8 research outputs found
Linear Logic and Strong Normalization
Strong normalization for linear logic requires elaborated rewriting techniques. In this paper we give a new presentation of MELL proof nets, without any commutative cut-elimination rule. We show how this feature induces a compact and simple proof of strong normalization, via reducibility candidates. It is the first proof of strong normalization for MELL which does not rely on any form of confluence, and so it smoothly scales up to full linear logic. Moreover, it is an axiomatic proof, as more generally it holds for every set of rewriting rules satisfying three very natural requirements with respect to substitution, commutation with promotion, full composition, and Kesner\u27s IE property. The insight indeed comes from the theory of explicit substitutions, and from looking at the exponentials as a substitution device
Collapsing non-idempotent intersection types
We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi
Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion
We study the semantics of a resource-sensitive extension of the lambda
calculus in a canonical reflexive object of a category of sets and relations, a
relational version of Scott's original model of the pure lambda calculus. This
calculus is related to Boudol's resource calculus and is derived from Ehrhard
and Regnier's differential extension of Linear Logic and of the lambda
calculus. We extend it with new constructions, to be understood as implementing
a very simple exception mechanism, and with a "must" parallel composition.
These new operations allow to associate a context of this calculus with any
point of the model and to prove full abstraction for the finite sub-calculus
where ordinary lambda calculus application is not allowed. The result is then
extended to the full calculus by means of a Taylor Expansion formula. As an
intermediate result we prove that the exception mechanism is not essential in
the finite sub-calculus
Proof-Net as Graph, Taylor Expansion as Pullback
We introduce a new graphical representation for multiplicative and exponential linear logic proof-structures, based only on standard labelled oriented graphs and standard notions of graph theory. The inductive structure of boxes is handled by means of a box-tree. Our proof-structures are canonical and allows for an elegant definition of their Taylor expansion by means of pullbacks
Gluing resource proof-structures: inhabitation and inverting the Taylor expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be
expanded into a set of resource proof-structures: its Taylor expansion. We
introduce a new criterion characterizing those sets of resource
proof-structures that are part of the Taylor expansion of some MELL
proof-structure, through a rewriting system acting both on resource and MELL
proof-structures. As a consequence, we also prove semi-decidability of the type
inhabitation problem for cut-free MELL proof-structures.Comment: arXiv admin note: substantial text overlap with arXiv:1910.0793
An introduction to Differential Linear Logic: proof-nets, models and antiderivatives
Differential Linear Logic enriches Linear Logic with additional logical rules
for the exponential connectives, dual to the usual rules of dereliction,
weakening and contraction. We present a proof-net syntax for Differential
Linear Logic and a categorical axiomatization of its denotational models. We
also introduce a simple categorical condition on these models under which a
general antiderivative operation becomes available. Last we briefly describe
the model of sets and relations and give a more detailed account of the model
of finiteness spaces and linear and continuous functions
Gluing resource proof-structures: inhabitation and inverting the Taylor expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be
expanded into a set of resource proof-structures: its Taylor expansion. We
introduce a new criterion characterizing (and deciding in the finite case)
those sets of resource proof-structures that are part of the Taylor expansion
of some MELL proof-structure, through a rewriting system acting both on
resource and MELL proof-structures. We also prove semi-decidability of the type
inhabitation problem for cut-free MELL proof-structures