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.