The Cut-Elimination Theorem for Differential Nets with Promotion

  title={The Cut-Elimination Theorem for Differential Nets with Promotion},
  author={Michele Pagani},
Recently Ehrhard and Regnier have introduced Differential Linear Logic, DiLL for short -- an extension of the Multiplicative Exponential fragment of Linear Logic that is able to express non-deterministic computations. The authors have examined the cut-elimination of the promotion-free fragment of DiLL by means of a proofnet-like calculus: differential interaction nets. We extend this analysis to exponential boxes and prove the Cut-Elimination Theorem for the whole DiLL: every differential net… 

The conservation theorem for differential nets

The conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL) is proved, which turns the quest for strong normalisation into one for non-erasing weak normalisation (WN), and indeed this result is used to prove SN of simply typed DiLL.

An introduction to differential linear logic: proof-nets, models and antiderivatives

  • T. Ehrhard
  • Mathematics, Computer Science
    Mathematical Structures in Computer Science
  • 2017
A proof-net syntax for differential linear logic and a categorical axiomatization of its denotational models and a simple categorical condition on these models under which a general antiderivative operation becomes available are introduced.

Realizability Proof for Normalization of Full Differential Linear Logic

This work provides an extension of this proof that embrace Full Differential Linear Logic (a logic that can describe both single-use resources and inexhaustible resources) and is modular enough so that further extensions (to second order, to additive constructs or to any other independent feature that can be dealt with using realizability) come for free.

MELL proof-nets in the category of graphs

This work presents a formalization of proof-nets (and more generally, proofstructures) for the multiplicative-exponential fragment of linear logic, with a novel treatment of boxes, and aims to contribute to the way proofs can be represented in LL, pushing forward with Girard’s graphical spirit.

An application of the extensional collapse of the relational model of linear logic

This work construction of a new model, which features a new duality, is presented, and how to use it for reducing normalization results in idempotent intersection types to purely combinatorial methods is explained.

Collapsing non-idempotent intersection types

This work construction of a new model, which features a new duality, is presented, and how to use it for reducing normalization results in idempotent intersection types to purely combinatorial methods is explained.

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

This work introduces a finiteness structure on resource terms, which is such that a λ-term is strongly normalizing iff the support of its Taylor expansion is finitary, and proves the existence of a normal form for the Taylor expansion of any strongly normalizable non-deterministic κ-term.

A Logical Account for Linear Partial Differential Equations

D-DiLL is introduced, a deterministic refinement of DiLL with a D-exponential, for which it exhibits a cut-elimination procedure, and a categorical semantics.

Linear Logic and Strong Normalization

This paper gives a new presentation of MELL proof nets, without any commutative cut-elimination rule, which 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.

Proof-Net as Graph, Taylor Expansion as Pullback

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 is introduced, which allows for an elegant definition of their Taylor expansion by means of pullbacks.



Acyclicity and Coherence in Multiplicative Exponential Linear Logic

It turns out that visible acyclicity has also nice computational properties, especially it is stable under cut reduction.

Strong normalization property for second order linear logic

Computational Adequacy in an Elementary Topos

It is proved that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true Σ\(^{\rm 0}_{\rm 1}\)-sentences).

Linear logic

This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.

Differential Interaction Nets

The structure of multiplicatives

Investigating Girard's 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

Interaction nets

A new kind of programming language, with the following features: a simple graph rewriting semantics, a complete symmetry between constructors and destructors, and a type discipline for deterministic and deadlock-free (microscopic) parallelism.

La Logique Linéaire appliquée à l'étude de divers processus de normalisation (principalement du Lambda-calcul)

Certaines preuves formelles peuvent aussi bien etre vues comme des programmes. D'ou vient qu'il y ait un recoupement entre theorie de la demonstration et informatique. Un evenement d'importance dans