Abstract
Automata, Logic and Semantics Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.
Highlights
Codes, i.e., subsets X of a monoid such that every product of the elements decomposes uniquely over X, are a common object of study
A limited version of the defect theorem holds in this case and codicity is decidable for very restricted classes of figures; see a comprehensive study in Harju and Karhumaki (2004) and Moczurad (2000, 2007)
We consider directed figures defined as labelled polyominoes with designated start and end points
Summary
I.e., subsets X of a monoid such that every product of the elements decomposes uniquely over X, are a common object of study. Standard variable-length word codes (subsets of a free monoid Σ∗, where Σ is an alphabet) are the best-known among them (see e.g. the classic monograph Berstel and Perrin (1985)), with properties that have become folklore This includes the defect effect and the decidability of codicity. We consider directed figures defined as labelled polyominoes with designated start and end points This setting is similar to one of the models, symbolic pixel pictures, described in Costagliola et al (2005) and admits a natural definition of catenation. We prove that verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm (constructive in the sense that it finds a double factorization of a figure for a non-code) This is a significant change in comparison to previously mentioned picture models. We conclude with some observations and directions for possible further research
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