Christian Retoré

Pomset logic: A non-commutative extension of classical linear logic

We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae.

The Logic of Categorial Grammars

A method for harvesting invention fowl which includes the steps of horizontally extending beneath the fowl, in a confined area, a plurality of lifting fingers; raising and pivoting the fingers to lift the fowl and supporting them at least in part upon a continuously moving structure; continuing to said the fowl on the continuously moving structure to convey the fowl to a cooping location; and moving the fowl into a coop from the continuously moving structure.

Perfect matchings and series-parallel graphs: multiplicatives proof nets as R&B-graphs

Abstract A graph-theoretical look at multiplicative proof nets lead us to two new descriptions of a proof net, both as a graph endowed with a perfect matching. The first one is a rather conventional encoding of the connectives which nevertheless allows us to unify various sequentialisation techniques as the corollaries of a single graph theoretical result. The second one is more exciting: a proof net simply consists in the set of its axioms — the perfect matching — plus one single series-parallel graph which encodes the whole syntactical forest of the sequent. We thus identify proof nets which only differ because of the commutativity or associativity of the connectives, or because final par have been performed or not. We thus push further the program of proof net theory which is to get closer to the proof itself, ignoring as much as possible the syntactical “bureaucracy”.

A semantic characterisation of the correctness of a proof net

The purpose of this note is to show that the correctness of a multiplicative proof net with mix is equivalent to its semantic correctness: a proof structure is a proof net if and only if its semantic interpretation is a clique, where one given finite coherence space interprets all propositional variables.This is just an example of what can be done with these kinds of semantic techniques; for more information and further results, the reader is referred to Retore (1994).

A Complete Axiomatisation for the Inclusion of Series-Parallel Partial Orders

Series-parallel orders are defined as the least class of partial orders containing the one-element order and closed by ordinal sum and disjoint union. From this inductive definition, it is almost immediate that any series-parallel order may be represented by an algebraic expression, which is unique up to the associativity of ordinal sum and to the associativivity and commutativity of disjoint union. In this paper, we introduce a rewrite system acting on these algebraic expressions that axiomatises completely the sub-ordering relation for the class of series-parallel orders.

A Logic for Categorial Grammars: Lambek’s Syntactic Calculus

Our second chapter is a rather complete study of the Lambek calculus, which enables a completely logical treatment of categorial grammar.

Logical Aspects of Computational Linguistics: An Introduction

The papers in this collection are all devoted to single theme: logic and its applications in computational linguistics. They share many themes, goals and techniques, and any editorial classification is bound to highlight some connections at the expense of other. Nonetheless, we have found it useful to divide these papers (somewhat arbitrarily) into the following four categories: logical semantics of natural language, grammar and logic, mathematics with linguistic motivations, and computational perspectives. In this introduction, we use this four-way classification as a guide to the papers, and, more generally, to the research agenda that underlies them. We hope that the reader will find it a useful starting point to the collection.

A Faithful Representation of Non-Associative Lambek Grammars in Abstract Categorial Grammars

This paper solves a natural but still open question: can abstract categorial grammars (ACGs) respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcome of such a representation are discussed.

Classifiers, Sorts, and Base Types in the Montagovian Generative Lexicon and Related Type Theoretical Frameworks for Lexical Compositional Semantics

Type-theoretic lexical semantics have gained a lot of interest since the inception of the Generative Lexicon Theory and the development of tools for such approaches. Conceived as an extensions of Montagovian compositional semantics, they represent a satisfying mean to deal with complex lexical phenomena without losing compositionality — and without losing the logical information that a proposition is asserted, refuted or supposed. All formalisms that have been proposed for that purpose rely on a system with many base types and relations between those; we use many-sorted logic, and will thus call those base types lexical sorts. This chapter details why this system of sorts (or base types) is needed and what features it should provide. We propose that turning to classifiers, in languages that possess such a part of speech, provides a linguistically and cognitively motivated starting point for such a system.

Classical Categorial Grammars: AB Grammars

This first chapter deals with material from the late fifties and early sixties, but which nevertheless introduces the design of categorial grammars, which are lexcalized grammars, as opposed to the phrase structure grammars like context-free grammars that were introduced afterwards.