for computer architecture, inApplied Computation Theory: Analysis, Design, Modeling, R. T. Yeh, ed., Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ, 1976, pp. 374–447.Google Scholar
  32. [32]
    D. Matula, Basic digit sets for radix representation,J. Assoc. Comput. Mach. 29 (1982), 1131–1143.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    W. Penney, A “binary” system for complex numbers,J. Assoc. Comput. Mach. 12 (1965), 247–248.zbMATHCrossRefGoogle Scholar
  34. [34]
    G. Rauzy, Systèmes de numération,Journées Math. SMF-CNRS, Théorie élémentaire et analytique des nombres, 8–9 Mars 1982, Valenciennes, pp. 137–145.Google Scholar
  35. [35]
    A. Robert, A good basis for computing with complex numbers,Elem. Math. 49 (1994), 111–117.MathSciNetzbMATHGoogle Scholar
  36. [36]
    O. Salon, Suites automatiques à multi-indices,Séminaire de Théorie des Nombres, Bordeaux, Exp. 4 (1986–1987), pp. 4-01–4-27; followed by an Appendix by J. Shallit, 4-29A–4-36A.Google Scholar
  37. [37]
    O. Salon, Suites automatiques à multi-indices et algébricité,C. R. Acad. Sci. Paris Sér. I 305 (1987), 501–504.MathSciNetzbMATHGoogle Scholar
  38. [38]
    O. Salon, Propriétés arithmétiques des automates multidimensionnels, Thèse, Université Bordeaux I, 1989.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • J. P. Allouche
    • 1
  • E. Cateland
    • 2
  • W. J. Gilbert
    • 3
  • H. O. Peitgen
    • 4
  • J. O. Shallit
    • 5
  • G. Skordev
    • 4
  1. 1.CNRS, LRIOrsay CedexFrance
  2. 2.Département de MathématiquesUniversité Bordeaux ITalence CedexFrance
  3. 3.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  4. 4.Institut für Dynamische SystemeUniversität BremenBremenGermany
  5. 5.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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