1,823,924 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)
Really Natural Linear Indexed Type Checking
Recent works have shown the power of linear indexed type systems for
enforcing complex program properties. These systems combine linear types with a
language of type-level indices, allowing more fine-grained analyses. Such
systems have been fruitfully applied in diverse domains, including implicit
complexity and differential privacy. A natural way to enhance the
expressiveness of this approach is by allowing the indices to depend on runtime
information, in the spirit of dependent types. This approach is used in DFuzz,
a language for differential privacy. The DFuzz type system relies on an index
language supporting real and natural number arithmetic over constants and
variables. Moreover, DFuzz uses a subtyping mechanism to make types more
flexible. By themselves, linearity, dependency, and subtyping each require
delicate handling when performing type checking or type inference; their
combination increases this challenge substantially, as the features can
interact in non-trivial ways. In this paper, we study the type-checking problem
for DFuzz. We show how we can reduce type checking for (a simple extension of)
DFuzz to constraint solving over a first-order theory of naturals and real
numbers which, although undecidable, can often be handled in practice by
standard numeric solvers
On the factor systems of the Shubnikov space groups
The procedure of Backhouse for the determination of a complete set of inequivalent factor systems of the symmorphic Shubnikov space groups of type I is generalized for all Shubnikov space groups of types I and 111. The results are used to obtain a method for the construction of a set of inequivalent factor systems of the Shubnikov space groups of types I1 and IV
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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