Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

Daniel Ventura
(Universidade de Brasilia)
Mauricio Ayala-Rincón
(Universidade de Brasilia)
Fairouz Kamareddine
(Heriot-Watt University, Edinburgh)

The lambda-calculus with de Bruijn indices assembles each alpha-class of lambda-terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable lambda-terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in beta-contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In previous work, we introduced a de Bruijn version of the lambda-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for beta-normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it.

In Maribel Fernández: Proceedings Ninth International Workshop on Reduction Strategies in Rewriting and Programming (WRS 2009), Brasilia, Brazil, 28th June 2009, Electronic Proceedings in Theoretical Computer Science 15, pp. 69–82.
Published: 26th January 2010.

ArXived at: bibtex PDF

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