k-valued non-associative Lambek categorial grammars are not learnable from strings

This paper is concerned with learning categorial grammars in Gold's model. In contrast to k-valued classical categorial grammars, k-valued Lambek grammars are not learnable from strings. This result was shown for several variants but the question was left open for the weakest one, the non-associative variant NL.We show that the class of rigid and k-valued NL grammars is unlearnable from strings, for each k; this result is obtained by a specific construction of a limit point in the considered class, that does not use product operator.Another interest of our construction is that it provides limit points for the whole hierarchy of Lambek grammars, including the recent pregroup grammars.Such a result aims at clarifying the possible directions for future learning algorithms: it expresses the difficulty of learning categorial grammars from strings and the need for an adequate structure on examples.