THE SYNTAX OF LINEAR LOGIC

The connectives of linear logic

Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. Since there is no hope to modify the extant classical or intuitionistic connectives, linear logic introduces new connectives.

Exponentials : actions vs situations

Classical and intuitionistic logics deal with stable truths:

if A and A ⇒ B, then B, but A still holds.

This is perfect in mathematics, but wrong in real life, since real implication is causal. A causal implication cannot be iterated since the conditions are modified after its use ; this process of modification of the premises (conditions) is known in physics as reaction. For instance, if A is to spend $1 on a pack of cigarettes and B is to get them, you lose $1 in this process, and you cannot do it a second time. The reaction here was that $1 went out of your pocket. The first objection to that view is that there are in mathematics, in real life, cases where reaction does not exist or can be neglected : think of a lemma which is forever true, or of a Mr. Soros, who has almost an infinite amount of dollars. Such cases are situations in the sense of stable truths. Our logical refinements should not prevent us to cope with situations, and there will be a specific kind of connectives (exponentials, “!” and “?”) which shall express the iterability of an action, i.e. the absence of any reaction ; typically!A means to spend as many dollars as one needs.

Abstract

The paper studies the properties of the subnets of proof-nets. Very simple proofs are obtained of known results on proof-nets for MLL-, Multiplicative Linear Logic without propositional constants.

Preface

The theory of proof-nets for MLL-, multiplicative linear logic without the propositional constants 1 and ⊥, has been extensively studied since Girard's fundamental paper [5]. The improved presentation of the subject given by Danos and Regnier [3] for propositional MLL- and by Girard [7] for the first-order case has become canonical: the notions are defined of an arbitrary proof-structure and of a ‘contex-forgetting’ map (·)- from sequent derivations to proof-structures which preserves cut-elimination; correctness conditions are given that characterize proof-nets, the proof-structures R such that R = (D)-, for some sequent calculus derivation D. Although Girard's original correctness condition is of an exponential computational complexity over the size of the proof-structure, other correctness conditions are known of quadratic computational complexity.

A further simplification of the canonical theory of proof-nets has been obtained by a more general classification of the subnet of a proof-net. Given a proof-net R and a formula A in R, consider the set of subnets that have A among their conclusions, in particular the largest and the smallest subnet in this set, called the empire and the kingdom of A, respectively. One must give a construction proving that such a set is not empty: in Girard's fundamental paper a construction of the empires is given which is linear in the size of the proof-net.

Introduction

The aim of this paper is to give a purely graph-theoretical definition of noncommutative proof nets, i.e. graphs coming from proofs in MNLL (multiplicative noncommutative linear logic, the (⊗, ℘)-fragment of the one-sided sequent calculus for classical noncommutative linear logic, introduced in [Abr91]). Analogously, one of the aims of [Gir87] was to give a purely graph-theoretical definition of proof nets, i.e. graphs coming from the proofs in MLL (multiplicative linear logic, the (⊗, ℘)-fragment of the one-sided sequent calculus for classical linear logic - better, for classical commutative linear logic). - The relevance of the purely graph-theoretical definition of proof nets for the development of commutative linear logic is well-know; thus we hope the results of this paper will be useful for a similar development of noncommutative linear logic.

The language for MNLL is an extension of the language for MLL, obtained simply adding, as atomic formulas, propositional letters with an arbitrary finite number of negations written after the propositional letter (linear post-negation) or before the propositional letter (linear retronegation). Every formula A of MNLL may be translated into a formula Tv(A) of MLL (simply by replacing each propositional letter with an even number of negations by the propositional letter without negations, and each propositional letter with an odd number of negations by the propositional letter with only one negation after the propositional letter).

Abstract

The syntactic calculus, a fragment of noncommutative linear logic, was introduced in 1958 because of its hoped for linguistic application. Working with a Gentzen style presentation, one was led to the problem of finding all derivations f : A1 … An → B in the free syntactic calculus generated by a context free grammar g (with arrows reversed) and to the problem of determining all equations f = g between two such derivations. The first problem was solved by showing that f is equal to a derivation in normal form, whose construction involves no identity arrows and no cuts (except those in g) and the second problem is solved by reducing both f and g to normal form.

The original motivation for the syntactic calculus came from multilinear algebra and a categorical semantics was given by the calculus of bimodules. Bimodules RFS may be viewed as additive functors R → Mod S, where R and S are rings (of several objects). It is now clear that Lawvere's generalized bimodules will also provide a semantics for what may be called labeled bilinear logic.

Introduction.

