Journal ArticleDOI
Defect theorem in the plane
TLDR
This work establishes the defect property in the case of three dominoes ( n × 1 or 1 × n rectangles) in the context of labelled polyominoes, i.e. two-dimensional figures.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).read more
Citations
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Journal ArticleDOI
Periodicity in rectangular arrays
TL;DR: It is shown that one can test primitivity and compute the primitive root of an array in linear time.
Journal ArticleDOI
Directed figure codes are decidable
TL;DR: It is proved that in this setting verification whether a given finite set of directed Figures is a code is decidable and a constructive algorithm is given and the status of the defect effect for directed figures is clarified.
Dissertation
Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes
TL;DR: In this article, the authors examined four well-known and traditional concepts of combinatorics on words, including the defect theorem, the satisfiability question and the compactness property with respect to this kind of equations.
Journal Article
A note on defect theorems for 2-dimensional words and trees
TL;DR: This note completes the analysis of the defect property with 2- dimensional words in the classes of dominoes, squares, rectangles and figures with unrestricted shape and gives two counterexamples to show that the defectproperty fails in both situations.
Journal ArticleDOI
The code problem for directed figures
TL;DR: It is shown that depending on the catenation type the question whether a given set of directed figures is a code is decidable or not, and a constructive proof is given which leads to a straightforward algorithm.
References
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Journal ArticleDOI
Polyomino tilings, cellular automata and codicity
TL;DR: The class of polyominoe families, which are called neighbourhood codes, which generate tilings which are recognizable by cellular automata using only neighbourhood relations are studied.
Journal ArticleDOI
A codicity undecidable problem in the plane
Danièle Beauquier,Maurice Nivat +1 more
TL;DR: A new undecidability result about tiling problems is given, given a finite set of polyomino types, the problem whether this set is a code, is undecidable.
Journal ArticleDOI
Many aspects of defect theorems
Tero Harju,Juhani Karhumäki +1 more
TL;DR: A survey and a unified presentation of the defect theorem, its generalizations and recent aspects of interest, which are related to equations of words, and in this way to the compactness theorem for systems of word equations.
Journal ArticleDOI
Codes and equations on trees
Sabrina Mantaci,Antonio Restivo +1 more
TL;DR: The notion of stability for sets of trees closed under concatenation is introduced and this allows for a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids.
Book ChapterDOI
Some Open Problems in Decidability of Brick (Labelled Polyomino) Codes
TL;DR: The codicity problem is decidable for sets with keys of size n when n = 1 and, under obvious constraints, for every n, and it is proved that it is undecidable in the general case of sets with Key n, when n≥ 6.