scholarly journals Proof Nets for Additive Linear Logic with Units

2011 ◽  
Author(s):  
Willem Heijltjes
Keyword(s):  
Linear Logic ◽  
Proof Nets

scholarly journals Proof nets and semi-star-autonomous categories

2014 ◽  
Vol 26 (5) ◽  
pp. 789-828 ◽  
Author(s):  
WILLEM HEIJLTJES ◽  
LUTZ STRAßBURGER
Keyword(s):  
Linear Logic ◽  
Proof Nets

In this paper, it is proved that Girard's proof nets for multiplicative linear logic characterize free semi-star-autonomous categories.

scholarly journals Preface

2016 ◽  
Vol 28 (7) ◽  
pp. 991-994
Author(s):  
LORENZO TORTORA DE FALCO
Keyword(s):  
Taylor Expansion ◽  
Linear Logic ◽  
Research Area ◽  
Theoretical Computer Science ◽  
Quantitative Approach ◽  
Algebraic Setting ◽  
Use Of Resources ◽  
Natural Notion ◽  
Proof Nets ◽  
New Ideas

This special issue is devoted to some aspects of the new ideas that recently arose from the work of Thomas Ehrhard on the models of linear logic (LL) and of the λ-calculus. In some sense, the very origin of these ideas dates back to the introduction of LL in the 80s by Jean-Yves Girard. An obvious remark is that LL yielded a first logical quantitative account of the use of resources: the logical distinction between linear and non-linear formulas through the introduction of the exponential connectives. As explicitly mentioned by Girard in his first paper on the subject, the quantitative approach, to which he refers as ‘quantitative semantics,’ had a crucial influence on the birth of LL. And even though, at that time, it was given up for lack of ‘any logical justification’ (quoting the author), it contained rough versions of many concepts that were better understood, precisely introduced and developed much later, like differentiation and Taylor expansion for proofs. Around 2003, and thanks to the developments of LL and of the whole research area between logic and theoretical computer science, Ehrhard could come back to these fundamental intuitions and introduce the structure of finiteness space, allowing to reformulate this quantitative approach in a standard algebraic setting. The interpretation of LL in the category Fin of finiteness spaces and finitary relations suggested to Ehrhard and Regnier the differential extensions of LL and of the simply typed λ-calculus: Differential Linear Logic (DiLL) and the differential λ-calculus. The theory of LL proof-nets could be straightforwardly extended to DiLL, and a very natural notion of Taylor expansion of a proof-net (and of a λ-term) was introduced: an element of the Taylor expansion of the proof-net/term α is itself a (differential) proof-net/term and an approximation of α.

Planar and braided proof-nets for multiplicative linear logic with mix

1998 ◽  
Vol 37 (5-6) ◽  
pp. 309-325 ◽  
Author(s):  
G. Bellin ◽  
A. Fleury
Keyword(s):  
Linear Logic ◽  
Proof Nets

scholarly journals Proof nets for multiplicative cyclic linear logic and Lambek calculus

2019 ◽  
Vol 29 (06) ◽  
pp. 733-762
Author(s):  
V. Michele Abrusci ◽  
Roberto Maieli
Keyword(s):  
Linear Logic ◽  
Lambek Calculus ◽  
Correctness Criterion ◽  
Proof Nets ◽  
Technical Achievement ◽  
Cyclic Fragment

AbstractThis paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

scholarly journals A coding theoretic study of MLL proof nets

2012 ◽  
Vol 22 (3) ◽  
pp. 409-449 ◽  
Author(s):  
SATOSHI MATSUOKA
Keyword(s):  
Error Detection ◽  
Coding Theory ◽  
Linear Logic ◽  
List Type ◽  
Type Number ◽  
Intelligent Search ◽  
Novel Approach ◽  
Proof Nets ◽  
Enumeration Algorithms ◽  
Definition Of

In this paper we propose a novel approach for analysing proof nets of Multiplicative Linear Logic (MLL) using coding theory. We define families of proof structures called PS-families and introduce a metric space for each family. In each family: (1)an MLL proof net is a true code element; and(2)a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics elegantly reflects the duality of the multiplicative connectives. We show that in our framework one-error-detection is always possible but one-error-correction is always impossible. We also demonstrate the importance of our main result by presenting two proof-net enumeration algorithms for a given PS-family: the first searches proof nets naively and exhaustively without help from our main result, while the second uses our main result to carry out an intelligent search. In some cases, the first algorithm visits proof structures exponentially, while the second does so only polynomially.

Causal dependencies in multiplicative linear logic with MIX

1995 ◽  
Vol 5 (3) ◽  
pp. 351-380 ◽  
Author(s):  
Andrea Asperti
Keyword(s):  
Distributed Systems ◽  
Linear Logic ◽  
Correctness Criterion ◽  
Proof Nets ◽  
Original Interpretation ◽  
Flow Of Information

A new correctness criterion for discriminating Proof Nets among Proof Structures of Multiplicative Linear Logic with the MIX rule is provided. This criterion is inspired by an original interpretation of Proof Structures as distributed systems, and logical formulae as processes. The computation inside a system corresponds to the logical flow of information inside a proof, that is, roughly speaking, a distributed version of Girard's token trip. Proof Nets are then characterised as deadlock free Proof Structures (deadlock free distributed systems). This result follows by explicitly considering the causal dependencies among logical formulae inside proofs, and it provides a new understanding of notions such as acyclicity, chains and empires in terms of concurrent computations.

scholarly journals Proofs, denotational semantics and observational equivalences in Multiplicative Linear Logic

2007 ◽  
Vol 17 (2) ◽  
pp. 341-359 ◽  
Author(s):  
MICHELE PAGANI
Keyword(s):  
Linear Logic ◽  
Denotational Semantics ◽  
Cut Elimination ◽  
Relational Semantics ◽  
Observational Equivalence ◽  
Proof Nets ◽  
General Method

We study full completeness and syntactical separability of MLL proof nets with the mix rule. The general method we use consists of first addressing these two questions in the less restrictive framework of proof structures, and then adapting the results to proof nets.At the level of proof structures, we find a semantical characterisation of their interpretations in relational semantics, and define an observational equivalence that is proved to be the equivalence induced by cut elimination. Hence, we obtain a semantical characterisation (in coherent spaces) and an observational equivalence for the proof nets with the mix rule.

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