Modelling Coeffects in the Relational Semantics of Linear Logic

Breuvart, Flavien; Pagani, Michele

doi:10.4230/lipics.csl.2015.567

Cited by 7 publications

(11 citation statements)

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“…More recently, researchers started to design calculi with types governing more general notions of resource consumptions [Brunel et al 2014;Ghica and Smith 2014], quantitative aspects of code usage [Atkey 2018;Orchard et al 2019;Reed and Pierce 2010], and environmental requirements [Orchard 2014;Petricek et al 2014], this way obtaining general modal-like type systems [Abel and Bernardy 2020;Gaboardi et al 2016;Orchard et al 2019]. From a semantic perspective, such systems have been investigated by means of (comonadic) denotational semantics [Breuvart and Pagani 2015;Gaboardi et al 2016;Ghica and Smith 2014] and (mostly heap-based) resource sensitive operational semantics [Abel and Bernardy 2020;Bernardy et al 2018;Brunel et al 2014;Choudhury et al 2021;Orchard et al 2019]. From a denotational perspective, we can think about our program refinements (and equivalences) as defining a denotational model in a category of modal relational spaces and relational homomorphism.…”

Section: Discussionmentioning

confidence: 99%

“…Due to their many applications, there is a growing interest in the programming language community for modal and coeffectful languages, especially when combined with computational effects [Gaboardi et al 2016;Orchard et al 2019]. From a semantic perspective, however, coeffectful languages have been mostly studied by means of denotational semantics [Breuvart and Pagani 2015;Brunel et al 2014;Gaboardi et al 2016] and almost nothing has been done from the point of view of relational reasoning, with the exception of the recent work by Abel and Bernardy [2020] on logical relations which, however, deals with coeffects only and does not support computational effects. This gap in the literature is quite surprising, since many cornerstone results Ð such as non-interference [Abadi et al 1999], metric preservation [Reed and Pierce 2010], and proof irrelevance [Pfenning 2001] Ð on concrete coeffects are inherently relational.…”

Section: Introductionmentioning

confidence: 99%

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A relational theory of effects and coeffects

Lago

Gavazzo

2022

Proc. ACM Program. Lang.

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Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent, usage-sensitive computations, especially when combined with computational effects. From a semantic perspective, effectful and coeffectful languages have been studied mostly by means of denotational semantics and almost nothing has been done from the point of view of relational reasoning. This gap in the literature is quite surprising, since many cornerstone results — such as non-interference , metric preservation , and proof irrelevance — on concrete coeffects are inherently relational. In this paper, we fill this gap by developing a general theory and calculus of program relations for higher-order languages with combined effects and coeffects. The relational calculus builds upon the novel notion of a corelator (or comonadic lax extension ) to handle coeffects relationally. Inside such a calculus, we define three notions of effectful and coeffectful program refinements: contextual approximation , logical preorder , and applicative similarity . These are the first operationally-based notions of program refinement (and, consequently, equivalence) for languages with combined effects and coeffects appearing in the literature. We show that the axiomatics of a corelator (together with the one of a relator) is precisely what is needed to prove all the aforementioned program refinements to be precongruences, this way obtaining compositional relational techniques for reasoning about combined effects and coeffects.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A relational theory of effects and coeffects

Lago

Gavazzo

2022

Proc. ACM Program. Lang.

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“…While B S LL is presented in [4] as an intuitionistic system, we consider here its (one-sided) classical version. Everything we discuss in this paper could be done in an intuitionistic setting in a very similar way.…”

Section: Stratified Bounded Linear Logicmentioning

confidence: 99%

“…Some closure properties of the parameters (with respect to the pre-order) are required for cut elimination to hold. The idea of indexing the exponential modalities is also at the heart of Stratified Bounded Linear Logic (B S LL) [4]. Indexing is there based on a semi-ring endowed with a compatible partial order.The new system we consider is called Superexponential Linear Logic (superLL).…”

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confidence: 99%

Super Exponentials in Linear Logic

Bauer

Olivier

2021

Electron. Proc. Theor. Comput. Sci.

