503 research outputs found
A Theory of Explicit Substitutions with Safe and Full Composition
Many different systems with explicit substitutions have been proposed to
implement a large class of higher-order languages. Motivations and challenges
that guided the development of such calculi in functional frameworks are
surveyed in the first part of this paper. Then, very simple technology in named
variable-style notation is used to establish a theory of explicit substitutions
for the lambda-calculus which enjoys a whole set of useful properties such as
full composition, simulation of one-step beta-reduction, preservation of
beta-strong normalisation, strong normalisation of typed terms and confluence
on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference
"International Colloquium on Automata, Languages and Programming 2008" edited
by Giuseppe Castagna and Igor Walukiewic
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
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