Annie Foret

On Limit Points for Some Variants of Rigid Lambek Grammars

In this paper we give some learnability results in the field of categorial grammars. We show that in contrast to k-valued classical categorial grammars, different classes of Lambek grammars are not learnable from strings following Golds model. The results are obtained by the construction of limit points in each considered class: non associative Lambek grammars with empty sequences and Lambek grammars without empty sequences and without product. Such results express the difficulty of learning categorial grammars from unstructured strings and the need for structured examples.

Dependency structure grammars

In this paper, we define Dependency Structure Grammars (DSG), which are rewriting rule grammars generating sentences together with their dependency structures, are more expressive than CF-grammars and non-equivalent to mildly context-sensitive grammars. We show that DSG are weakly equivalent to Categorial Dependency Grammars (CDG) recently introduced in [6,3]. In particular, these dependency grammars naturally express long distance dependencies and enjoy good mathematical properties.

Two models of learning iterated dependencies

We study the learnability problem in the family of Categorial Dependency Grammars (CDG), a class of categorial grammars defining unlimited dependency structures. CDG satisfying a reasonable condition on iterated (i.e., repeatable and optional) dependencies are shown to be incrementally learnable in the limit.

On Mixing Deduction and Substitution in Lambek Categorial Grammars

Recently, learning algorithms in Golds model [Gol67] have been proposed for some particular classes of classical categorial grammars [Kan98]. We are interested here in learning Lambek categorial grammars.In general grammatical inference uses unification and substitution. In the context of Lambek categorial grammars it seems appropriate to incorporate an operation on types based both on deduction (Lambek derivation) and on substitution instead of standard substitution and standard unification.The purpose of this paper is to investigate such operations defined both in terms of deduction and substitution in categorial grammars and to study a modified unification that may serve as a basis for learning in this framework. We consider some variants of definition : in particular we show that deduction and substitution do not permute. We then consider a modified unification, here called ||= -unification :we give a criterion for the existence and construction of ||= -unifiers in terms of group issues.

Fully lexicalized pregroup grammars

Pregroup grammars are a context-free grammar formalism introduced as a simplification of Lambek calculus. This formalism is interesting for several reasons: the syntactical properties of words are specified by a set of types like the other type-based grammar formalisms; as a logical model, compositionality is easy ; a polytime parsing algorithm exists. However, this formalism is not completely lexicalized because each pre-group grammar is based on the free pregroup built from a set of primitive types together with a partial order, and this order is not lexical information. In fact, only the pregroup grammars that are based on primitive types with an order that is equality can be seen as fully lexicalized. We show here how we can transform, using a morphism on types, a particular pregroup grammar into another pregroup grammar that uses the equality as the order on primitive types. This transformation is at most quadratic in size (linear for a fixed set of primitive types), it preserves the parse structures of sentences and the number of types assigned to a word.

k-Valued Non-Associative Lambek Grammars are Learnable from Function-Argument Structures

This paper is concerned with learning categorial grammars in the model of Gold. We show that rigid and k-valued non-associative Lambek grammars are learnable from function-argument structured sentences. In fact, function-argument structures are natural syntactical decompositions of sentences in sub-components with the indication of the head of each sub-component. This result is interesting and surprising because for every k, the class of k-valued NL grammars has innite elasticity and one could think that it is not learnable, which is not true. Moreover, these classes are very close to unlearnable classes like k-valued associative Lambek grammars learned from function-argument sentences or k-valued non-associative Lambek calculus grammars learned from well-bracketed list of words or from strings. Thus, the k-valued non-associative Lambek grammars learned from function-argument sentences is at the frontier between learnable and unlearnable classes of languages.

On categorial grammars as logical information systems

We explore different perspectives on how categorial grammars can be considered as Logical Information Systems (LIS) both theoretically, and practically. Categorial grammars already have close connections with logic. We discuss the advantages of integrating both approaches. We consider more generally different ways of connecting computational linguistic data and LIS as an application of Formal Concept Analysis.

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 nonassociative 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.

k-valued non-associative lambek grammars (without product) form a strict hierarchy of languages

The notion of k-valued categorial grammars where a word is associated to at most k types is often used in the field of lexicalized grammars as a fruitful constraint for obtaining several properties like the existence of learning algorithms. This principle is relevant only when the classes of k-valued grammars correspond to a real hierarchy of languages. This paper establishes the relevance of this notion for two related grammatical systems. In the first part, the classes of k-valued non-associative Lambek (NL) grammars without product is proved to define a strict hierarchy of languages. The second part introduces the notion of generalized functor argument for non-associative Lambek (NL∅) calculus without product but allowing empty antecedent and establishes also that the classes of k-valued (NL∅) grammars without product form a strict hierarchy of languages.

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.