The Lambek calculus with iteration: two variants

Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations

Learning categorial grammars

In 1967 E. M. Gold published a paper in which the language classes from the Chomsky-hierarchy were analyzed in terms of learnability, in the technical sense of identification in the limit. His results were mostly negative, and perhaps because of this his work had little impact on linguistics. In the early eighties there was renewed interest in the paradigm, mainly because of work by Angluin and Wright. Around the same time, Arikawa and his co-workers refined the paradigm by applying it to so-called Elementary Formal Systems. By making use of this approach Takeshi Shinohara was able to come up with an impressive result; any class of context-sensitive grammars with a bound on its number of rules is learnable. Some linguistically motivated work on learnability also appeared from this point on, most notably Wexler & Culicover 1980 and Kanazawa 1994. The latter investigates the learnability of various classes of categorial grammar, inspired by work by Buszkowski and Penn, and raises some interesting questions. We follow up on this work by exploring complexity issues relevant to learning these classes, answering an open question from Kanazawa 1994, and applying the same kind of approach to obtain (non)learnable classes of Combinatory Categorial Grammars, Tree Adjoining Grammars, Minimalist grammars, Generalized Quantifiers, and some variants of Lambek Grammars. We also discuss work on learning tree languages and its application to learning Dependency Grammars. Our main conclusions are: - formal learning theory is relevant to linguistics, - identification in the limit is feasible for non-trivial classes, - the `Shinohara approach' -i.e., placing a numerical bound on the complexity of a grammar- can lead to a learnable class, but this completely depends on the specific nature of the formalism and the notion of complexity. We give examples of natural classes of commonly used linguistic formalisms that resist this kind of approach, - learning is hard work. Our results indicate that learning even `simple' classes of languages requires a lot of computational effort, - dealing with structure (derivation-, dependency-) languages instead of string languages offers a useful and promising approach to learnabilty in a linguistic contex

Sequents and link graphs: contraction criteria for refinements of multiplicative linear logic

In this thesis we investigate certain structural refinements of multiplicative linear logic, obtained by removing structural rules like commutativity and associativity, in addition to the removal of weakening and contraction, which characterizes linear logic. We define a notion of sequent that is able to capture these subtle structural distinctions. For each of our calculi (MLL, NCLL, CNL, and NLR) we introduce a theory of two-sided proof structures, which, in many respects, turns out to be more appropriate than the standard one-sided approach. We prove correctness criteria, stating which among those proof structures correspond to proofs, i.e. are proof nets. For this we introduce a contraction relation defined on the space of link graphs, a notion sufficiently general to capture both proof structures and sequents, and the key-concept in this work, which is a step towards a unification of the logical core of many distinct calculi

Apprentissage de grammaires catégorielles (transducteurs d'arbres et clustering pour induction de grammaires catégorielles)

