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Star-free picture expressions are strictly weaker than first-order logic

  • Thomas Wilke
Session 7: Formal Languages II
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)

Abstract

We exhibit a first-order definable picture language which we prove is not expressible by any star-free picture expression, i. e., it is not star-free. Thus first-order logic over pictures is strictly more powerful than star-free picture expressíons are. This is in sharp contrast with the situation with words: the well-known McNaughton-Papert theorem states that a word language is expressible by a first-order formula if and only if it is expressible by a star-free (word) expression.

The main ingredients of the non-expressibility result are a Fraïssé-style algebraic characterization of star freeness for picture languages and combinatorics on words.

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References

  1. 1.
    J. R. Büchi. Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 6:66–92, 1960.Google Scholar
  2. 2.
    H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical Logic. Springer-Verlag, New York, 1984.Google Scholar
  3. 3.
    R. Fraïssé. Sur quelques classifications des relations, basés sur des isomorphismes restreints, Publ. Sci. de l'Univ. Alger, Sér. A 1, pp. 35–182, 1954.Google Scholar
  4. 4.
    D. Giammarresi and A. Restivo. Two-dimensional languages, in G. Rozenberg and A. Salomaa, ed., Handbook of Formal Languages. Springer-Verlag, Berlin. To appear. Preprint, available as:http://www.dsi.unive.it/%7Edora/Papers/chap96.ps.Z.Google Scholar
  5. 5.
    D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae. To appear.Google Scholar
  6. 6.
    D. Giammarresi, A. Restivo, S. Seibert, and W. Thomas. Monadic second-order logic over rectangular pictures and recognizability by tiling systems. Inform. and Comput., 125(1):32–45, 1996.CrossRefGoogle Scholar
  7. 7.
    N. Immerman and D. Kozen. Definability with bounded number of bound variables. Inform. and Comput., 83(2):121–139, 1989.CrossRefGoogle Scholar
  8. 8.
    H. A. Maurer, G. Rozenberg, and E. Welzl. Using string languages to describe picture languages. Inform. and Control, 54(3):155–185, 1982.CrossRefGoogle Scholar
  9. 9.
    R. McNaughton and S. Papert. Counter-Free Automata, vol. 69 of Research Monograph. MIT Press, Cambridge, Mass., 1971.Google Scholar
  10. 10.
    W. Thomas. A concatenation game and the dot-depth hierarchy. In E. Börger, editor, Computation Theory and Logic, volume 270 of Lecture Notes in Comput. Science, pages 415–426. Springer-Verlag, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Thomas Wilke
    • 1
  1. 1.Institut für Informatik und Praktische MathematikChristian-Albrechts-Universität zu KielKielGermany

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