I was asked to talk about one precursor of linear logic that I happened to be involved in, even though it anticipated only a small fraction of what goes on in the linear logic enterprise. I would now call this system “bilinear logic”, meaning “non-commutative linear logic” or “logic without Gentzen's three structural rules”.

Abstract

We present stochastic interactive semantics for propositional linear logic without modalities. The framework is based on interactive protocols considered in computational complexity theory, in which a prover with unlimited power interacts with a verifier that can only toss fair coins or perform simple tasks when presented with the given formula or with subsequent messages from the prover. The additive conjunction &, is described as random choice, which reflects the intuitive idea that the verifier can perform only “random spot checks”. This stochastic interactive semantic framework is shown to be sound and complete. Furthermore, the prover's winning strategies are basically proofs of the given formula. In this framework the multiplicative and additive connectives of linear logic are described by means of probabilistic operators, giving a new basis for intuitive reasoning about linear logic and a potential new tool in automated deduction.

Introduction

Linear logic arose from the semantic study of the structure of proofs in intuitionistic logic. Girard presented the coherence space and phase space semantics of linear logic in his original work on linear logic [Gir87]. While these models provide mathematical tools for the study of several aspects of linear logic, they do not oifer a simple intuitive way of reasoning about linear logic. More recently, Blass [Bla92], Abramsky and Jagadeesan [AJ94], Lamarche, and Hyland and Ong have developed semantics of linear logic by means of games and interaction. These new approaches have already proven fruitful in providing an evocative semantic paradigm for linear logic and have found a striking application to programming language theory in the work of Abramsky, Jagadeesan, and Malacaria [AJM93] and in the work of Hyland and Ong [HO93].

Abstract

The problems of inheritance reasoning in taxonomical networks are crucial in object-oriented languages and in artificial intelligence. A taxonomical network is a graph that enables knowledge to be represented. This paper focuses on the means linear logic offers to represent these networks and is a follow-up to the note on exceptions by Girard [Gir92a]. It is first proved that all compatible nodes of a taxonomical network can be deduced in the taxonomical linear theory associated to the network. Moreover, this theory can be integrated in the Unified Logic LU [Gir92b] and so taxonomical and classical reasoning can be combined.

Introduction

The problems of inheritance reasoning in taxonomical networks are crucial in object-oriented languages and in artificial intelligence. A taxonomical network is a graph that enables knowledge to be represented. The nodes represent concepts or properties of a set of individuals whereas the edges represent relations between concepts. The network can be viewed as a hierarchy of concepts according to levels of generality. A more specific concept is said to inherit informations from its subsumers. There are two kinds of edges: default and exception. A default edge between A and B means that A is generally a B or A has generally the property B. An exception edge between A and B means that there is an exception between A and B, namely A is not a B or A has not the property B. Nonmonotonic systems were developed in the last decade in order to attempt to represent defaults and exceptions in a logical way: the set of inferred grounded facts is the set of properties inherited by concepts.

Abstract

Linear Logic was introduced by Girard [3] as a resource-sensitive refinement of classical logic. Lincoln, Mitchell, Scedrov, and Shankar [13] have proved the undecidability of full propositional Linear Logic. This implies that Linear Logic is more expressive than traditional classical or intuitionistic logic, even if we consider the modalized versions of those logics. In [9, 10] we prove that standard many-counter Minsky machines [17] can be simulated directly in propositional Linear Logic. Here we are going to present a more transparent and fruitful simulation of many-counter Minsky machines in Linear Logic.

Simulating one system of concepts in terms of another system is known to consist of two procedures: (A) Suggesting an encoding of the first system in terms of the second one, and (B) Proving that the encoding suggested is correct and fair.

Here, based on a computational interpretation of Linear Logic [9, 10], we present: (A) A direct and natural encoding of many-counter Minsky machines in Linear Logic, and (B) Transparent proof of the correctness and fairness of this encoding.

As a corollary, we prove that all partial recursive relations are directly definable in propositional Linear Logic.

Introduction and Summary

Linear Logic was introduced by Girard [3] as a resource-sensitive refinement of classical logic. Lincoln, Mitchell, Scedrov, and Shankar [13] have proved the undecidability of full propositional Linear Logic. In [13] the proof of undecidability of propositional Linear Logic consists of a reduction from the Halting Problem for And-Branching Two Counter Machines Without Zero-Test (specified in the same [13]) to a decision problem in Linear Logic.

Introduction

If we consider the interpretation of proofs as programs, say in intuitionistic logic, the question of equality between proofs becomes crucial: The syntax introduces meaningless distinctions whereas the (denotational) semantics makes excessive identifications. This question does not have a simple answer in general, but it leads to the notion of proof-net, which is one of the main novelties of linear logic. This has been already explained in [Gir87] and [GLT89].

