François Lamarche

Naming proofs in classical propositional logic

We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. This gives us a “Boolean” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.

Classical non-associative Lambek calculus

We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

On Proof Nets for Multiplicative Linear Logic with Units

In 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. Furthermore, the identifications enforced on proofs are such that the proof nets, as they are presented here, form the arrows of the free (symmetric) *-autonomous category.

From Proof Nets to Games

Abstract We give a class of proof nets for Intuitionistic Linear Logic with the connectives -o, !, prove a correctness criterion for them and show that a games semantics can be directly derived from these nets, along with a full completeness theorem.

Multiplicative Linear Logics and Fibrations

we define a notion of fibration on generalized operads that automatically gives the categorical axiomatizion of a large class of multiplicative deductive systems we described in a previous paper. We illustrate it by showing examples taken from previously descirbed logics. Also we show interesting properties of the ctegory of structas, including the fact that it contains many well-known categories as subcategories, including the category of categories.

Modeling Martin-Löf type theory in categories

Abstract We present a model of Martin-Lof type theory that includes both dependent products and the identity type. It is based on the category of small categories, with cloven Grothendieck bifibrations used to model dependent types. The identity type is modeled by a path functor that seems to have independent interest from the point of view of homotopy theory. We briefly describe this modelʼs strengths and limitations.