Weakly maximal decidable structures
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 137-145.

We prove that there exists a structure M whose monadic second order theory is decidable, and such that the first-order theory of every expansion of M by a constant is undecidable.

DOI : 10.1051/ita:2007044
Classification : 03B25, 03C57, 03D05
Mots-clés : decidability, first-order theories, monadic second-order theories, maximality, automata, rich words 
@article{ITA_2008__42_1_137_0,
     author = {B\`es, Alexis and C\'egielski, Patrick},
     title = {Weakly maximal decidable structures},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {137--145},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     doi = {10.1051/ita:2007044},
     mrnumber = {2382548},
     zbl = {1149.03015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2007044/}
}
TY  - JOUR
AU  - Bès, Alexis
AU  - Cégielski, Patrick
TI  - Weakly maximal decidable structures
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2008
SP  - 137
EP  - 145
VL  - 42
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita:2007044/
DO  - 10.1051/ita:2007044
LA  - en
ID  - ITA_2008__42_1_137_0
ER  - 
%0 Journal Article
%A Bès, Alexis
%A Cégielski, Patrick
%T Weakly maximal decidable structures
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2008
%P 137-145
%V 42
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita:2007044/
%R 10.1051/ita:2007044
%G en
%F ITA_2008__42_1_137_0
Bès, Alexis; Cégielski, Patrick. Weakly maximal decidable structures. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 137-145. doi : 10.1051/ita:2007044. http://archive.numdam.org/articles/10.1051/ita:2007044/

[1] J.R. Büchi, On a decision method in the restricted second-order arithmetic. In Proc. Int. Congress Logic, Methodology and Philosophy of science, Berkeley 1960. Stanford University Press (1962) 1-11. | MR

[2] K.J. Compton, On rich words. In M. Lothaire, editor, Combinatorics on words. Progress and perspectives, Proc. Int. Meet., Waterloo, Canada (1982). Encyclopedia of Mathematics 17, Addison-Wesley (1983) 39-61. | MR | Zbl

[3] C.C. Elgot and M.O. Rabin. Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. J. Symbolic Logic 31 (1966) 169-181. | Zbl

[4] S. Feferman and R.L. Vaught, The first order properties of products of algebraic systems. Fund. Math. 47 (1959) 57-103. | MR | Zbl

[5] D. Perrin and J.-É. Pin, Infinite Words. Pure Appl. Math. 141 (2004). | Zbl

[6] V.S. Harizanov, Computably-theoretic complexity of countable structures. Bull. Symbolic Logic 8 (2002) 457-477. | MR | Zbl

[7] S. Shelah, The monadic theory of order. Ann. Math. 102 (1975) 379-419. | MR | Zbl

[8] S. Soprunov, Decidable expansions of structures. Vopr. Kibern. 134 (1988) 175-179 (in Russian). | MR | Zbl

[9] W. Thomas, The theory of successor with an extra predicate. Math. Ann. 237 (1978) 121-132. | MR | Zbl

[10] W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht. Lect. Notes Comput. Sci. 1261 (1997) 118-143. | MR | Zbl

Cité par Sources :