The notion of interaction net introduced in [Laf90] comes from an attempt to implement the reduction of these proof-nets. It happens to be a simple model of parallel computation, and so it can be presented independently of linear logic, as in [Laf94]. However, we think that it is also useful to relate the exact origin of interaction nets, especially for readers with some knowledge in linear logic. We take this opportunity to give a survey of the theory of proof-nets, including a new proof of the sequentialization theorem.

Multiplicatives

First we consider the kernel of linear logic, with only two connectives: ⊗ (times or tensor product) and its dual ℘ (par or tensor sum). The first one can be seen as a conjunction and the second one as a disjunction. Each atom has a positive form p and a negative one p⊥ (the linear negation of p).

This volume is based to a large extent on the Linear Logic Workshop held June 14–18, 1993 at the MSI and partially supported by the US Army Research Office and the US Office of Naval Research. The workshop was attended by about 70 participants from the USA, Canada, Europe, and Japan. The workshop program committee was chaired by A. Scedrov (University of Pennsylvania) and included S. Abramsky (Imperial College, London), J.-Y. Girard (CNRS, Marseille), D. Miller (University of Pennsylvania), and J. Mitchell (Stanford). The principal speakers at the workshop were J.-M. Andreoli, A. Blass, V. Danos, J.-Y. Girard, A. Joyal, Y. Lafont, J. Lambek, P. Lincoln, M. Moortgat, R. Pareschi, and V. Pratt. There were also a number of invited 30 minute talks and several software demonstration sessions.

Our intention was not only to publish a volume of proceedings. We also wanted to give an overview of a topic that started almost 10 years ago and that is of interest for mathematicians as well as for computer scientists. For these reasons, the book is divided into 5 parts:

1. Categories and Semantics

2. Complexity and Expressivity

3. Proof Theory

4. Proof Nets

5. Geometry of Interaction

The five parts are preceded by a general introduction to Linear Logic by J.-Y. Girard. Furthermore, parts 2 and 4 start with survey papers by P. Lincoln and Y. Lafont. We hope this book can be useful for those who work in this area as well as for those who want to learn about it. All papers have been refereed and the editors are grateful to A. Scedrov who took care of the refereeing process for the papers written by the the editors themselves.

Abstract

We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli.

Introduction

The present article is a new version of an article already published, with the same title, in Mathematical Structures in Computer Science (1993), vol. 3, pp. 365–385. It is identical to this previous version, except for a few minor details.

In the denotational semantics of purely functional languages (such as PCF [P, BCL]), types are interpreted as objects and programs as morphisms in a cartesian closed category (CCC for short). Usually, the objects of this category are at least Scott domains, and the morphisms are at least continuous functions. The goal of denotational semantics is to express, in terms of “abstract” properties of these functions, some interesting computational properties of the language.

One of these abstract properties is “continuity”. It corresponds to the basic fact that any computation that terminates can use only a finite amount of data. The corresponding semantics of PCF is the continuous one, where objects are Scott domains, and morphisms continuous functions.

But the continuous semantics does not capture an important property of computations in PCF, namely “determinism”. Vuillemin and Milner are at the origin of the first (equivalent) definitions of sequentiality, a semantic notion corresponding to determinism. Kahn and Plotkin ([KP]) generalized this notion of sequentiality. More precisely, they defined a category of “concrete domains” (represented by “concrete data structures”) and of sequential functions.

Abstract

Girard's execution formula (given in [Gir88a]) is a decomposition of usual β-reduction (or cut-elimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a λ-term or a net, as the sum of maximal paths on the λ-term/net that are not cancelled by the algebra L* (as was done in [Dan90, Reg92]).

It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the λ-term/net. Remarkably, the proof puts to use for the first time the interpretation of λ-terms/nets as operators on the Hilbert space.

Presentation

λ-Calculus is simple but not completely convincing as a real machine-language. Real machine instructions have a fixed run-time; a β-reduction step does not. Some implementations do map-reductions into sequences of real elementary steps (as in environment machines for example) but they use a global time to achieve this. The “geometry of interaction” (GOI) is an attempt to find a low-level combinatorial code within which β-reduction could be implemented and such that:

• elementary reduction steps are local;

• parallelism shows up and global time disappears;

• some mathematics dealing with syntax is uncovered.

— Goal and organization of this paper.

A persistent path is a path on a λ-term which survives the action of any reduction (defined in [Reg92]). A regular path is a path which is not cancelled by Girard's algebraic device L* (defined in [Gir88a]).