Categorial grammars with iterated types form a strict hierarchy of k-valued languages

The notion of k-valued categorial grammars in which every word is associated to at most k types is often used in the field of lexicalized grammars as a fruitful constraint for obtaining interesting properties like the existence of learning algorithms. This constraint is reasonable only when the classes of k-valued grammars correspond to a real hierarchy of generated languages. Such a hierarchy has been established earlier for the classical categorial grammars.In this paper the hierarchy by the k-valued constraint is established in the class of categorial grammars extended with iterated types adapted to express the so called projective dependency structures.