De nos jours, il n est pas rare d utiliser des logiciels capables d avoir une conversation, d interagir avec nous (systèmes questions/réponses pour les SAV, gestion d interface ou simplement Intelligence Artificielle - IA - de discussion). Ceux-ci doivent comprendre le contexte ou réagir par mot-clefs, mais générer ensuite des réponses cohérentes, aussi bien au niveau du sens de la phrase (sémantique) que de la forme (syntaxe). Si les premières IA se contentaient de phrases toutes faites et réagissaient en fonction de mots-clefs, le processus s est complexifié avec le temps. Pour améliorer celui-ci, il faut comprendre et étudier la construction des phrases. Nous nous focalisons sur la syntaxe et sa modélisation avec des grammaires catégorielles. L idée est de pouvoir aussi bien générer des squelettes de phrases syntaxiquement correctes que vérifier l appartenance d une phrase à un langage, ici le français (il manque l aspect sémantique). On note que les grammaires AB peuvent, à l exception de certains phénomènes comme la quantification et l extraction, servir de base pour la sémantique en extrayant des -termes. Nous couvrons aussi bien l aspect d extraction de grammaire à partir de corpus arborés que l analyse de phrases. Pour ce faire, nous présentons deux méthodes d extraction et une méthode d analyse de phrases permettant de tester nos grammaires. La première méthode consiste en la création d un transducteur d arbres généralisé, qui transforme les arbres syntaxiques en arbres de dérivation d une grammaire AB. Appliqué sur les corpus français que nous avons à notre disposition, il permet d avoir une grammaire assez complète de la langue française, ainsi qu un vaste lexique. Le transducteur, même s il s éloigne peu de la définition usuelle d un transducteur descendant, a pour particularité d offrir une nouvelle méthode d écriture des règles de transduction, permettant une définition compacte de celles-ci. Nous transformons actuellement 92,5% des corpus en arbres de dérivation. Pour notre seconde méthode, nous utilisons un algorithme d unification en guidant celui-ci avec une étape préliminaire de clustering, qui rassemble les mots en fonction de leur contexte dans la phrase. La comparaison avec les arbres extraits du transducteur donne des résultats encourageants avec 91,3% de similarité. Enfin, nous mettons en place une version probabiliste de l algorithme CYK pour tester l efficacité de nos grammaires en analyse de phrases. La couverture obtenue est entre 84,6% et 92,6%, en fonction de l ensemble de phrases pris en entrée. Les probabilités, appliquées aussi bien sur le type des mots lorsque ceux-ci en ont plusieurs que sur les règles, permettent de sélectionner uniquement le meilleur arbre de dérivation.Tous nos logiciels sont disponibles au téléchargement sous licence GNU GPL.Nowadays, we have become familiar with software interacting with us using natural language (for example in question-answering systems for after-sale services, human-computer interaction or simple discussion bots). These tools have to either react by keyword extraction or, more ambitiously, try to understand the sentence in its context. Though the simplest of these programs only have a set of pre-programmed sentences to react to recognized keywords (these systems include Eliza but also more modern systems like Siri), more sophisticated systems make an effort to understand the structure and the meaning of sentences (these include systems like Watson), allowing them to generate consistent answers, both with respect to the meaning of the sentence (semantics) and with respect to its form (syntax). In this thesis, we focus on syntax and on how to model syntax using categorial grammars. Our goal is to generate syntactically accurate sentences (without the semantic aspect) and to verify that a given sentence belongs to a language - the French language. We note that AB grammars, with the exception of some phenomena like quantification or extraction, are also a good basis for semantic purposes. We cover both grammar extraction from treebanks and parsing using the extracted grammars. On this purpose, we present two extraction methods and test the resulting grammars using standard parsing algorithms. The first method focuses on creating a generalized tree transducer, which transforms syntactic trees into derivation trees corresponding to an AB grammar. Applied on the various French treebanks, the transducer s output gives us a wide-coverage lexicon and a grammar suitable for parsing. The transducer, even if it differs only slightly from the usual definition of a top-down transducer, offers several new, compact ways to express transduction rules. We currently transduce 92.5% of all sen- tences in the treebanks into derivation trees.For our second method, we use a unification algorithm, guiding it with a preliminary clustering step, which gathers the words according to their context in the sentence. The comparision between the transduced trees and this method gives the promising result of 91.3% of similarity.Finally, we have tested our grammars on sentence analysis with a probabilistic CYK algorithm and a formula assignment step done with a supertagger. The obtained coverage lies between 84.6% and 92.6%, depending on the input corpus. The probabilities, estimated for the type of words and for the rules, enable us to select only the best derivation tree. All our software is available for download under GNU GPL licence.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF

The Conjoinability Relation in Lambek Calculus and Linear Logic

In 1958 J. Lambek introduced a calculus L of syntactic types and defined an equivalence relation on types: "x # y means that there exists a sequence x = x 1 , . . . , x n = y (n # 1), such that x i # x i+1 or x i+1 # x i (1 # i < n)". He pointed out that x # y if and only if there is join z such that x # z and y # z. This paper gives an e#ective characterization of this equivalence for the Lambek calculi L and LP , and for the multiplicative fragments of Girard's and Yetter's linear logics. Moreover, for the non-directed Lambek calculus LP and the multiplicative fragment of Girard's linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are