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Asian Symposium on Programming Languages and Systems

APLAS 2018: Programming Languages and Systems pp 24–44Cite as

Factoring Derivation Spaces via Intersection Types

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Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11275))

Abstract

In typical non-idempotent intersection type systems, proof normalization is not confluent. In this paper we introduce a confluent non-idempotent intersection type system for the \(\lambda \)-calculus. Typing derivations are presented using proof term syntax. The system enjoys good properties: subject reduction, strong normalization, and a very regular theory of residuals. A correspondence with the \(\lambda \)-calculus is established by simulation theorems. The machinery of non-idempotent intersection types allows us to track the usage of resources required to obtain an answer. In particular, it induces a notion of garbage: a computation is garbage if it does not contribute to obtaining an answer. Using these notions, we show that the derivation space of a \(\lambda \)-term may be factorized using a variant of the Grothendieck construction for semilattices. This means, in particular, that any derivation in the \(\lambda \)-calculus can be uniquely written as a garbage-free prefix followed by garbage.

Work partially supported by CONICET.

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Notes

  1. 1.

    The notion of projection defined by means of residuals is the standard one, see e.g. [4, Chap. 12] or [33, Sect. 8.7].

  2. 2.

    Problem 2 in the RTA List of Open Problems [14] poses the open-ended question of investigating the properties of “spectra”, i.e. derivation spaces.

  3. 3.

    In [29], Autoerasure is called Axiom A, Finite Residuals is called Axiom B, and Semantic Orthogonality is called PERM. We follow the nomenclature of [1].

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Acknowledgements

To Eduardo Bonelli and Delia Kesner for introducing the first author to these topics. To Luis Scoccola and the anonymous reviewers for helpful suggestions.

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Authors and Affiliations

  1. Departamento de Computación, FCEyN, UBA, Buenos Aires, Argentina

    Pablo Barenbaum & Gonzalo Ciruelos

  2. IRIF, Université Paris 7, Paris, France

    Pablo Barenbaum

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  1. Pablo Barenbaum
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  2. Gonzalo Ciruelos
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Correspondence to Pablo Barenbaum .

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  1. Korea Advanced Institute of Science and Technology, Daejeon, Korea (Republic of)

    Sukyoung Ryu

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Barenbaum, P., Ciruelos, G. (2018). Factoring Derivation Spaces via Intersection Types. In: Ryu, S. (eds) Programming Languages and Systems. APLAS 2018. Lecture Notes in Computer Science(), vol 11275. Springer, Cham. https://doi.org/10.1007/978-3-030-02768-1_2

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  • DOI: https://doi.org/10.1007/978-3-030-02768-1_2

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