Abstract

Logic programming can be given a foundation in sequent calculus by viewing computation as the process of building a cut-free sequent proof bottom-up. The first accounts of logic programming as proof search were given in classical and intuitionistic logic. Given that linear logic allows richer sequents and richer dynamics in the rewriting of sequents during proof search, it was inevitable that linear logic would be used to design new and more expressive logic programming languages. We overview how linear logic has been used to design such new languages and describe briefly some applications and implementation issues for them.

Introduction

It is now commonplace to recognize the important role of logic in the foundations of computer science. When a major new advance is made in our understanding of logic, we can thus expect to see that advance ripple into many areas of computer science. Such rippling has been observed during the years since the introduction of linear logic by Girard in 1987. Since linear logic embraces computational themes directly in its design, it often allows direct and declarative approaches to computational and resource sensitive specifications. Linear logic also provides new insights into the many computational systems based on classical and intuitionistic logics since it refines and extends these logics.

There are two broad approaches by which logic, via the theory of proofs, is used to describe computation. One approach is the proof reduction paradigm, which can be seen as a foundation for functional programming. Here, programs are viewed as natural deduction or sequent calculus proofs and computation is modeled using proof normalization.

This volume is published in honour of the Azores summer school on Linear Logic and Computer Science held August 30 – Sept. 7, 2000 in St. Miguel, Azores. It can be considered as the third in a series, following a volume dedicated to the Cornell Linear Logic workshop of 1993 published as vol. 222 in the LMSLNS series of Cambridge University Press, and volumes dedicated to the Tokyo meeting of 1996 published as vol. 227(1–2) and 294(3) of Theoretical Computer Science. The summer school was attended by students and researchers from the different sites of the EU Training and Mobility of Researchers project “Linear Logic in Computer Science” (ERBFMRX-CT-97-0170, 1998 – 2002). The Organizing Committee consisted of: V. Michele Abrusci (University of Rome), Nuno Barreiro (University of Lisbon), and Jose Luiz Fiadeiro (University of Lisbon). The school included a series of tutorials, together with thematic sessions covering applications and new directions.

The main purpose of this book is twofold: to give a detailed overview of some well-established developments of Linear Logic under the guise of four tutorials, and to present some of the more recent advances and new directions in the subject through refereed contributions and invited papers. This book does not pretend to exhaustively cover the field of Linear Logic. In the spirit of the TMR “Linear” network, of which the Azores summer school was the climax, we decided to pay particular attention in this volume to the connections of Linear Logic with Computer Science.

We thank the authors of the various contributions for their wideranging and accessible presentations.

Abstract

To attack the problem of “computing with the additives”, we introduce a notion of sliced proof-net for the polarized fragment of linear logic. We prove that this notion yields computational objects, sequentializable in the absence of cuts. We then show how the injectivity property of denotational semantics guarantees the “canonicity” of sliced proof-nets, and prove injectivity for the fragment of polarized linear logic corresponding to the simply typed λ-calculus with pairing.

Introduction

The question of equality of proofs is an important one in the “proofsas-programs” paradigm. Traditional syntaxes (sequent calculus, natural deduction, …) distinguish proofs which are clearly the same as computational processes. On the other hand, denotational semantics identifies “too many” proofs (two different stages of the same computation are always identified). The seek of an object sticking as much as possible to the computational nature of proofs led to the introduction of a new syntax for logic: proof-nets, a graph-theoretic presentation which gives a more geometric account of proofs (see). This discovery was achieved by a sharp (syntactical and semantical) analysis of the cut-elimination procedure.

Any person with a little knowledge of the multiplicative framework of linear logic (LL), has no doubt that proof-nets are the canonical representation of proofs. But as soon as one moves from such a fragment, the notion of proof-net appears “less pure”. A reasonable solution for the multiplicative and exponential fragment of LL (with quantifiers) does exist (combining and, like in).

Abstract

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic.

An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and *-autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic.

No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus.

Introduction

Category theory arose as an organizing framework for expressing the naturality of certain constructions in algebraic topology. Its subsequent applicability, both as a language for simply expressing complex relationships between mathematical structures and as a mathematical theory in its own right, is remarkable. Categorical principles have been put to good use in virtually every branch of mathematics, in most cases leading to profound new understandings.

Roughly a category is an abstraction of the principle that the morphisms between objects of interest are just as important as the objects themselves.