Joining k- and l-recognizable sets of natural numbers

  • Roger Villemaire
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)


We show that the first order theory of < IN, +, Vk, Vl >, where Vr: IN{0} → IN is the function which sends x to Vr(x), the greatest power of r which divides x and k, l are multiplicatively independent (i.e. they have no common power) is undecidable. Actually we prove that multiplication is definable in < IN, +, Vk, Vl>. This shows that the theorem of Büchi cannot be generalized to a class containing all k- and all l-recognizable sets.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    V. Bruyère, Entiers et automates finis, U.E. Mons (mémoire de licence en mathématiques) 1984–85Google Scholar
  2. [2]
    J.R. Büchi, Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlagen Math. 6 (1960), pp 66–92Google Scholar
  3. [3]
    A. Cobham, On the Base-Dependence of Sets of Numbers Recognizable by Finite-Automata, Math. Systems Theory 3, 1969, pp 186–192.Google Scholar
  4. [4]
    S. Eilenberg, Automata, Languages and Machines, Academic Press 1974.Google Scholar
  5. [5]
    C.C. Elgot, M.O. Rabin, Decidability and undecidability of extensions of second (first) order theory of (generalized) successor, J.S.L. vol. 31 (2) 1966, pp 169–181.Google Scholar
  6. [6]
    G. Hansel, A propos d'un théorème de Cobham, in: D. Perrin, ed., Actes de la FÊte des Mots, Greco de Programmation, CNRS, Rouen (1982).Google Scholar
  7. [7]
    R. McNaughton, Review of [2], J. Symbolic Logic 28 (1963), pp 100–102.Google Scholar
  8. [8]
    C. Michaux, F. Point, Les ensembles k-reconnaissables sont définissables dans < IN, +, Vk >, C.R. Acad. Sc. Paris t. 303, Série I, no 19, 1986, p.---Google Scholar
  9. [9]
    D. Perrin, Finite Automata, in: J. van Leeuwen, Handbook of Theoretical Computer Science, Elsevier 1990.Google Scholar
  10. [10]
    W.V. Quine. Concatenation as a basis for arithmetic. J.S.L. (1946) vol. 11 pp 105–114.Google Scholar
  11. [11]
    A.L. Semenov, On certain extensions of the arithmetic of addition of natural numbers, Math. USSR. Izvestiya, vol 15 (1980), 2, p.401–418.Google Scholar
  12. [12]
    J.W. Thatcher, Decision Problems for multiple successor arithmetics, J.S.L. (1966) vol. 11 pp 182–190.Google Scholar
  13. [13]
    W. Thomas, A note on undecidable extensions of monadic second order successor arithmetic, Arch. math. Logik 17 (1975), pp 43–44.Google Scholar
  14. [14]
    R. Villemaire, < IN, +,V k,Vl> is undecidable. (preprint).Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Roger Villemaire
    • 1
  1. 1.Département de mathématiques et d'informatiqueU.Q.A.M.MontréalCanada

Personalised recommendations