Multiset, Set and Numerically Decipherable Codes over Directed Figures

  • Michał Kolarz
  • Włodzimierz Moczurad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)


Codes with various kinds of decipherability, weaker than the usual unique decipherability, have been studied since multiset decipherability was introduced in mid-1980s. We consider decipherability of directed figure codes, where directed figures are defined as labelled polyominoes with designated start and end points, equipped with catenation operation that may use a merging function to resolve possible conflicts. This is one of possible extensions generalizing words and variable-length codes to planar structures.

Here, verification whether a given set is a code is no longer decidable in general. We study the decidability status of figure codes depending on catenation type (with or without a merging function), decipherability kind (unique, multiset, set or numeric) and code geometry (several classes determined by relative positions of start and end points of figures). We give decidability or undecidability proofs in all but two cases that remain open.


Direct Figure Decidability Result Translation Vector Weak Equality Catenation Type 
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  1. 1.
    Aigrain, P., Beauquier, D.: Polyomino tilings, cellular automata and codicity. Theoretical Computer Science 147(1-2), 165–180 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Beauquier, D., Nivat, M.: A codicity undecidable problem in the plane. Theoretical Computer Science 303(2-3), 417–430 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Blanchet-Sadri, F.: On unique, multiset, set decipherability of three-word codes. IEEE Transactions on Information Theory 47(5), 1745–1757 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blanchet-Sadri, F., Morgan, C.: Multiset and set decipherable codes. Computers and Mathematics with Applications 41(10-11), 1257–1262 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Burderi, F., Restivo, A.: Coding partitions. Discrete Mathematics and Theoretical Computer Science 9(2), 227–240 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Burderi, F., Restivo, A.: Varieties of codes and kraft inequality. Theory of Computing Systems 40(4), 507–520 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Costagliola, G., Ferrucci, F., Gravino, C.: Adding symbolic information to picture models: definitions and properties. Theoretical Computer Science 337, 51–104 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Giammarresi, D., Restivo, A.: Two-dimensional finite state recognizability. Fundamenta Informaticae 25(3), 399–422 (1996)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Guzmán, F.: Decipherability of codes. Journal of Pure and Applied Algebra 141(1), 13–35 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Head, T., Weber, A.: The Finest Homophonic Partition and Related Code Concepts. In: Privara, I., Ružička, P., Rovan, B. (eds.) MFCS 1994. LNCS, vol. 841, pp. 618–628. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  11. 11.
    Head, T., Weber, A.: Deciding multiset decipherability. IEEE Transactions on Information Theory 41(1), 291–297 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kolarz, M.: The code problem for directed figures. Theoretical Informatics and Applications RAIRO 44(4), 489–506 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kolarz, M.: Directed Figure Codes: Decidability Frontier. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 530–539. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Kolarz, M., Moczurad, W.: Directed figure codes are decidable. Discrete Mathematics and Theoretical Computer Science 11(2), 1–14 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lempel, A.: On multiset decipherable codes. IEEE Transactions on Information Theory 32(5), 714–716 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Mantaci, S., Restivo, A.: Codes and equations on trees. Theoretical Computer Science 255, 483–509 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Moczurad, W.: Brick codes: families, properties, relations. International Journal of Computer Mathematics 74, 133–150 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Moczurad, W.: Directed Figure Codes with Weak Equality. In: Fyfe, C., Tino, P., Charles, D., Garcia-Osorio, C., Yin, H. (eds.) IDEAL 2010. LNCS, vol. 6283, pp. 242–250. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Restivo, A.: A note on multiset decipherable code. IEEE Transactions on Information Theory 35(3), 662–663 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Salomaa, A., Salomaa, K., Yu, S.: Variants of codes and indecomposable languages. Information and Computation 207(11), 1340–1349 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michał Kolarz
    • 1
  • Włodzimierz Moczurad
    • 1
  1. 1.Institute of Computer Science, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland

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