On the Cobham-Semenov theorem

Theory of Computing Systems

Volumn 30, Issue 2, 1997, Pages 197-220

  Point, F ^a   Bruyere V ^a  

^a UNIVERSITY OF MONS   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SCIENCE; MATHEMATICAL TECHNIQUES;

NUMERATION SYSTEMS;

NUMBER THEORY;

EID: 0031081935     PISSN: 14324350     EISSN: 14330490     Source Type: Journal    
DOI: 10.1007/BF02679449     Document Type: Article

References (30)

1. n≥0, langages codés et θ-shift, Bull. Soc. Math. France 114 (1986), 271-323.
2. A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui n'est pas entière, Acta Math. Acad. Sci. Hungar. 54 (1989), 237-241.
3. V. Bruyère, G. Hansel, Recognizable Sets of Numbers in Nonstandard Bases, Lecture Notes in Computer Science, Vol. 911, Springer-Verlag, Berlin, 1995, pp. 167-179.
4. V. Bruyère, G. Hansel, C. Michaux, R. Villemaire, Logic and p-recognizable sets of integers Bull. Belg. Math. Soc. 1 (1994), 191-238.
5. J. R. Büchi, Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlag. Math. 6 (1960), 66-92.
6. A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3 (1969), 186-192.
7. S. Eilenberg, Automata, Languages and Machines, Vol. A, Academic Press, New York, 1974.
8. S. Fabre, Une généralisation du théorème de Cobham, Acta Arith. 67 (1994), 197-208.
9. L. Flatto, J. C. Lagarias, B. Poonen, The zeta function of the beta transformation, Ergodic Theory Dynamical Systems 14 (1994), 237-266.
10. A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105-114.
11. C. Frougny, Representations of numbers and finite automata, Math. Systems Theory 25 (1992), 37-60.
12. C. Frougny, B. Solomyak, On representation of integers in linear numeration systems, in Ergodic Theory & Dynamical Systems, M. Pollicott and K. Schmidt, Eds., London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, to appear, and LITP Report 94/54, University Paris 6, University Paris 7, 1995, 18 pp.
13. G. Hansel, A propos d'un théorème de Cobham, in Actes de la fête des mots, D. Perrin, Ed., Greco de Programmation, CNRS, Rouen, 1982.
14. G. Hansel, Private communication (1995).
15. G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford University Press, Oxford, 1979.
16. B. R. Hodgson, Décidabilité par automate fini, Ann. Sci. Math. Québec 7 (1983), 39-57.
17. M. Hollander, Greedy numeration systems and regularity, Preprint (1995), 21 pp.
18. D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynamical Systems 4 (1984), 283-300.
19. N. Loraud, Beta-shift, systèmes de numération et automates. Preprint (1995).
20. R. MacNaughton, review of [Bu], J. Symbolic Logic 28 (1963), 100-102.
21. k), C. R. Acad. Sci. Paris 303 (1986), 939-942.
22. C. Michaux, R. Villemaire, Cobham's theorem seen through Büchi theorem, Proc. ICALP '93, Lecture Notes in Computer Science, Vol. 700, Springer-Verlag, Berlin, 1993, pp. 325-334.
23. C. Michaux, R. Villemaire, Presburger arithmetic and recognizability of natural numbers by automata: new proofs of Cobham's and Semenov's theorems, Ann. Pure Appl. Logic 77 (1996), 251-277.
24. A. Muchnik, Definable criterion for definability in Presburger arithmetic and its application, Preprint, in Russian, Institute of New Technologies (1991).
25. W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.
26. D. Perrin, Finite automata, in Handbook of Theoretical Computer Science, Vol. B,J. Van Leeuwen, Ed., Elsevier, Amsterdam, 1990, pp. 2-57.
27. A. L. Semenov, Presburgerness of predicates regular in two number systems (in Russian), Sibirsk. Mat. Zh. 18 (1977), 403-418, English translation, Siberian Math. J. 18 (1977), 289-299.
28. A. L. Semenov, Presburgerness of predicates regular in two number systems (in Russian), Sibirsk. Mat. Zh. 18 (1977), 403-418, English translation, Siberian Math. J. 18 (1977), 289-299.
29. J. Shallit, Numeration Systems, Linear Recurrences and Regular Sets, Lecture Notes in Computer Science, Vol. 623, Springer-Verlag, Berlin, 1992, pp. 89-100.
30. B. Solomyak, Conjugates of beta-numbers and the zero-free for a class of analytic functions, Proc. London Math. Soc. 68 (1994), 477-498.