A completeness result for a realisability semantics for an intersection type system

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In this paper we consider a type system with a universal type ? where any term (whether open or closed, ß-normalising or not) has type ?. We provide this type system with a realisability semantics where an atomic type is interpreted as the set of ?-terms saturated by a certain relation. The variation of the saturation relation gives a number of interpretations to each type. We show the soundness and completeness of our semantics and that for different notions of saturation (based on weak head reduction and normal ß-reduction) we obtain the same interpretation for types. Since the presence of ? prevents typability and realisability from coinciding and creates extra difficulties in characterizing the interpretation of a type, we define a class U+ of the so-called positive types (where ? can only occur at specific positions). We show that if a term inhabits a positive type, then this term is ß-normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for U+ becomes interesting indeed since it establishes a perfect equivalence between typable terms and terms that inhabit a type. In other words, typability and realisability coincide on U+. We give a number of examples to explain the intuition behind the definition of U+ and to show that this class cannot be extended while keeping its desired properties. © 2007 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)180-198
Number of pages19
JournalAnnals of Pure and Applied Logic
Issue number2-3
Publication statusPublished - May 2007


  • Completeness
  • Intersection type systems
  • Normalisation
  • Positive types
  • Realisability semantics
  • Soundness
  • Subject expansion
  • Subject reduction


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