Skip to main content Accessibility help

On countable chains having decidable monadic theory

  • Alexis Bés (a1) and Alexander Rabinovich (a2)


Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.



Hide All
[1] Bés, A. and Cégielski, P., Weakly maximal decidable structures, RAIRO Theoretical Informatics and Applications, vol. 42 (2008). no. 1. pp. 137145.
[2] Bés, A. and Cégielski, P., Nonmaximal decidable structures. Journal of Mathematical Sciences, vol. 158 (2009), pp. 615622.
[3] Bés, A. and Rabinovkh, A.. Decidable expansions of labelled linear orderings, Logical Methods in Computer Science, vol. 7 (2011), no. 2.
[4] Blumensath, A., Colcombet, T.. and Löding, C., Logical theories and compatible operations. Logic and automata: History and Perspectives (Flum, J., Grädel, E.. and Wilke, T.. editors), Amsterdam University Press. 2007. pp. 72106.
[5] Büchi, J. R., On a decision method in the restricted second-order arithmetic. Proceedings of the International Congress on Logic, Methodology and Philosophy of Science, Berkeley 1960. Stanford University Press, 1962, pp. 111.
[6] Büchi, J. R., Transfinite automata recursions and weak second order theory of ordinals. Proceedings of the Internation Congress on Logic, Methodology, and Philosophy of Science, Jerusalem 1964, North Holland, 1965, pp. 223.
[7] Büchi, J. R. and Zaiontz, C., Deterministic automata and the monadic theory of ordinals ω2 , Zeitschrift für mathematische Logik und Grundlagen der Mathematik. vol. 29 (1983), pp. 313336.
[8] 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. 169181.
[9] Feferman, S. and Vaught, R. L.. The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57103.
[10] Gurevich, Y., Modest theory of short chains. I, this Journal, vol. 44 (1979). no. 4. pp. 481490.
[11] Gurevich, Y., Monadic second-order theories. Model-theoretic logics (Barwise, J. and Feferman, S.. editors). Perspectives in Mathematical Logic, Springer-Verlag. 1985. pp. 479506.
[12] Gurevich, Y., Magidor, M., and Shelah, S.. The monadic theory of ω2 . this Journal, vol. 48 (1983), no. 2, pp. 387398.
[13] Gurevich, Y. and Shelah, S., Modest theory of short chains. II. this Journal, vol. 44 (1979), no. 4. pp. 491502.
[14] Gurevich, Y. and Shelah, S., Interpreting second-order logic in the monadic theory of order, this Journal, vol. 48 (1983), no. 3. pp. 816828.
[15] Läuchli, H., A decision procedure for the weak second-order theory of linear order. Contributions to Mathematical Logic (Schmidt, H. A., Schütte, K., and Thiele, H.-J.. editors), North-Holland Publishing Company, 1968, pp. 189197.
[16] Makowsky, J. A., Algorithmic uses of the Feferman–Vaught theorem, Annals of Pure and Applied Logic, vol. 126 (2004), no. 1–3. pp. 159213.
[17] Rabin, M. O., Decidability of second-order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.
[18] Rabinovich, A.. The full binary tree cannot be interpreted in a chain, this Journal, vol. 75 (2010), no. 4, pp. 14891498.
[19] Rosenstein, J. G., Linear orderings. Academic Press. New York, 1982.
[20] Shelah, S., The monadic theory of order. Annals of Mathematics, vol. 102 (1975), pp. 379419.
[21] Soprunov, S., Decidable expansions of structures, Voprosy Kibernet, vol. 134 (1988), pp. 175179, (in Russian).
[22] Thomas, W., Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht. Lecture Notes in Computer Science, no. 1261, Springer, 1997, pp. 118143.
[23] Thomas, W., Languages, automata, and logic, Handbook of formal languages (Rozenberg, G. and Salomaa, A., editors), vol. III, Springer, 1997, pp. 389455.
[24] Thomas, W., Model transformations in decidability proofs for monadic theories, Computer science logic 2008 (Kaminski, Michael and Martini, Simone, editors). Lecture Notes in Computer Science, vol. 5213, Springer, 2008, pp. 2331.
[25] Zeitman, R. S., The composition method, Ph.D. thesis, Wayne State University, Michigan, 1994.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed