One quantifier will do in existential monadic second-order logic over pictures

  • Oliver Matz
Contributed Papers Picture Languages - Function Systems/Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1450)


We show that every formula of the existential fragment of monadic second-order logic over picture models (i.e., finite, two-dimensional, coloured grids) is equivalent to one with only one existential monadic quantifier.

The corresponding claim is true for the class of word models ([Tho82]) but not for the class of graphs ([Ott95]).

The class of picture models is of particular interest because it has been used to show the strictness of the different (and more popular) hierarchy of quantifier alternation.


Existential Quantifier Fixed Height Boundary Symbol Word Model Quantifier Alternation 
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  1. [GR96]
    D. Giammarresi and A. Restivo. Two-dimensional languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Language Theory, volume III. Springer-Verlag, New York, 1996.Google Scholar
  2. [GRST96]
    D. Giammarresi, A. Restivo, S. Seibert, and W. Thomas. Monadic second-order logic and recognizability by tiling systems. Information and Computation, 125:32–45, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [LS94]
    M. Latteux and D. Simplot. Recognizable picture languages and domino tiling. Internal Report IT-94-264, Laboratoire d'Informatique Fondamentale de Lille, Université de Lille, France, 1994.Google Scholar
  4. [Mat95]
    O. Matz. Klassifizierung von Bildsprachen mit rationalen Ausdrücken, Grammatiken und Logik-Formeln. Diploma thesis, Christian-Albrechts-Universität Kiel, 1995. (German).Google Scholar
  5. [MT97]
    O. Matz and W. Thomas. The monadic quantifier alternation hierarchy over graphs is infinite. In Twelfth Annual IEEE Symposium on Logic in Computer Science, pages 236–244, Warsaw, Poland, 1997. IEEE.Google Scholar
  6. [Ott95]
    M. Otto. Note on the number of monadic quantifiers in monadic σ11. Information Processing Letters, 53:337–339, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [Tho82]
    W. Thomas. Classifying regular events in symbolic logic. Journal of Computer and System Sciences, 25:360–376, 1982.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Oliver Matz
    • 1
  1. 1.Institut für Informatik und Praktische MathematikChristian-Albrechts-Universität KielKielGermany

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