Composition with Algebra at the Background

On a Question by Gurevich and Rabinovich on the Monadic Theory of Linear Orderings
  • Thomas Colcombet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7913)


Gurevich and Rabinovich raised the following question: given a property of a set of rational numbers definable in the real line by a monadic second-order formula, is it possible to define it directly in the rational line? In other words, is it true that the presence of reals at the background do not increase the expressive power of monadic second-order logic?

In this paper, we answer positively this question. The proof in itself is a simple application of classical and more recent techniques. This question will guide us in a tour of results and ideas related to monadic theories of linear orderings.


Linear Ordering Expressive Power Regular Language Countable Word Rational Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Logic, Methodology and Philosophy of Science (Proc. 1960 Internat. Congr.), pp. 1–11. Stanford Univ. Press, Stanford (1962)Google Scholar
  3. 3.
    Carton, O., Colcombet, T., Puppis, G.: Regular languages of words over countable linear orderings. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 125–136. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–51 (1961)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feferman, S., Vaught, R.L.: The first order properties of products of algebraic systems. Fund. Math. 47, 57–103 (1959)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gurevich, Y., Rabinovich, A.M.: Definability and undefinability with real order at the background. J. Symb. Log. 65(2), 946–958 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gurevich, Y., Shelah, S.: Monadic theory of order and topology in zfc. In: Ann. of Math. Logic, vol. 23, pp. 179–198. North-Holland Publishing Compagny (1982)Google Scholar
  8. 8.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Shelah, S.: The monadic theory of order. Ann. of Math. (2) 102(3), 379–419 (1975)Google Scholar
  10. 10.
    Trakhtenbrot, B.A.: Finite automata and monadic second order logic (Russian). Siberian Math. J 3, 103–131 (1962)zbMATHGoogle Scholar
  11. 11.
    Trakthenbrot, B.A.: The impossibility of an algorithm for the decision problem for finite domains (Russian). Doklady Academii Nauk SSSR 70, 569–572 (1950)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Thomas Colcombet
    • 1
  1. 1.LIAFA/CNRS/Université Denis DiderotParis 7France

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