Published online by Cambridge University Press: 28 February 2011
The paper deals with some classes of two-dimensional recognizablelanguages of “high complexity”,in a sense specified in the paper and motivated by some necessary conditions holding for recognizable andunambiguous languages. For such classes we can solve someopen questions related to unambiguity, finite ambiguityand complementation. 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 arenot sufficient. The proof relies on a family of languages defined by functions.