Intersection Type Systems and Explicit Substitutions Calculi

  • Daniel Lima Ventura
  • Mauricio Ayala-Rincón
  • Fairouz Kamareddine
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6188)


The λ-calculus with de Bruijn indices, called λ dB , assembles each α-class of λ-terms into a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism satisfying important properties like principal typing, which allows the type system to include features such as data abstraction (modularity) and separate compilation. To be closer to computation and to simplify the formalisation of the atomic operations involved in β-contractions, several explicit substitution calculi were developed most of which are written with de Bruijn indices. Although untyped and simply types versions of explicit substitution calculi are well investigated, versions with more elaborate type systems (e.g., with intersection types) are not. In previous work, we presented a version for λ dB of an intersection type system originally introduced to characterise principal typings for β-normal forms and provided the characterisation for this version. In this work we introduce intersection type systems for two explicit substitution calculi: the λσ and the λs e . These type system are based on a type system for λ dB and satisfy the basic property of subject reduction, which guarantees the preservation of types during computations.


Type System Intersection Type Inference Rule Typing Rule Principal Typing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Daniel Lima Ventura
    • 1
  • Mauricio Ayala-Rincón
    • 1
  • Fairouz Kamareddine
    • 2
  1. 1.Grupo de Teoria da ComputaçãoDep. de Matemática Universidade de BrasíliaBrasília D.F.Brasil
  2. 2.School of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghScotland UK

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