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Some Results on the Full Abstraction Problem for Restricted Lambda Calculi
 IN PROC. OF THE 18 TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE, MFCS'93, SPRINGER LNCS
, 1993
"... Issues in the mathematical semantics of two restrictions of the calculus, i.e. Icalculus and V calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scottcontinuous functions. A correct, albeit nonfully abs ..."
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Issues in the mathematical semantics of two restrictions of the calculus, i.e. Icalculus and V calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scottcontinuous functions. A correct, albeit nonfully abstract, model for the SECD evaluation of the latter is defined using Girard's coherence spaces and stable functions. These results are used to illustrate the interest of the analysis of the fine structure of mathematical models of programming languages.
IOS Press On Realisability Semantics for Intersection Types with Expansion Variables
"... Abstract. Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, Evariables were introduced in order to make the calculations easier to mechanise and reason about. Recently, Evariables have been ..."
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Abstract. Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, Evariables were introduced in order to make the calculations easier to mechanise and reason about. Recently, Evariables have been further simplified and generalised to also allow calculating other type operators than just intersection. There has been much work on semantics for type systems with intersection types, but none whatsoever before our work, on type systems with Evariables. In this paper we expose the challenges of building a semantics for Evariables and we provide a novel solution. Because it is unclear how to devise a space of meanings for Evariables, we develop instead a space of meanings for types that is hierarchical. First, we index each type with a natural number and show that although this intuitively captures the use of Evariables, it is difficult to index the universal type ω with this hierarchy and it is not possible to obtain completeness of the semantics if more than one Evariable is used. We then move to a more complex semantics where each type is associated with a list of natural numbers and establish that both ω and an arbitrary number of Evariables can be represented without losing any of the desirable properties of a realisability semantics. Keywords: Realisability semantics, expansion variables, intersection types, completeness