8,171 research outputs found
Curry-style type Isomorphisms and Game Semantics
Curry-style system F, ie. system F with no explicit types in terms, can be
seen as a core presentation of polymorphism from the point of view of
programming languages. This paper gives a characterisation of type isomorphisms
for this language, by using a game model whose intuitions come both from the
syntax and from the game semantics universe. The model is composed of: an
untyped part to interpret terms, a notion of game to interpret types, and a
typed part to express the fact that an untyped strategy plays on a game. By
analysing isomorphisms in the model, we prove that the equational system
corresponding to type isomorphisms for Curry-style system F is the extension of
the equational system for Church-style isomorphisms with a new, non-trivial
equation: forall X.A = A[forall Y.Y/X] if X appears only positively in A.Comment: Accept\'e \`a Mathematical Structures for Computer Science, Special
Issue on Type Isomorphism
Mixin Composition Synthesis based on Intersection Types
We present a method for synthesizing compositions of mixins using type
inhabitation in intersection types. First, recursively defined classes and
mixins, which are functions over classes, are expressed as terms in a lambda
calculus with records. Intersection types with records and record-merge are
used to assign meaningful types to these terms without resorting to recursive
types. Second, typed terms are translated to a repository of typed combinators.
We show a relation between record types with record-merge and intersection
types with constructors. This relation is used to prove soundness and partial
completeness of the translation with respect to mixin composition synthesis.
Furthermore, we demonstrate how a translated repository and goal type can be
used as input to an existing framework for composition synthesis in bounded
combinatory logic via type inhabitation. The computed result is a class typed
by the goal type and generated by a mixin composition applied to an existing
class
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
Refinement Types for Logical Frameworks and Their Interpretation as Proof Irrelevance
Refinement types sharpen systems of simple and dependent types by offering
expressive means to more precisely classify well-typed terms. We present a
system of refinement types for LF in the style of recent formulations where
only canonical forms are well-typed. Both the usual LF rules and the rules for
type refinements are bidirectional, leading to a straightforward proof of
decidability of typechecking even in the presence of intersection types.
Because we insist on canonical forms, structural rules for subtyping can now be
derived rather than being assumed as primitive. We illustrate the expressive
power of our system with examples and validate its design by demonstrating a
precise correspondence with traditional presentations of subtyping. Proof
irrelevance provides a mechanism for selectively hiding the identities of terms
in type theories. We show that LF refinement types can be interpreted as
predicates using proof irrelevance, establishing a uniform relationship between
two previously studied concepts in type theory. The interpretation and its
correctness proof are surprisingly complex, lending support to the claim that
refinement types are a fundamental construct rather than just a convenient
surface syntax for certain uses of proof irrelevance
A Typed Lambda Calculus with Intersection Types
AbstractIntersection types are well known to type theorists mainly for two reasons. Firstly, they type all and only the strongly normalizable lambda terms. Secondly, the intersection type operator is a meta-level operator, that is, there is no direct logical counterpart in the Curry–Howard isomorphism sense. In particular, its meta-level nature implies that it does not correspond to the intuitionistic conjunction.The intersection type system is naturally a type inference system (system à la Curry), but the meta-level nature of the intersection operator does not allow to easily design an equivalent typed system (system à la Church). There are many proposals in the literature to design such systems, but none of them gives an entirely satisfactory answer to the problem. In this paper, we will review the main results in the literature both on the logical interpretation of intersection types and on proposed typed lambda calculi.The core of this paper is a new proposal for a true intersection typed lambda calculus, without any meta-level notion. Namely, any typable term (in the intersection type inference) has a corresponding typed term (which is the same as the untyped term by erasing the type decorations and the typed term constructors) with the same type, and vice versa.The main idea is to introduce a relevant parallel term constructor which corresponds to the intersection type constructor, in such a way that terms in parallel share the same resources, that is, the same context of free typed variables. Three rules allow us to generate all typed terms. The first two rules, Application and Lambda-abstraction, are performed on all the components of a parallel term in a synchronized way. Finally, via the third rule of Local Renaming, once a free typed variable is bounded by lambda-abstraction, each of the terms in parallel can do its local renaming, with type refinement, of that particular resource
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
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
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