Intersection types and lambda models

Alessi, Fabio; Barbanera, Franco; Dezani‐Ciancaglini, Mariangiola

doi:10.1016/j.tcs.2006.01.004

Cited by 16 publications

(15 citation statements)

References 28 publications

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“…Finally, the appropriate sub-reduction relation for the λ-calculus, which we have used to prove Subject Expansion as generally as possible, also helps understanding how and when the semantics of terms is preserved, see Appendix B. This is similar to [ABDC06], and future work should adapt their methodology to accommodate our non-idempotent intersections.…”

Section: Resultsmentioning

confidence: 99%

“…One of the motivations was to characterise strongly normalising (SN) λ-terms, namely the property that a λ-term can be typed if and only if it is strongly normalising. Variants of systems using intersection types have been studied to characterise other evaluation properties of λ-terms and served as the basis of corresponding semantics [BCDC83, Lei86, Kri90, vB95, Ghi96, AC98, Gal98, DCGL04, DCHM05, ABDC06,CS07].…”

mentioning

confidence: 99%

“…2 In particular, the explicit contractions of λlxr relate to the left-introduction of intersections. 3 For instance, [ABDC06] reveals how the notion of intersection type filter can be tuned so that the corresponding filter models identify those λ-terms that are convertible by various restrictions of β-and η-conversion. 4 This also departs from [ABDC06].…”

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confidence: 99%

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Non-idempotent intersection types and strong normalisation

Bernadet¹,

Lengrand²

2013

Logical Methods in Computer Science

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Abstract. We present a typing system with non-idempotent intersection types, typing a term syntax covering three different calculi: the pure λ-calculus, the calculus with explicit substitutions λS, and the calculus with explicit substitutions, contractions and weakenings λlxr.In each of the three calculi, a term is typable if and only if it is strongly normalising, as it is the case in (many) systems with idempotent intersections.Non-idempotency brings extra information into typing trees, such as simple bounds on the longest reduction sequence reducing a term to its normal form. Strong normalisation follows, without requiring reducibility techniques.Using this, we revisit models of the λ-calculus based on filters of intersection types, and extend them to λS and λlxr. Non-idempotency simplifies a methodology, based on such filter models, that produces modular proofs of strong normalisation for well-known typing systems (e.g. System F ). We also present a filter model by means of orthogonality techniques, i.e. as an instance of an abstract notion of orthogonality model formalised in this paper and inspired by classical realisability. Compared to other instances based on terms (one of which rephrases a now standard proof of strong normalisation for the λ-calculus), the instance based on filters is shown to be better at proving strong normalisation results for λS and λlxr.Finally, the bounds on the longest reduction sequence, read off our typing trees, are refined into an exact measure, read off a specific typing tree (called principal ); in each of the three calculi, a specific reduction sequence of such length is identified. In the case of the λ-calculus, this complexity result is, for longest reduction sequences, the counterpart of de Carvalho's result for linear head-reduction sequences.

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Section: Resultsmentioning

confidence: 99%

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confidence: 99%

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Non-idempotent intersection types and strong normalisation

Bernadet¹,

Lengrand²

2013

Logical Methods in Computer Science

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“…The set D is called the domain of the λ-model. The intersection type assignment system allows us to build λ-models by considering as domains sets of filters induced by type theories [4]. Filters of types are sets closed under subtyping and intersection.…”

Section: λ-Modelsmentioning

confidence: 99%

A tale of intersection types

Bono

Dezani‐Ciancaglini

2020

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

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Intersection types have come a long way since their introduction in the Seventies. They have been exploited for characterising behaviours of λ-terms and π-calculus processes, building λ-models, verifying properties of higher-order programs, synthesising code, and enriching the expressivity of programming languages. This paper is a light overview of intersection types and some of their applications. CCS Concepts: • Theory of computation → Type theory.

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“…In Section 2, we recall standard syntactic proofs [ABDC06,Lau12] of preservation of typing by β -reduction and β -expansion for the BCD system. Our presentation stresses the fact that, starting from intersection (and Ω) only, type constructors can be added in a modular way.…”

Section: Introductionmentioning

confidence: 99%

Intersection Subtyping with Constructors

Olivier

2019

Electron. Proc. Theor. Comput. Sci.

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We study the question of extending the BCD intersection type system with additional type constructors. On the typing side, we focus on adding the usual rules for product types. On the subtyping side, we consider a generic way of defining a subtyping relation on families of types which include intersection types. We find back the BCD subtyping relation by considering the particular case where the type constructors are intersection, omega and arrow. We obtain an extension of BCD subtyping to product types as another instance. We show how the preservation of typing by both reduction and expansion is satisfied in all the considered cases. Our approach takes benefits from a "subformula property" of the proposed presentation of the subtyping relation.

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Intersection types and lambda models

Cited by 16 publications

References 28 publications

Non-idempotent intersection types and strong normalisation

Non-idempotent intersection types and strong normalisation

A tale of intersection types

Intersection Subtyping with Constructors

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