LATA 2008: Language and Automata Theory and Applications pp 385-396 | Cite as
How Many Figure Sets Are Codes?
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.
Keywords
Polyominoes codes asymptotic densityPreview
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References
- 1.Beauquier, D., Nivat, M.: A codicity undecidable problem in the plane. Theoret. Comp. Sci. 303, 417–430 (2003)MATHCrossRefMathSciNetGoogle Scholar
- 2.Berstel, J., Perrin, D.: Theory of Codes. Academic Press, London (1985)MATHGoogle Scholar
- 3.Harju, T., Karhumäki, J.: Many aspects of the defect effect. Theoret. Comp. Sci. 324, 35–54 (2004)MATHCrossRefGoogle Scholar
- 4.Karhumäki, J., Mantaci, S.: Defect Theorems for Trees. Fundam. Inform. 38, 119–133 (1999)MATHGoogle Scholar
- 5.Karhumäki, J., Maňuch, J.: Multiple factorizations of words and defect effect. Theoret. Comp. Sci. 273, 81–97 (2002)MATHCrossRefGoogle Scholar
- 6.Karhumäki, J., Maňuch, J., Plandowski, W.: A defect theorem for bi-infinite words. Theoret. Comp. Sci. 292, 237–243 (2003)MATHCrossRefGoogle Scholar
- 7.Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
- 8.Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
- 9.Mantaci, S., Restivo, A.: Codes and equations on trees. Theoret. Comp. Sci. 255, 483–509 (2001)MATHCrossRefMathSciNetGoogle Scholar
- 10.Maňuch, J.: Defect Effect of Bi-infinite Words in the Two-element Case. Discrete Mathematics & Theoretical Computer Science 4, 273–290 (2001)MATHMathSciNetGoogle Scholar
- 11.Moczurad, M., Moczurad, W.: Asymptotic density of brick and word codes. Ars Combinatoria 83, 169–177 (2007)MathSciNetGoogle Scholar
- 12.Moczurad, M., Tyszkiewicz, J., Zaionc, M.: Statistical properties of simple types. Math. Struct. in Comp. Science 10, 575–594 (2000)MATHCrossRefMathSciNetGoogle Scholar
- 13.Moczurad, W.: Defect theorem in the plane. Theoret. Informatics Appl. 41, 403–409 (2007)MATHCrossRefMathSciNetGoogle Scholar
- 14.Wilf, H.: Generatingfunctionology. Academic Press, London (1994)MATHGoogle Scholar
- 15.Yeats, K.: Asymptotic Density in Combined Number Systems. New York J. Math. 8, 63–83 (2002)MATHMathSciNetGoogle Scholar