Double categories: a modular model of multiplicative linear logic

2002 ◽  
Vol 12 (4) ◽  
pp. 449-479 ◽  
Author(s):  
PAUL-ANDRÉ MELLIÈS
Keyword(s):  
Side Effects ◽  
Linear Logic ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Modular Model ◽  
Categorical Model ◽  
Double Categories ◽  
Proof Nets ◽  
Composite Module

We construct a double category [Dscr ] of proof-nets in multiplicative linear logic (MLL). Its horizontal arrows are MLL modules (subnets of well-formed nets), its vertical arrows model side-effects, and its double cells interpret the cut-elimination procedure. The categorical model is modular in the sense that every computation of a composite module (π1; π2) factors out as the separate and interacting computations of the two subcomponents π1 and π2. This enables us to trace MLL modules in the course of cut-elimination, and analyze their behaviour in time.

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.

Multi-focused cut elimination

2017 ◽  
Vol 28 (5) ◽  
pp. 614-650
Author(s):  
TAUS BROCK-NANNESTAD ◽  
NICOLAS GUENOT
Keyword(s):  
Simple Proof ◽  
Linear Logic ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Elimination Procedure ◽  
The Impact

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

Multiple conclusion linear logic: cut elimination and more

2020 ◽  
Vol 30 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Harley Eades III ◽  
Valeria de Paiva
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Direct Proof ◽  
Cut Elimination ◽  
Tensor Model ◽  
Double Negation ◽  
Proof Nets ◽  
Small Change ◽  
Definition Of ◽  
Intuitionistic Linear Logic

Abstract Full intuitionistic linear logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par and implication, into an intuitionistic linear logic. Bierman showed that their formalization of FILL did not enjoy cut elimination as such, but Bellin proposed a small change to the definition of FILL regaining cut elimination and using proof nets. In this note we adopt Bellin’s proposed change and give a direct proof of cut elimination for the sequent calculus. Then we show that a categorical model of FILL in the basic dialectica category is also a linear/non-linear model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. We give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor. Lastly, we introduce a new library to be used in the proof assistant Agda for proving properties of dialectica categories.

Minimality of the correctness criterion for multiplicative proof nets

1998 ◽  
Vol 8 (6) ◽  
pp. 543-558 ◽  
Author(s):  
DENIS BECHET
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Correctness Criterion ◽  
Proof Nets ◽  
Static Criterion

Almost a decade ago, Girard invented linear logic with the notion of a proof-net. Proof-nets are special graphs built from formulas, links and boxes. However, not all nets are proof-nets. First, they must be well constructed (we say that such graphs are proof-structures). Second, a proof-net is a proof-structure that corresponds to a sequential proof. It must satisfy a correctness criterion. One may wonder what this static criterion means for cut-elimination. We prove that every incorrect proof-structure (without cut) can be put in an environment where reductions lead to two kinds of basically wrong configurations: deadlocks and disconnected proof-structures. Thus, this proof says that there does not exist a bigger class of proof-structures than proof-nets where normalization does not lead to obviously bad configurations.

scholarly journals A MALL geometry of interaction based on indexed linear logic

2021 ◽  
pp. 1-29
Author(s):  
Masahiro Hamano
Keyword(s):  
Monoidal Category ◽  
Linear Logic ◽  
Dynamic Modelling ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Dynamic Level

Abstract We construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

scholarly journals An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Jules Chouquet ◽  
Lionel Vaux Auclair
Keyword(s):  
Arbitrary Number ◽  
Taylor Expansion ◽  
Linear Logic ◽  
Cut Elimination ◽  
Sufficient Condition ◽  
Weighted Sum ◽  
Combinatorial Properties ◽  
Proof Nets ◽  
Infinite Sums

We examine some combinatorial properties of parallel cut elimination in multiplicative linear logic (MLL) proof nets. We show that, provided we impose a constraint on some paths, we can bound the size of all the nets satisfying this constraint and reducing to a fixed resultant net. This result gives a sufficient condition for an infinite weighted sum of nets to reduce into another sum of nets, while keeping coefficients finite. We moreover show that our constraints are stable under reduction. Our approach is motivated by the quantitative semantics of linear logic: many models have been proposed, whose structure reflect the Taylor expansion of multiplicative exponential linear logic (MELL) proof nets into infinite sums of differential nets. In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion. It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative. We moreover show that the set of differential nets that occur in the Taylor expansion of an MELL net automatically satisfies our constraints. Interestingly, our nets are untyped: we only rely on the sequentiality of linear logic nets and the dynamics of cut elimination. The paths on which we impose bounds are the switching paths involved in the Danos--Regnier criterion for sequentiality. In order to accommodate multiplicative units and weakenings, our nets come equipped with jumps: each weakening node is connected to some other node. Our constraint can then be summed up as a bound on both the length of switching paths, and the number of weakenings that jump to a common node.

scholarly journals Jump from parallel to sequential proofs: exponentials

2016 ◽  
Vol 28 (7) ◽  
pp. 1204-1252
Author(s):  
PAOLO DI GIAMBERARDINO
Keyword(s):  
Distinctive Feature ◽  
Linear Logic ◽  
The Other ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Structural Rules ◽  
New Class ◽  
Game Semantics ◽  
Proof Nets

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

scholarly journals Expansion trees with cut

2019 ◽  
Vol 29 (8) ◽  
pp. 1009-1029 ◽  
Author(s):  
Federico Aschieri ◽  
Stefan Hetzl ◽  
Daniel Weller
Keyword(s):  
Sequent Calculus ◽  
Higher Order ◽  
Point Of View ◽  
Compact Representation ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Herbrand’S Theorem ◽  
Syntactic Approach ◽  
Proof Nets ◽  
Herbrand's Theorem

AbstractHerbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas.In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

scholarly journals Natural deduction via graphs: formal definition and computation rules

2007 ◽  
Vol 17 (3) ◽  
pp. 485-526 ◽  
Author(s):  
HERMAN GEUVERS ◽  
IRIS LOEB
Keyword(s):  
Type Theory ◽  
Natural Deduction ◽  
Linear Logic ◽  
Cut Elimination ◽  
Simple Type ◽  
Slight Variation ◽  
Formal Definition ◽  
Simple Type Theory ◽  
Proof Nets ◽  
Definition Of

In this paper, we introduce the formalism of deduction graphs as a generalisation of both Gentzen–Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that, as with flag deductions (but not natural deduction), subproofs can be shared, but the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. We give a precise definition of deduction graphs, together with some illustrative examples. Furthermore, we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation, we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure, so we also propose a translation to a context calculus with lets that faithfully captures the structure of deduction graphs. The proof nets of linear logic also offer a graph-like presentation of natural deduction, and we point out some similarities between the two formalisms.

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