8 research outputs found
Realisability Semantics for Intersection Types and Expansion Variables
Expansion was invented at the end of the 1970s for calculating principal
typings for -terms in type systems with intersection types. Expansion
variables (E-variables) were invented at the end of the 1990s to simplify and
help mechanise expansion. Recently, E-variables have been further simplified
and generalised to also allow calculating type operators other than just
intersection. There has been much work on denotational semantics for type
systems with intersection types, but none whatsoever before now on type systems
with E-variables. Building a semantics for E-variables turns out to be
challenging. To simplify the problem, we consider only E-variables, and not the
corresponding operation of expansion. We develop a realisability semantics
where each use of an E-variable in a type corresponds to an independent degree
at which evaluation occurs in the -term that is assigned the type. In
the -term being evaluated, the only interaction possible between
portions at different degrees is that higher degree portions can be passed
around but never applied to lower degree portions. We apply this semantics to
two intersection type systems. We show these systems are sound, that
completeness does not hold for the first system, and completeness holds for the
second system when only one E-variable is allowed (although it can be used many
times and nested). As far as we know, this is the first study of a denotational
semantics of intersection type systems with E-variables (using realisability or
any other approach)
Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices
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
A complete realisability semantics for intersection types and arbitrary expansion variables
Expansion was introduced at the end of the 1970s for calculating principal
typings for -terms in intersection type systems. Expansion variables
(E-variables) were introduced at the end of the 1990s to simplify and help
mechanise expansion. Recently, E-variables have been further simplified and
generalised to also allow calculating other type operators than just
intersection. There has been much work on semantics for intersection type
systems, but only one such work on intersection type systems with E-variables.
That work established that building a semantics for E-variables is very
challenging. Because it is unclear how to devise a space of meanings for
E-variables, that work developed instead a space of meanings for types that is
hierarchical in the sense of having many degrees (denoted by indexes). However,
although the indexed calculus helped identify the serious problems of giving a
semantics for expansion variables, the sound realisability semantics was only
complete when one single E-variable is used and furthermore, the universal type
was not allowed. In this paper, we are able to overcome these
challenges. We develop a realisability semantics where we allow an arbitrary
(possibly infinite) number of expansion variables and where is
present. We show the soundness and completeness of our proposed semantics.Comment: 5th International Colloquium on Theoretical Aspects of Computing,
ICTAC 2008, 1-3 September 2008, Istanbul : Turquie (2008
A completeness result for a realisability semantics for an intersection type system
International audienceIn this paper we consider a type system with a universal type where any term (whether open or closed, -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 -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 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 -normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for 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 . We give a number of examples to explain the intuition behind the definition of and to show that this class cannot be extended while keeping its desired properties
A completeness result for a realisability semantics for an intersection type system
International audienceIn this paper we consider a type system with a universal type where any term (whether open or closed, -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 -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 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 -normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for 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 . We give a number of examples to explain the intuition behind the definition of and to show that this class cannot be extended while keeping its desired properties