k-Valued non-associative Lambek grammars are learnable from generalized functor-argument structures

This paper is concerned with learning categorial grammars from positive examples in the model of Gold. Functor-argument structures (written FA) are usual syntactical decompositions of sentences in sub-components distinguishing the functional parts from the argument parts defined in the case of classical categorial grammars also known as AB-grammars. In the case of non-associative type-logical grammars, we propose a similar notion that we call generalized functor-argument structures and we show that these structures capture the essence of non-associative Lambek (NL) calculus without product.We show that (i) rigid and k-valued non-associative Lambek (NL without product) grammars are learnable from generalized functor-argument structured sentences.We also define subclasses of k-valued grammars in terms of arity. We first show that (ii) for each k and each bound on arity the class of FA-arity bounded k-valued NL languages of FA structures is finite and (iii) that FA-arity bounded k-valued NL grammars are learnable both from strings and from FA structures as a corollary.Result (i) is obtained from (ii); this learnability result (i) is interesting and surprising when compared to other results: in fact we also show that (iv) this class has infinite elasticity. Moreover, these classes are very close to classes like rigid associative Lambek grammars learned from natural deduction structured sentences (that are different and much richer than FA or generalized FA) or to k-valued non-associative Lambek grammars unlearnable from strings or even from bracketed strings. Thus, the class of k-valued non-associative Lambek grammars learned from generalized functor-argument sentences is at the frontier between learnable and unlearnable classes of languages.