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.

SN and CR for free-style LKtq: linear decorations and simulation of normalization

2002 ◽  
Vol 67 (1) ◽  
pp. 162-196 ◽  
Author(s):  
Jean-Baptiste Joinet ◽  
Harold Schellinx ◽  
Lorenzo Tortora De Falco
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Present Report ◽  
Point Of View ◽  
Strong Normalization ◽  
Structural Rules ◽  
Specific Combination ◽  
Proof Nets ◽  
Classical Proof

AbstractThe present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e.. by only one specific combination, and structural rules.

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.

A new deconstructive logic: linear logic

1997 ◽  
Vol 62 (3) ◽  
pp. 755-807 ◽  
Author(s):  
Vincent Danos ◽  
Jean-Baptiste Joinet ◽  
Harold Schellinx
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Second Order ◽  
Main Concern ◽  
Simple Pattern ◽  
Strong Normalization ◽  
Viable Systems ◽  
Proof Nets

AbstractThe main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

On the unification of classical, intuitionistic and affine logics

2018 ◽  
Vol 29 (8) ◽  
pp. 1177-1216
Author(s):  
CHUCK LIANG
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Kripke Semantics ◽  
Cut Elimination ◽  
Structural Rules ◽  
Phase Semantics ◽  
Intuitionistic Logics

This article presents a unified logic that combines classical logic, intuitionistic logic and affine linear logic (restricting contraction but not weakening). We show that this unification can be achieved semantically, syntactically and in the computational interpretation of proofs. It extends our previous work in combining classical and intuitionistic logics. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. We define a phase semantics for this logic that naturally extends the Kripke semantics of intuitionistic logic. We present a sequent calculus with novel structural rules, which entail a more elaborate procedure for cut elimination. Computationally, this system allows affine-linear interpretations of proofs to be combined with classical interpretations, such as the λμ calculus. We show how cut elimination must respect the boundaries between classical and non-classical modes of proof that correspond to delimited control effects.

On bunched typing

2003 ◽  
Vol 13 (4) ◽  
pp. 747-796 ◽  
Author(s):  
PETER O'HEARN
Keyword(s):  
Additive Function ◽  
Linear Logic ◽  
Type System ◽  
The Other ◽  
Basic System ◽  
Interference Control ◽  
Joint Work ◽  
Function Type ◽  
Structural Rules ◽  
Single Category

We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to different varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most basic system has two context-combining operations, and the structural rules of Weakening and Contraction are allowed for one but not the other. This system includes a multiplicative, or substructural, function type −∗ alongside the usual (additive) function type $\rightarrow$; it is dubbed the $\alpha\lambda$-calculus after its binders, $\alpha$ for the $\alpha$dditive binder and $\lambda$ for the multiplicative, or $\lambda$inear, binder. We show that the features of this system are, in a sense, complementary to calculi based on linear logic; it is incompatible with an interpretation where a multiplicative function uses its argument once, but perfectly compatible with a reading based on sharing of resources. This sharing interpretation is derived from syntactic control of interference, a type-theoretic method of controlling sharing of storage, and we show how bunch-based management of Contraction can be used to provide a more flexible type system for interference control.

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 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 A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-28
Author(s):  
Delia Kesner
Keyword(s):  
Linear Logic ◽  
Operational Semantics ◽  
Functional Term ◽  
Strong Normalization ◽  
Fine Grained ◽  
Fundamental Properties ◽  
Structural Equivalence ◽  
Proof Nets ◽  
Intuitionistic Linear Logic ◽  
Single Exponential

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

Sign in / Sign up

Export Citation Format

Share Document