The Logic of Categorial Grammars: Lecture Notes

These lecture notes present categorial grammars as deductive systems, and include detailed proofs of their main properties. The first chapter deals with Ajdukiewicz and Bar-Hillel categorial grammars (AB grammars), their relation to context-free grammars and their learning algorithms. The second chapter is devoted to the Lambek calculus as a deductive system; the weak equivalence with context free grammars is proved; we also define the mapping from a syntactic analysis to a higher-order logical formula, which describes the semantics of the parsed sentence. The third and last chapter is about proof-nets as parse structures for Lambek grammars; we show the linguistic relevance of these graphs in particular through the study of a performance question. Although definitions, theorems and proofs have been reformulated for pedagogical reasons, these notes contain no personnal result but in the proofnet chapter

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

From Proof Nets to the Free *-Autonomous Category

In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free *-autonomous category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph

On the Algebra of Structural Contexts

Article dans revue scientifique avec comité de lecture.We discuss a general way of defining contexts in linear logic, based on the observation that linear universal algebra can be symmetrized by assigning an additional variable to represent the output of a term. We give two approaches to this, a syntactical one based on a new, reversible notion of term, and an algebraic one based on a simple generalization of typed operads. We relate these to each other and to known examples of logical systems, and show new examples, in particular discussing the relationship between intuitionistic and classical systems. We then present a general framework for extracting deductive system from a given theory of contexts, and prove that all these systems have cut-elimination by the means of a generic argument