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

Volumn 77, Issue 3, 1996, Pages 251-277

  Michaux, Christian ^a   Villemaire, Roger ^b  

^a UNIVERSITY OF MONS   (Belgium)

^b UNIVERSITÉ DU QUÉBEC À MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001994111     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/0168-0072(95)00022-4     Document Type: Article

References (21)

1. J. Berstel and C. Reutenauer, Rational Series and their Languages, EATCS Monographs on Theoretical Computer Science, Vol. 12 (Springer, Berlin, 1988).
2. V. Bruyère, Entiers et automates finis, U.E. Mons (mémoire de licence en mathématiques) (1984-85).
3. V. Bruyère, C. Michaux, G. Hansel and R. Villemaire, logic and p-recognizable sets of integers, Bull. Belgian Math. Soc. Simon Stevin 1(2) (1994) (Proceedings of the Congress Journées Montoises 1992), 191-238.
4. J.R. Büchi, Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlagen Math. 6 (1960), 66-92.
5. A. Cobham, On the base-dependence of sets of numbers reecognizable by finite-automata, Math. Systems Theory 3 (1969) 186-192.
6. S. Eilenberg, Automata, Languages and Machines (Academic Press, New York, 1974).
7. H.E. Enderton, An Introduction to Mathematical Logic (Academic Press, New York, 1972).
8. 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).
9. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th ed. (Clarendon Press, Oxford, 1962).
10. B. Hodgson, Décidabilité par automate fini, Ann. Sci. Math. Québec 7 (1983) 39-57.
11. R. McNaughton, Review of [4], J. Symbolic Logic 28 (1963) 100-102.
12. k〉, C.R. Acad. Sc. Paris t. 303, Série I, 19 (1986) 939-942.
13. C. Michaux and R. Villemaire, Open Questions around Büchi and Presburger Arithmetic, to appear in the Proceedings of the Logic Colloquium '93, Association for Symbolic Logic (Oxford University Press, Oxford, 1995).
14. A. Muchnik, Definable criterion for definability in Presburger Arithmetic and its applications (preprint in Russian), Institute of new technologies (1991).
15. D. Perrin, Finite automata, in: J. van Leeuwen, ed., Handbook of Theoretical Computer Science (Elsevier, Amsterdam, 1990).
16. M. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, C. R. 1er congrès des Mathématiciens des pays slaves, Varsovie (1929) 92-101.
17. O. Salon, Suites automatiques à multi-indices, C. R. Acad. Sci. Paris 305 (1987) 501-504.
18. A.L. Semenov, The Presburger nature of predicates that are regular in two number systems, Siberian Math. J. 18 (1977) 289-299.
19. A.L. Semenov, On certain extensions of the arithmetic of addition of natural numbers, Math. USSR Izvestiya, 15 (1980) 401-418.
20. l〉 is undecidable, Theoret. Comput. Sci. 106 (1992) 337-349.
21. R. Villemaire, Joining k- and l-recognizable sets of natural numbers, Proc. Stacs '92, Lecture Notes in Computer Science, Vol. 577 (Springer, Berlin, 1992) 83-94.