A Translation of Intersection and Union Types for the λμ-Calculus

Kikuchi, Kentaro; Sakurai, Takafumi

doi:10.1007/978-3-319-12736-1_7

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…Different qualitative and/or quantitative models for classical calculi were proposed in [45,48,51,3], thus limiting the characterization of operational properties to head-normalization. Intersection and union types were also studied in the framework of classical logic [36,47,34,20], but no work addresses the problem from a quantitative perspective. Type-theoretical characterization of strong-normalization for classical calculi were provided both for λµ [49] and λµμ-calculus [20], but the (idempotent) typing systems do not allow to construct decreasing measures for reduction, thus a resource aware semantics cannot be extracted from those interpretations.…”

Section: Related Workmentioning

confidence: 99%

Non-idempotent types for classical calculi in natural deduction style

Kesner¹,

Vial²

2020

Logical Methods in Computer Science ; Volume 16

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In the first part of this paper, we define two resource aware typing systems for the λµ-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments -based on decreasing measures of type derivations-to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head-reduction and the maximal reduction sequences to normal-form.In the second part of this paper, the λµ-calculus is refined to a small-step calculus called λµs, which is inspired by the substitution at a distance paradigm. The λµs-calculus turns out to be compatible with a natural extension of the non-idempotent interpretations of λµ, i.e. λµs-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly λµs-normalizing terms by means of typing.

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Section: Related Workmentioning

confidence: 99%

Non-idempotent types for classical calculi in natural deduction style

Kesner¹,

Vial²

2020

Logical Methods in Computer Science ; Volume 16

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“…The translation in Sect. 7 that maps simple types into our extension of intersection types is a form of negative translation; in [37] it is extended to the system in [9], thereby relating the original intersection and union type assignment system for λµ to ours.…”

Section: Characterisation Of Strong Normalisationmentioning

confidence: 99%

Intersection Types for the lambda-mu Calculus

Bakel¹,

Barbanera²,

de’Liguoro³

2018

Logical Methods in Computer Science ; Volume 14

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We introduce an intersection type system for the λµ-calculus that is invariant under subject reduction and expansion. The system is obtained by describing Streicher and Reus's denotational model of continuations in the category of ω-algebraic lattices via Abramsky's domain-logic approach. This provides at the same time an interpretation of the type system and a proof of the completeness of the system with respect to the continuation models by means of a filter model construction.We then define a restriction of our system, such that a λµ-term is typeable if and only if it is strongly normalising. We also show that Parigot's typing of λµ-terms with classically valid propositional formulas can be translated into the restricted system, which then provides an alternative proof of strong normalisability for the typed λµ-calculus.

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Consuming and Persistent Types for Classical Logic

Kesner

Vial

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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A Translation of Intersection and Union Types for the λμ-Calculus

Cited by 3 publications

References 18 publications

Non-idempotent types for classical calculi in natural deduction style

Non-idempotent types for classical calculi in natural deduction style

Intersection Types for the lambda-mu Calculus

Consuming and Persistent Types for Classical Logic

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