Skip to main content
SpringerLink
Log in

Monadic second-order logic over pictures and recognizability by tiling systems

  • Conference paper
  • First Online:
STACS 94 (STACS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 775))

Included in the following conference series:

Abstract

We show that a set of pictures (rectangular arrays of symbols) is recognized by a finite tiling system if and only if it is definable in existential monadic second-order logic. As a consequence, finite tiling systems constitute a notion of recognizability over two-dimensional inputs which at the same time generalizes finite-state recognizability over strings and matches a natural logic. The proof is based on the Ehrenfeucht-FraÏssé technique for first-order logic and an implementation of “threshold counting” within tiling systems.

Work supported by the ESPRIT Basic Research Actions Program of the EC under Project ASMICS 2 (contract No. 6317).

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Blum and C. Hewitt. Automata on a two-dimensional tape. IEEE Symposium on Switching and Automata Theory, 1967,pp. 155–160.

    Google Scholar 

  2. D. Beauquier and J.E. Pin. Factors of words, In Proc. 16th Int. Colloquium on Automata, Languages and Programming, G. Ausiello et al., (eds.), Lecture Notes in Computer Science, 372, Springer-Verlag, Berlin 1989, pp. 190–200.

    Google Scholar 

  3. H.D. Ebbinghaus and J. Flum and W. Thomas. Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, New York 1984.

    Google Scholar 

  4. R. Fagin Monadic generalized spectra, Z. math. Logik Grundl. Math. 21, 1975, pp. 89–96.

    Google Scholar 

  5. Fagin, R., Stockmeyer, L., Vardi, M.Y.: On monadic NP vs. monadic co-NP. Proc. 8th IEEE Conf. on Structure in Complexity Theory, 1993, pp. 19–30.

    Google Scholar 

  6. D. Giammarresi and A. Restivo. Recognizable picture languages. In Proc. First International Colloquium on Parallel Image Processing, 1991. International Journal Pattern Recognition and Artificial Intelligence. Vol. 6, No. 2 & 3, 1992.

    Google Scholar 

  7. D. Giammarresi and A. Restivo. On recognizable two-dimensional languages. Tech. Report Dipartimento di matematica ed applicazioni, Università di Palermo.

    Google Scholar 

  8. D. Giammarresi, A. Restivo, S. Seibert, W. Thomas. Monadic second-order logic over pictures and recognizability by tiling systems. Tech. Report 9318 Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität Kiel.

    Google Scholar 

  9. J. E. Hopcroft, and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, MA, 1979.

    Google Scholar 

  10. K. Inoue and A. Nakamura. Some properties of two-dimensional on-line tessellation acceptors. Information Sciences, Vol. 13, 1977, pp. 95–121.

    Google Scholar 

  11. K. Inoue and A. Nakamura. Nonclosure properties of two-dimensional on-line tessellation acceptors and one-way parallel/sequential array acceptors. Transaction of IECE of Japan, Vol. 6, 1977, pp. 475–476.

    Google Scholar 

  12. K. Inoue and I. Takanami. A survey of two-dimensional automata theory. In Proc. 5th Int. Meeting of Young Computer Scientists, J. Dasson and J. Kelemen (Eds.), Lecture Notes in Computer Science 381, Springer-Verlag, Berlin 1990, pp. 72–91.

    Google Scholar 

  13. K. Inoue and I. Takanami. A characterization of recognizable picture languages. In Proc. Second International Colloquium on Parallel Image Processing, A. Nakamura et al. (Eds.), Lecture Notes in Computer Science 654, Springer, Berlin 1992.

    Google Scholar 

  14. W. Thomas. On Logics, Tilings, and Automata. In Proc. 18th Int. Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, 510, Springer-Verlag, Berlin 1991, pp. 441–453.

    Google Scholar 

  15. W. Thomas. On the Ehrenfeucht-Fraissé Game in Theoretical Computer Science. In Proc. TAPSOFT '93, M.C. Gaudel, J.P. Jouannaud (Eds.) Lecture Notes in Computer Science, 668, Springer-Verlag, Berlin 1993, pp. 559–568.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica ed Applicazioni, Università di Palermo, via Archirafi, 34, 90123, Palermo, Italy

    Dora Giammarresi & Antonio Restivo

  2. Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität, Olshausenstr. 40, D 24118, Kiel

    Sebastian Seibert & Wolfgang Thomas

Authors
  1. Dora Giammarresi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sebastian Seibert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Restivo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wolfgang Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Patrice Enjalbert Ernst W. Mayr Klaus W. Wagner

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Giammarresi, D., Seibert, S., Restivo, A., Thomas, W. (1994). Monadic second-order logic over pictures and recognizability by tiling systems. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_155

Download citation

  • DOI: https://doi.org/10.1007/3-540-57785-8_155

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57785-0

  • Online ISBN: 978-3-540-48332-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation