On Intermediate Structures for Non-Associative Lambek Grammars and Learnability

This paper is concerned with learning categorial grammars in the model of Gold. We show that rigid and k-valued non-associative Lambek (NL) grammars are not learnable from well-bracketed sentences. In contrast to k-valued classical categorial grammars, k-valued Lambek grammars are not learnable from strings. This result was shown for several variants including the non-associative variant NL. Whereas k-valued Lambek NL grammars have been shown learnable from functor-argument structures, we show that non-learnability still holds for the languages of well-bracketed strings ; this result is obtained for two variants NL and NL∅ by exhibiting a limit point in the considered class. Such a result aims at clarifying the possible directions for future learning algorithms: it expresses the difficulty of learning categorial grammars from strings or intermediate structures and helps to draw the frontier between learnability and non-learnability.