Intuitionistic Light Affine Logic

Asperti, Andrea; Roversi, Luca

doi:10.1145/504077.504081

Cited by 91 publications

(131 citation statements)

References 11 publications

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“…In order to handle weakening we will use proof-nets with polarities, following [AR02]. Note that proof-nets with polarities had been considered before, e.g.…”

Section: Proof-nets and Complexity Soundnessmentioning

confidence: 99%

“…Note that proof-nets with polarities had been considered before, e.g. in [Lam96], but here we will follow the conventions and notations of [AR02].…”

Section: Proof-nets and Complexity Soundnessmentioning

confidence: 99%

“…It had been observed from the beginning [Gir98] that this feature does not modify the dynamics of ELL and its complexity bounds. We will also allow unrestricted weakening, which is innocuous and common practise since [AR02]. Given some types W for binary words and B for booleans, we will then show that for k ≥ 0, the proofs of conclusion !W ⊸ !…”

Section: Introductionmentioning

confidence: 99%

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Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Baillot

2011

Programming Languages and Systems

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Abstract. Elementary linear logic is a simple variant of linear logic, introduced by Girard and which characterizes in the proofs-as-programs approach the class of elementary functions (computable in time bounded by a tower of exponentials of fixed height). Other systems, like light linear logic have then been defined to capture in a similar way polynomial time functions, but at the price of either a more complicated syntax or of more involved encodings. Such logical systems can then be the basis of type systems to guarantee statically complexity properties on lambdacalculus. Our goal here is to show that despite its simplicity, elementary linear logic can nevertheless be used as a common framework to characterize the different levels of a hierarchy of deterministic time complexity classes, within elementary time. We consider a variant of this logic with type fixpoints and weakening (elementary affine logic with fixpoints). The key ingredients are then the choice of specific types and a finer study of the normalization procedure on proof-nets. We characterize in this way the class P of polynomial time predicates and more generally the hierarchy of classes k-EXP, for k ≥ 0, where k-EXP is the union of DTIME(2 n i k ), for i ≥ 1.

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“…In order to handle weakening we will use proof-nets with polarities, following [AR02]. Note that proof-nets with polarities had been considered before, e.g.…”

Section: Proof-nets and Complexity Soundnessmentioning

confidence: 99%

“…Note that proof-nets with polarities had been considered before, e.g. in [Lam96], but here we will follow the conventions and notations of [AR02].…”

Section: Proof-nets and Complexity Soundnessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Baillot

2011

Programming Languages and Systems

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“…The coming lemmas will imply Proposition 1. Firstly, a generalization of the polynomial soundness in [2], provable by induction on l, is:…”

Section: Canonical Reductions a Reduction Sequencementioning

confidence: 99%

“…We are interested in polynomial time computations and in their characterizations by means of restrictions of Linear Logic (LL) [1]. Specifically, we focus on Light Affine Logic (LAL) [2], a simplification of Light Linear Logic (LLL) [3]. LAL is: (i) strongly polynomial time sound, and (ii) polynomial time complete, under the Curry-Howard (CH) correspondence.…”

Section: Introductionmentioning

confidence: 99%

Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

Roversi¹,

Vercelli

2009

Lecture Notes in Computer Science

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Abstract. We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: "More modalities means more expressiveness". MS is a set of building-rule schemes that depend on parameters. We interpret the values of the parameters as modalities. Fixing the parameters yields deductive systems as instances of MS, that we call subsystems. Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems. A first result is a sufficient condition for determining when a subsystem is strongly polynomial time sound. A second one shows that the ability to choose which modalities are used and how can be rewarding. We give a family of subsystems as complex as Multiplicative Linear Logic -they are linear time and space sound -that can represent Church numerals and some common combinators on them.

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Elementary Affine Logic and the Call-by-Value Lambda Calculus

Coppola

Lago

Rocca

2005

Lecture Notes in Computer Science

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Abstract. Light and elementary linear logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic, as discussed in [1]. In this paper, we show that shifting from usual call-by-name to call-by-value lambda calculus allows to regain strong connections with the underlying logic. We will do this in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.

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Intuitionistic Light Affine Logic

Cited by 91 publications

References 11 publications

Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

Elementary Affine Logic and the Call-by-Value Lambda Calculus

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