Lionel Vaux

The algebraic lambda calculus

We introduce an extension of the pure lambda calculus by endowing the set of terms with the structure of a vector space, or, more generally, of a module, over a fixed set of scalars. Moreover, terms are subject to identities similar to the usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda calculus. We also provide normalisation results for a simple type system.

The differential λμ-calculus

We define a differential @l@m-calculus which is an extension of both Parigots @l@m-calculus and Ehrhard-Regniers differential @l-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing.

The differential lambda-mu-calculus

We define a differential @l@m-calculus which is an extension of both Parigots @l@m-calculus and Ehrhard-Regniers differential @l-calculus. We prove some basic properties of the system: reduction enjoys Church-Rosser and simply typed terms are strongly normalizing.

Differential Linear Logic and Polarization

We extend Ehrhard---Regniers differential linear logic along the lines of Laurents polarization. We provide a denotational semantics of this new system in the well-known relational model of linear logic, extending canonically the semantics of both differential and polarized linear logics: this justifies our choice of cut elimination rules. Then we show this polarized differential linear logic refines the recently introduced convolution

Transport of finiteness structures and applications

{\bar\lambda}\mu

Convolution \(\bar\lambda\mu\)-Calculus

-calculus, the same as linear logic decomposes *** -calculus.

Strong Normalizability as a Finiteness Structure via the Taylor Expansion of λ -terms

We describe a general construction of finiteness spaces which subsumes the interpretations of all positive connectors of linear logic. We then show how to apply this construction to prove the existence of least fixpoints for particular functors in the category of finiteness spaces: these include the functors involved in a relational interpretation of lazy recursive algebraic datatypes along the lines of the coherence semantics of system T.

A non-uniform finitary relational semantics of system T

We define an extension of Herbelin’s \(\bar\lambda\mu\)-calculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of Ehrhard-Regnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambda-model in the category of sets and relations.We define an extension of Herbelins ¯ λμ-calculus, introduc- ing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new seman- tical constructions, extending models of Ehrhard-Regniers differential interaction nets, along the lines of Laurents polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambda-model in the category of sets and relations.

The differential λ μ -calculus

In the folklore of linear logic, a common intuition is that the structure of finiteness spaces, introduced by Ehrhard, semantically reflects the strong normalization property of cut-elimination.

On linear combinations of lambda-terms

We study iteration and recursion operators in the denotational semantics of typed λ-calculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhards finiteness spaces.