Laurent Regnier

The structure of multiplicatives

Investigating Girards new propositionnal calculus which aims at a large scale study of computation, we stumble quickly on that question: What is a multiplicative connective? We give here a detailed answer together with our motivations and expectations.

The differential Lambda-calculus

We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

Differential interaction nets

We introduce interaction nets for a fragment of the differential lambda-calculus and exhibit in this framework a new symmetry between the of course and the why not modalities of linear logic, which is completely similar to the symmetry between the tensor and par connectives of linear logic. We use algebraic intuitions for introducing these nets and their reduction rules, and then we develop two correctness criteria (weak typability and acyclicity) and show that they guarantee strong normalization. Finally, we outline the correspondence between this interaction nets formalism and the resource lambda-calculus.

Game semantics and abstract machines

The interaction processes at work by M. Hyland and L. Ong (1994) (HO) and S. Abramsky et al. (1994) (AJM) new game semantics are two preexisting paradigmatic implementations of linear head reduction: respectively Krivines abstract machine and Girards interaction abstract machine. There is a simple and natural embedding of AJM-games to HO-games, mapping strategies to strategies and reducing AJM definability (or full abstraction) property to HOs one.

Uniformity and the Taylor expansion of ordinary lambda-terms

We define the complete Taylor expansion of an ordinary lambda-term as an infinite linear combination-with rational coefficients-of terms of a resource calculus similar to Boudols lambda-calculus with multiplicities (or with resources). In our resource calculus, all applications are (multi)linear in the algebraic sense, i.e. commute with linear combinations of the function or the argument. We study the collective behaviour of the beta-reducts of the terms occurring in the Taylor expansion of any ordinary lambda-term, using, in a surprisingly crucial way, a uniformity property that they enjoy. As a corollary, we obtain (the main part of) a proof that this Taylor expansion commutes with Bohm tree computation, syntactically.

Believe it or not, AJM's games model is a model of classical linear logic

A general category of games is constructed. A subcategory of saturated strategies, closed under all possible codings in copy games, is shown to model reduction in classical linear logic.

Böhm trees, krivine's machine and the taylor expansion of lambda-terms

We introduce and study a version of Krivines machine which provides a precise information about how much of its argument is needed for performing a computation. This information is expressed as a term of a resource lambda-calculus introduced by the authors in a recent article; this calculus can be seen as a fragment of the differential lambda-calculus. We use this machine to show that Taylor expansion of lambda-terms (an operation mapping lambda-terms to generally infinite linear combinations of resource lambda-terms) commutes with Bohm tree computation.

Local and asynchronous beta-reduction (an analysis of Girard's execution formula)

The authors build a confluent, local, asynchronous reduction on lambda -terms, using infinite objects (partial injections of Girards (1988) algebra L*), which is simple (only one move), intelligible (semantic setting of the reduction), and general (based on a large-scale decomposition of beta ), and may be mechanized.<<ETX>>

About translations of classical logic into polarized linear logic

We show that the decomposition of intuitionistic logic into linear logic along the equation A /spl rarr/ B = !A /spl rarr/ B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly, we build a categorical model of classical logic (a control category) from a categorical model of linear logic by a construction similar to the co-Kleisli category. Secondly, we analyze two standard continuation-passing style (CPS) translations, the Plotkin and the Krivines translations, which are shown to correspond to two embeddings of LLP into LL.

Some results on the interpretation of lambda -calculus in operator algebras

J.-Y. Girard (Proc. ASL Meeting, 1988) proposed an interpretation of second order lambda -calculus in a C algebra and showed that the interpretation of a term is a nilpotent operator. By extending to untyped lambda -calculus the functional analysis interpretation for typed lambda -terms, V. Danos (Proc. 3rd Italian Conf. on Theor. Comput. Sci., 1989) showed that all and only strongly normalizable terms are interpreted by nilpotent operators; in particular all and only nonstrongly normalizable terms are interpreted by infinite sums of operators. It is shown that interpretation of lambda -terms always makes sense, by showing that lambda -terms are interpreted by weakly nilpotent operators in the sense of Girard. This result is obtained as a corollary of an aperiodicity property of execution of lambda -terms, which seems to be related to some basic property of environment machines.<<ETX>>