Abstract

This paper proposes a generic, axiomatic framework to express and study structural rules in resource conscious logics derived from Linear Logic. The proposed axioms aim at capturing minimal concepts, operations and relations in order to build an inference system which extends that of Linear Logic by the introduction of structure and structural rules, but still preserves in a very natural way the essential properties of any logical inference system: Cut elimination and Focussing. We consider here finite but unbounded structures, generated from elementary structures called “places”. The set of places is “isotropic”, in that no single place has a distinguished role in the structures, and each structure can be “transported” into an isomorphic structure by applying a permutation of places to the places from which it is built. The essential role of these places in the definition of Logic has been shown in Ludics, and leads to a locative reading of the traditional logical concepts (formulas, sequents, proofs), which is adopted here. All the logical connectives (multiplicatives, additives, exponentials) are expressed here in a locative manner.

Introduction

Linear Logic essentially differs from Classical Logic by the removal of the structural axioms of Contraction and Weakening. However, it retains other structural axioms, such as Exchange, which determine the essential properties of the whole system. Thus, from a sequent calculus point of view, Linear Logic sequents are built from a single constructor (usually represented as the “comma”) which is structurally considered associative-commutative, and hence, so are all the binary connectives. The structure of commutative monoid underlying phase semantics also results from this choice.

Abstract

We formulate Girard's long trip criterion for multiplicative linear logic (MLL) in a topological way, by associating a ribbon diagram to every switching, and requiring that it is homeomorphic to the disk. Then, we extend the well-known planarity criterion for multiplicative cyclic linear logic (McyLL) to multiplicative non-commutative logic (MNL) and show that the resulting planarity criterion is equivalent to Abrusci and Ruet's original long trip criterion for MNL.

Introduction

In his seminal article on linear logic, Jean-Yves Girard develops two alternative notations for proofs:

• a sequential syntax where proofs are expressed as derivation trees in a sequent calculus,

• a parallel syntax where proofs are expressed as bipartite graphs called proof-nets.

The proof-net notation plays the role of natural deduction in intuitionistic logic. It exhibits more of the intrinsic structure of proofs than the derivation tree notation, and is closer to denotational semantics. Typically, a derivation tree defines a unique proof-net, while a proof-net may represent several derivation trees, each derivation tree witnessing a particular order of sequentialization of the proof-net.

The parallel notation requires to separate “real proofs” (proof-nets) from “proof alikes” (called proof-structures) using a correctness criterion. Intuitively, the criterion reveals the “geometric” essence of the logic, beyond its “grammatical” presentation as a sequent calculus. In the case of MLL, the (unit-free) multiplicative fragment of (commutative) linear logic, Girard introduces a “long trip condition” which characterizes proof-nets among proof-structures. The criterion is then extended to full linear logic in.

Abstract

The copying of processes is limited in the context of distributed computation, either as a fact of life, often because remote networks are simply too complicated to have control over, or deliberately, as in the design of security protocols. Roughly, linearity is about how to manage without a presumed ability to copy. The meaning and mathematical consequences of linearity are studied for path-based models of processes which are also models of affine-linear logic. This connection yields an affine-linear language for processes in which processes are typed according to the kind of computation paths they can perform. One consequence is that the affine-linear language automatically respects open-map bisimulation. A range of process operations (from CCS, CCS with process-passing, mobile ambients, and dataflow) can be expressed within the affine-linear language showing the ubiquity of linearity. Of course, process code can be sent explicitly to be copied. Following the discipline of linear logic, suitable nonlinear maps are obtained as linear maps whose domain is under an exponential. Different ways to make assemblies of processes lead to different choices of exponential; the nonlinear maps of only some of which will respect bisimulation.

Introduction

In the film “Groundhog Day” the main character comes to relive and remember the same day repeatedly, until finally he gets it right (and the girl). Real life isn't like that. The world moves on and we cannot rehearse, repeat or reverse the effects of the more important decisions we take.

Abstract

In this survey we shall present the main results on proof nets for the Multiplicative and Exponential fragment of Linear Logic (MELL) and discuss their connections with λ-calculus. The survey ends with a short introduction to sharing reduction. The part on proof nets and on the encoding of λ-terms is self-contained and the proofs of the main theorems are given in full details. Therefore, the survey can be also used as a tutorial on that topics.

Introduction

In his seminal paper on Linear Logic, Girard introduced proof nets in order to overcome some of the limitations of the sequent calculus for Linear Logic. At the price of the loss of the symmetries of sequents, proof nets allowed to equate proofs that in sequent calculus differ by useless details, and gave topological tools for the characterization and analysis of Linear Logic proofs. Moreover, because of the encoding of λ-calculus in the Multiplicative and Exponential fragment of Linear Logic (MELL), it was immediately clear that proof nets might have become a key tool in the fine analysis of the reduction mechanism, the dynamics, of Acalculus. Such a property was even clearer after that Girard introduced GOI (Geometry of Interaction) and after the work by Danos and Regnier on the so-called local and asynchronous β-reduction. However, the key step towards the full exploitation of proof nets in the analysis of λ-calculus dynamics was the discover by Gonthier et al. that Lamping's algorithm for the implementation of λ-calculus optimal reduction might be reformulated on proof nets using GOI.

Abstract

Reporting on applications of bicategories to algebra and linguistics led me to take a new look at multicategories and polycategories: to replace free monoids by free categories and to introduce a new notation for Gentzen's cuts. This makes it clear that the equations holding in a multior polycategory are just those of the 2-category which contains it. Thus, a polycategory is almost the same as a 2-category whose underlying 1-category is freely generated by a graph, except that the class of 2-cells need not be closed under composition, but only under planar cuts.

Summary of contents

In Section 9.1 we point out that multicategories, slightly generalized, will do for bicategories what they originally did for monoidal categories, i.e. bicategories with one object. At the same time we introduce a new notation for Gentzen's “cut”, to present it as a special case of composition in a 2-category.

In Section 9.2 we look at adjunctions in 2-categories and bicategories, with the aim of studying those bicategories in which each 1-cell has both a left and a right adjoint, namely compact noncommutative *-autonomous categories with several 0-cells.

In Section 9.3 we give a short exposition of some applications of bicategories to linguistics that were developed by Claudia Casadio and the present author. These touch on three deductive systems: the syntactic calculus, classical bilinear logic and compact bilinear logic.

In Section 9.4 we take a new look at polycategories, which are to classical bilinear logic as multicategories are to the syntactic calculus. Equations in a polycategory are explained by viewing the latter as contained in a 2-category.