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A set X ⊆ Σ** of pictures is a code if every picture over Σ is tilable in at most one way with pictures in X. The definition of strong prefix code is introduced. The family of finite strong prefix codes is decidable and it has a polynomial time decoding algorithm. Maximality for finite strong prefix codes is also studied and related to the notion of completeness. We prove that any finite strong prefix code can be embedded in a unique maximal strong prefix code that has minimal size and cardinality. A complete characterization of the structure of maximal finite strong prefix codes completes the paper.
The paper deals with some classes of two-dimensional recognizable
languages of “high complexity”,
in a sense specified in the paper and motivated by some necessary conditions holding for recognizable and
For such classes we can solve some
open questions related to unambiguity, finite ambiguity
Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function.
We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement are
not sufficient. The proof relies on a family of languages defined by functions.
We consider the family UREC of unambiguous recognizable
two-dimensional languages. We prove that there are recognizable
languages that are inherently ambiguous, that is UREC family is a
proper subclass of REC family. The result is obtained by showing a
necessary condition for unambiguous recognizable languages.
Further UREC family coincides with the class of picture languages
defined by unambiguous 2OTA and it strictly contains its
deterministic counterpart. Some closure and non-closure properties
of UREC are presented. Finally we show that it is undecidable
whether a given tiling system is unambiguous.
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