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The Delta-calculus: Syntax and Types

Authors Luigi Liquori, Claude Stolze

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Luigi Liquori
  • Université Côte d'Azur, Inria, Sophia Antipolis, France
Claude Stolze
  • Université Côte d'Azur, Inria, Sophia Antipolis, France

Funding

Work supported by the COST Action CA15123 EUTYPES "The European research network on types for programming and verification".

Acknowledgements

We are grateful to Benjamin Pierce, Joe Wells, Furio Honsell, and the anonymous reviewers for the useful comments and remarks.

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Luigi Liquori and Claude Stolze. The Delta-calculus: Syntax and Types. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.FSCD.2019.28

Abstract

We present the Delta-calculus, an explicitly typed lambda-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T, e.g. the Coppo-Dezani, the Coppo-Dezani-Sallé, the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of Delta-calculi with related intersection typed systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems à la Curry and the corresponding intersection typed systems à la Church by means of an essence function translating an explicitly typed Delta-term into a pure lambda-term one. We finally translate a Delta-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic Delta-calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Type theory
Keywords
  • intersection types
  • lambda calculus à la Church and à la Curry
  • proof-functional logics

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