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How Many Figure Sets Are Codes?

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Language and Automata Theory and Applications (LATA 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5196))

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Abstract

Defect theorem, which provides a kind of dimension property for words, does not hold for two-dimensional figures (labelled polyominoes), except for some small sets. We thus turn to the analysis of asymptotic density of figure codes. Interestingly, it can often be proved to be 1, even in those cases where the defect theorem fails. Hence it reveals another weak dimension property which does hold for figures, i.e., non-codes are rare.

We show that the asymptotic densities of codes among the following sets are all equal to 1: (ordinary) words, square figures and small sets of dominoes, where small refers to cardinality ≤ 3. The latter is a borderline case for the defect theorem and additionally exhibits interesting properties at different alphabet sizes.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Jagiellonian University, Nawojki 11, 30-072, Kraków, Poland

    Małgorzata Moczurad & Włodzimierz Moczurad

Authors
  1. Małgorzata Moczurad
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  2. Włodzimierz Moczurad
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Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Rovira i Virgili University, Plaza Imperial Tàrraco 1, 43005, Tarragona, Spain

    Carlos Martín-Vide

  2. Fachbereich Elektrotechnik/Informatik, Universität Kassel, Wilhelmshöher Allee 73, 34121, Kassel, Germany

    Friedrich Otto

  3. Fachbereich 4, Abteilung Informatik/Wirtschaftsinformatik, Universität Trier, Campus II, Gebäude H, 54286, Trier, Germany

    Henning Fernau

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Moczurad, M., Moczurad, W. (2008). How Many Figure Sets Are Codes?. In: Martín-Vide, C., Otto, F., Fernau, H. (eds) Language and Automata Theory and Applications. LATA 2008. Lecture Notes in Computer Science, vol 5196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88282-4_35

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  • DOI: https://doi.org/10.1007/978-3-540-88282-4_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88281-7

  • Online ISBN: 978-3-540-88282-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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