Decidable Expansions of Labelled Linear Orderings

  • Alexis Bès
  • Alexander Rabinovich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6300)


Let \(M=(A,<,\overline{P})\) where (A, < ) is a linear ordering and \(\overline{P}\) denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or − ω, and the monadic second-order theory of M is decidable, then there exists a non-trivial expansion M′ of M by a monadic predicate such that the monadic second-order theory of M′ is still decidable.


monadic second-order logic decidability definability linear orderings 


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  1. 1.
    Bès, A., Cégielski, P.: Weakly maximal decidable structures. RAIRO-Theor. Inf. Appl. 42(1), 137–145 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bès, A., Cégielski, P.: Nonmaximal decidable structures. Journal of Mathematical Sciences 158, 615–622 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blumensath, A., Colcombet, T., Löding, C.: Logical theories and compatible operations. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and automata: History and Perspectives, pp. 72–106. Amsterdam University Press (2007)Google Scholar
  4. 4.
    Büchi, J.R.: On a decision method in the restricted second-order arithmetic. In: Proc. Int. Congress Logic, Methodology and Philosophy of science, Berkeley 1960, pp. 1–11. Stanford University Press, Stanford (1962)Google Scholar
  5. 5.
    Büchi, J.R.: Transfinite automata recursions and weak second order theory of ordinals. In: Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Jerusalem 1964, pp. 2–23. Holland (1965)Google Scholar
  6. 6.
    Büchi, J.R., Zaiontz, C.: Deterministic automata and the monadic theory of ordinals \(\omega\sb 2\). Z. Math. Logik Grundlagen Math. 29, 313–336 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carton, O., Thomas, W.: The monadic theory of morphic infinite words and generalizations. Inform. Comput. 176, 51–76 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Compton, K.J.: On rich words. In: Lothaire, M. (ed.) Combinatorics on words. Progress and perspectives, Proc. Int. Meet., Waterloo/Can. 1982. Encyclopedia of Mathematics, vol. 17, pp. 39–61. Addison, Reading (1983)Google Scholar
  9. 9.
    Elgot, C.C., Rabin, M.O.: Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. J. Symb. Log. 31(2), 169–181 (1966)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fratani, S.: The theory of successor extended with several predicates (2009) (preprint)Google Scholar
  11. 11.
    Gurevich, Y.: Modest theory of short chains.i. J. Symb. Log. 44(4), 481–490 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gurevich, Y.: Monadic second-order theories. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, Perspectives in Mathematical Logic, pp. 479–506. Springer, Heidelberg (1985)Google Scholar
  13. 13.
    Gurevich, Y., Magidor, M., Shelah, S.: The monadic theory of ω2. J. Symb. Log. 48(2), 387–398 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gurevich, Y., Shelah, S.: Modest theory of short chains. ii. J. Symb. Log. 44(4), 491–502 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gurevich, Y., Shelah, S.: Interpreting second-order logic in the monadic theory of order. J. Symb. Log. 48(3), 816–828 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Perrin, D., Pin, J.-E.: Infinite Words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004), ISBN 0-12-532111-2Google Scholar
  18. 18.
    Perrin, D., Schupp, P.E.: Automata on the integers, recurrence distinguishability, and the equivalence and decidability of monadic theories. In: Symposium on Logic in Computer Science (LICS 1986), Washington, D.C., USA, June 1986, pp. 301–305. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  19. 19.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society 141, 1–35 (1969)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rabinovich, A.: On decidability of monadic logic of order over the naturals extended by monadic predicates. Inf. Comput. 205(6), 870–889 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rabinovich, A., Thomas, W.: Decidable theories of the ordering of natural numbers with unary predicates. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 562–574. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Robinson, R.M.: Restricted set-theoretical definitions in arithmetic. Proc. Am. Math. Soc. 9, 238–242 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rosenstein, J.G.: Linear ordering. Academic Press, New York (1982)zbMATHGoogle Scholar
  24. 24.
    Semenov, A.L.: Decidability of monadic theories. In: Chytil, M.P., Koubek, V. (eds.) MFCS 1984. LNCS, vol. 176, pp. 162–175. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  25. 25.
    Semenov, A.L.: Logical theories of one-place functions on the set of natural numbers. Mathematics of the USSR - Izvestia 22, 587–618 (1984)CrossRefzbMATHGoogle Scholar
  26. 26.
    Shelah, S.: The monadic theory of order. Annals of Mathematics 102, 379–419 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Siefkes, D.: Decidable extensions of monadic second order successor arithmetic. In: Automatentheorie und Formale Sprachen, Tagung, Math. Forschungsinst, Oberwolfach (1969); Bibliograph. Inst., Mannheim, pp. 441–472 (1970)Google Scholar
  28. 28.
    Soprunov, S.: Decidable expansions of structures. Vopr. Kibern. 134, 175–179 (1988) (in Russian)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Thomas, W.: A note on undecidable extensions of monadic second order successor arithmetic. Arch. Math. Logik Grundlagenforsch. 17, 43–44 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Thomas, W.: Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht. LNCS, vol. 1261, pp. 118–143. Springer, Heidelberg (1997)Google Scholar
  31. 31.
    Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 389–455. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  32. 32.
    Thomas, W.: Model transformations in decidability proofs for monadic theories. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 23–31. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Alexis Bès
    • 1
  • Alexander Rabinovich
    • 2
  1. 1.University of Paris-Est Créteil, LACLFrance
  2. 2.The Blavatnik School of Computer ScienceTel-Aviv UniversityIsrael

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