Skip to main content
SpringerLink
Log in

The monadic second-order logic of graphs, II: Infinite graphs of bounded width

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

A countable graph can be considered as the value of a certain infinite expression, represented itself by an infinite tree. We establish that the set of finite or infinite (expression) trees constructed with a finite number of operators, the value of which is a graph satisfying a property expressed in monadic second-order logic, is itself definable in monadic second-order logic. From Rabin's theorem, the emptiness of this set of (expression) trees is decidable. It follows that the monadic second-order theory of an equational graph, or of the set of countable graphs of width less than an integerm, is decidable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauderon, M., On systems of equations defining infinite graphs,Proceedings of the Colloquium on Graph Theoretic Concepts in Computer Science (WG'88), June 1988, Amsterdam, Springer-Verlag, Berlin, to appear.

  2. Bauderon, M., Infinite hypergraphs, in preparation.

  3. Bauderon, M., and Courcelle, B., Graph expressions and graph rewritings,Math. Systems Theory,20 (1987), 83–127.

    Article  MathSciNet  Google Scholar 

  4. Courcelle, B., Fundamental properties of infinite trees,Theoret. Comput. Sci.,25 (1983), 95–169.

    Article  MATH  MathSciNet  Google Scholar 

  5. Courcelle, B., Equivalences and transformations of regular systems.Applications to recursive program schemes, Theoret. Comput. Sci.,42 (1986), 1–122.

    MATH  MathSciNet  Google Scholar 

  6. Courcelle, B., An axiomatic definition of context-free rewriting, and its application to NLC graph grammars,Theoret. Comput. Sci.,55 (1987), 141–181.

    Article  MATH  MathSciNet  Google Scholar 

  7. Courcelle, B., A representation of graphs by algebraic expressions and its use for graph rewriting systems,Proceedings of the Third Internal Workshop on Graph Grammars, Lecture Notes in Computer Science, Vol. 291, Springer-Verlag, Berlin, 1987, pp. 112–132.

    Google Scholar 

  8. Courcelle, B., On context-free sets of graphs and their monadic 2nd-order theory,Proceedings of the Third Internal Workshop on Graph Grammars, Lecture Notes in Computer Science, Vol. 291, Springer-Verlag, Berlin, 1987, pp. 133–146.

    Google Scholar 

  9. Courcelle, B., On using context-free graph grammars for analyzing recursive definitions, inProgramming of Future Generation Computers, II (K. Fuchi, L. Kott, eds.), Elsevier, Amsterdam, 1988, pp. 83–122.

    Google Scholar 

  10. Courcelle, B., The monadic second-order logic of graphs, I: Recognizable sets of finite graphs, Report I-8837, submitted.

  11. Courcelle, B., The monadic second-order logic of graphs, III: Tree-width, forbidden minors and complexity issues, Report I-8852, submitted.

  12. Courcelle, B., The monadic second-order logic of graphs, IV: Every equational graph is definable, Report I-8830, submitted.

  13. Courcelle, B., Recursive applicative program schemes, inHandbook of Theoretical Computer Science, J. Van Leeuwen, ed., Elsevier, Amsterdam, to appear.

  14. Doner, J., Tree acceptors and some of their applications,J. Comput. System Sci.,4 (1970), 406–451.

    Article  MATH  MathSciNet  Google Scholar 

  15. Goguen, J., Thatcher, J., and Wagner, E., An initial algebra approach to the specification, correctness and implementation of abstract data types, inCurrent Trends in Programming Methodology, R. Yeh, ed., Prentice-Hall, Englewood Cliffs, NJ, 1978, pp. 80–149.

    Google Scholar 

  16. Goguen, J., Thatcher, J., Wagner, E., and Wright, J., Initial algebra semantics and continuous algebras,Assoc. Comput. Mach.,24 (1977), 68–95.

    MATH  MathSciNet  Google Scholar 

  17. Guessarian, I.,Algebraic Semantics, Lecture Notes in Computer Science, Vol. 99, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  18. Gurevich, Y., Toward logic tailored for computational complexity, inComputation and Proof Theory, M. Richter, ed., Lecture Notes in Mathematics, Vol. 1104, Springer-Verlag, Berlin, 1984, pp. 175–216.

    Chapter  Google Scholar 

  19. Muller, D, and Schupp, P., The theory of ends, pushdown automata and second-order logic,Theoret. Comput. Sci.,37 (1985), 51–75.

    Article  MATH  MathSciNet  Google Scholar 

  20. Nivat, M., Genealogies, unpublished report, University of Paris 7, 1987.

  21. Rabin, M., Decidability of 2nd-order theories and automata on infinite trees,Trans. Amer. Math. Soc.,141 (1969), 1–35.

    Article  MATH  MathSciNet  Google Scholar 

  22. Rabin, M.,Automata on Infinite objects and Church's Problem, Regional Conference Series in Mathematics, Vol. 13, A.M.S., Providence, RI, 1972.

  23. Seese, D., The structure of the models of decidable monadic theories,J. Pure Appl. Logic, to appear.

  24. Seese, D., Ordered tree-representations of infinite graphs, Preprint, Akademie der DDR, Berlin, East Germany, 1987.

  25. Trahtenbrot, B., Impossibility of an algorithm for the decision problem on finite classes,Dokl. Akad. Nauk. SSSR,70 (1950), 569–572.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d'Informatique, Université Bordeaux I, 351 Cours de la Libération, 33 405, Talence, France

    Bruno Courcelle

Authors
  1. Bruno Courcelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work has been supported by the “Programme de Recherches Coordonnées: Mathématiques et Informatique.” Reprints can be requested by electronic mail at mcvax!inria!geocub!courcell (on UUCP network) or courcell@geocub.greco-prog.fr.

Unité de Recherche associée au C.N.R.S. no. 726.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Courcelle, B. The monadic second-order logic of graphs, II: Infinite graphs of bounded width. Math. Systems Theory 21, 187–221 (1988). https://doi.org/10.1007/BF02088013

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02088013

Keywords

Search

Navigation