Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T03:13:45.385Z Has data issue: false hasContentIssue false
  • English
  • Français

Defect theorem in the plane

Published online by Cambridge University Press:  17 August 2007

Włodzimierz Moczurad
Włodzimierz Moczurad*
Affiliation:
Institute of Computer Science, Jagiellonian University, Nawojki 11, 30-072 Kraków, Poland; [email protected]
Get access

Abstract

We consider the defect theorem in the context of labelled polyominoes, i.e., two-dimensional figures. The classical version of this property states that if a set of n words is not a code then the words can be expressed as a product of at most n - 1 words, the smaller set being a code. We survey several two-dimensional extensions exhibiting the boundaries where the theorem fails. In particular, we establish the defect property in the case of three dominoes (n × 1 or 1 × n rectangles).

Keywords

Defect theoremcodespolyominoes
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 4 , October 2007 , pp. 403 - 409
Copyright
© EDP Sciences, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aigrain, P., and Beauquier, D., Polyomino tilings, cellular automata and codicity. Theoret. Comput. Sci. 147 (1995) 165–180. CrossRef
Beauquier, D., and Nivat, M., A codicity undecidable problem in the plane. Theoret. Comput. Sci. 303 (2003) 417–430. CrossRef
J. Berstel, and D. Perrin, Theory of Codes. Academic Press (1985).
Bruyère, V., Cumulative defect. Theoret. Comput. Sci. 292 (2003) 97–109. CrossRef
Harju, T., and Karhumäki, J., Many aspects of the defect effect. Theoret. Comput. Sci. 324 (2004) 35–54. CrossRef
J. Karhumäki, Some open problems in combinatorics of words and related areas. TUCS Technical Report 359 (2000).
Karhumäki, J., and Mantaci, S., Defect Theorems for Trees. Fund. Inform. 38 (1999) 119–133.
Karhumäki, J., and Maňuch, J., Multiple factorizations of words and defect effect. Theoret. Comput. Sci. 273 (2002) 81–97. CrossRef
Karhumäki, J., Maňuch, J., and Plandowski, W., A defect theorem for bi-infinite words. Theoret. Comput. Sci. 292 (2003) 237–243. CrossRef
M. Lothaire, Combinatorics on Words. Cambridge University Press (1997).
M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press (2002).
Mantaci, S., and Restivo, A.: Codes and equations on trees. Theoret. Comput. Sci. 255 (2001) 483–509. CrossRef
Maňuch, J., Defect Effect of Bi-infinite Words in the Two-element Case. Discrete Math. Theor. Comput. Sci. 4 (2001) 273–290.
W. Moczurad, Algebraic and algorithmic properties of brick codes. Ph.D. Thesis, Jagiellonian University, Poland (2000).
Moczurad, W., Brick codes: families, properties, relations. Intern. J. Comput. Math. 74 (2000) 133–150. CrossRef
M. Moczurad, and W. Moczurad, Decidability of simple brick codes, in Mathematics and Computer Science III (Algorithms, Trees, Combinatorics and Probabilities), Trends in Mathematics. Birkhäuser (2004).
M. Moczurad, and W. Moczurad, Some open problems in decidability of brick (labelled polyomino) codes, in Cocoon 2004 Proceedings. Lect. Notes Comput. Sci. 3106 (2004) 72–81.