Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T17:58:54.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Undecidable extensions of Büchi arithmetic and Cobham-Semënov Theorem

Published online by Cambridge University Press:  12 March 2014

Alexis Bès
Alexis Bès*
Affiliation:
Equipe de Logique, Université Paris7, 2, Place Jussieu, 75251 Paris Cedex 05, France E-mail: bes@logique.jussieu.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Let k and l be two multiplicatively independent integers, and let L ⊆ ℕn be a l-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +, Vk, L〉, where Vk(x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1280 - 1296
Copyright
Copyright © Association for Symbolic Logic 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bruyère, V., Entiers et automates finis, U.E. Mons, memoire de licence, 1984–1985.Google Scholar
[2]Büchi, J. R., Weak second-order arithmetic and finite automata, Zeitschrift für Math. Logic Grundlagen Math., vol. 6 (1960), pp. 66–92.CrossRefGoogle Scholar
[3]Cherlin, G. and Point, F., On extensions of Presburger arithmetic, Proceedings of the 4th Easter Model Theory conference, Gross Köris 1986 Seminarberichte 86, Humboldt Universität zu Berlin, 1986, pp. 17–34.Google Scholar
[4]Cobham, A., On the base-dependance of sets of numbers recognizable by finite automata, Mathematical Systems Theory, vol. 3 (1969), pp. 186–192.CrossRefGoogle Scholar
[5]Eilenberg, S., Automata, languages and machines, vol. A, Academic Press, New York, 1974.Google Scholar
[6]Elgot, C. C. and Rabin, M. O., Decidability and undecidability of extensions of second (first) order theory of (generalized) successor, this Journal, vol. 31 (1966), no. 2, pp. 169–181.Google Scholar
[7]Hansel, G., A propos d'un théorème de Cobham, Actes de la Fête des Mots, Greco de Programmation, CNRS, Rouen, France (Perrin, D., editor), 1992.Google Scholar
[8]Hodgson, B., Décidabilité par automate fini, Les Annates des Sciences Mathematiques du Québec, vol. 7 (1983), pp. 39–57.Google Scholar
[9]Michaux, C. and Villemaire, R., Open problems on Büchi and Presburger arithmetics, Logic: From foundations to applications, European Logic Colloquium, Oxford University Press, 1996, pp. 353–383.CrossRefGoogle Scholar
[10]Michaux, C. and Villemaire, R., Presburger arithmetic and recognizability of sets of natural numbers by automata: new proofs of Cobham's and Semenov's theorems, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 251–277.CrossRefGoogle Scholar
[11]Muchnik, A., Definable criterion for definability in Presburger arithmetic and its application, Institute of new technologies, 1991, in Russian, preprint.Google Scholar
[12]Semenov, A. L., The Presburger nature of predicates that are regular in two number systems, Siberian Mathematics Journal, vol. 18 (1977), no. 2, pp. 289–299.CrossRefGoogle Scholar
[13]Thomas, W., A note on undecidable extensions of monadic second order successor arithmetic, Archive for Mathematical Logic, vol. 17 (1975), pp. 43–44.CrossRefGoogle Scholar
[14]Villemaire, R., Joining k- and l-recognizable sets of natural numbers, Proceedings of the 9th Symposium on Theoretical Aspects of Computer Science (STACS'92) Paris, France, Springer Lecture Notes in Computer Science, vol. 577, 1992, pp. 83–94.Google Scholar
[15]Villemaire, R., 〈ℕ; +, Vk, V1〉 is undecidable, Theoretical Computer Science, vol. 106 (1992), pp. 337–349.CrossRefGoogle Scholar