Exploring inside Tiling Recognizable Picture Languages to Find Deterministic subclasses

Tiling recognizable two-dimensional languages, also known as REC, generalize recognizable string languages to two dimensions and share with them several theoretical properties. Nevertheless family REC is not closed under complementation and this implies that it is intrinsically non-deterministic. We consider different notions of unambiguity and determinism and the corresponding REC subclasses: they define a hierarchy inside REC. We show that some definitions of unambiguity are equivalent to particular notions of determinism and therefore the corresponding classes have linear parsing algorithms and are closed under complementation.