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Following the idea of Subexponential Linear Logic and Stratified Bounded Linear Logic, we propose a new parameterized version of Linear Logic which subsumes other systems like ELL, LLL or SLL, by including variants of the exponential rules. We call this system Superexponential Linear Logic (superLL). Assuming some appropriate constraints on the parameters of superLL, we give a generic proof of cut elimination. This implies that each variant of Linear Logic which appears as a valid instance of superLL also satisfies cut elimination.Linear logic (LL) has been introduced by Jean-Yves Girard in 1987 [8]. Since then, it has become a pervasive tool in proof theory, in typing systems and semantics for programming languages, in computational complexity theory, etc. The key property which provides a computational meaning to this logic is cut elimination.During the years, many variants of LL have been introduced which differ in particular on some specific uses of exponential rules. Each time a dedicated proof of cut elimination is provided by the authors. We are interested in finding a generic cut-elimination proof for as many systems as possible.Proving the cut-elimination theorem for many systems at once is already the idea behind the parametric system of Subexponential Linear Logic (seLL) [7,14]. However it relies on a parameterized version of Girard's promotion rule, and thus rules out systems based on other kinds of promotions such as functorial promotion. Parameters of seLL allow to control ?-rules. Exponential connectives are indexed by some exponential signatures (instead of a single pair {!, ?}). These signatures are equipped with a pre-order structure used in extending Girard's promotion rule. Some closure properties of the parameters (with respect to the pre-order) are required for cut elimination to hold. The idea of indexing the exponential modalities is also at the heart of Stratified Bounded Linear Logic (B S LL) [4]. Indexing is there based on a semi-ring endowed with a compatible partial order.The new system we consider is called Superexponential Linear Logic (superLL). Its ?-rules are parameterized by predicates which provide the valid relations between the exponential signatures used in the premises and in the conclusion of each rule. In order to take into account variants of LL used in implicit computational complexity (ELL [9], LLL [9], SLL [12]), it is simpler to consider a system based on a functorial version of promotion together with an explicit digging rule. As a counter part, we have to understand how this is related with Girard's promotion rule.Under appropriate axioms on the parameters, we can describe various proof transformations on su-perLL including in particular cut elimination. Choosing specific instances of superLL leads to systems equivalent to a number of variants of LL from the literature (some light systems for complexity, but also seLL or B S LL).In Sections 1 and 2, we recall the definitions of LL and of the variants we are going to consider. The notion of E -formula which deals wi...

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“…Hence, we need to rephrase the substitutive property to explicitly consider such resources. To this end, we work in (fragments of) Linear Logic enriched by graded modalities [8]. Roughly, one adds to the calculus a family of unary connectives !…”

Section: Introductionmentioning

confidence: 99%

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

Dagnino¹,

Pasquali²

2021

Preprint

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Quantitative reasoning provides a flexible approach capable to deal with the uncertainty and imprecision that affects modern software systems due to their complexity. Its distinguishing feature is the use of distances, instead of equivalence relations, so that one can measure how much two objects are similar, rather than just saying whether they are equivalent or not. In this paper we aim at providing a solid logical ground to quantitative reasoning, using the categorical language of Lawvere's hyperdoctrines. The key idea is to see distances as equality predicates in Linear Logic. Adding equality to Linear Logic, however, is not as trivial as it might appear. The linear version of the usual rules for equality, asserting that it is reflexive and substitutive, has the consequence that equality collapses to an equivalence relation, as it can be used an arbitrary number of times. To overcome this issue, we consider the extension of Linear Logic with graded modalities and use them to write a resource sensitive substitution rule that keeps equality quantitative. We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give semantics to it. Main examples are based on metric spaces and Lipschitz maps and on quantitative realisability.

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Modelling Coeffects in the Relational Semantics of Linear Logic

Cited by 7 publications

References 0 publications

A relational theory of effects and coeffects

A relational theory of effects and coeffects

Super Exponentials in Linear Logic

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

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