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Rank 2 Type Systems and Recursive Definitions
, 1995
"... We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We int ..."
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Cited by 26 (1 self)
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We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We introduce a rank 2 system combining intersections and polymorphism, and prove that it types exactly the same terms as the other rank 2 systems. The combined system suggests a new rule for typing recursive definitions. The result is a rank 2 type system with decidable type inference that can type some interesting examples of polymorphic recursion. Finally,we discuss some applications of the type system in data representation optimizations such as unboxing and overloading.
Definability and full abstraction
 GDP FESTSCHRIFT
"... Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown sin ..."
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Cited by 16 (1 self)
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Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstraction, and we put a few old and recent results of this kind in perspective.
Intersection Type Systems and Logics Related to the Meyer–Routley System B +
, 2003
"... Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, rel ..."
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Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, related to the Meyer–Routley minimal logic B + (without ∨), is weaker than the → ∧ fragment of intuitionistic logic. In this paper we provide an introduction to the above work and also determine the →∧ logics that correspond to certain interesting subsystems of the full →∧ type theory. 1 Simple Typed Lambda Calculus In standard mathematical notation “f: α → β ” stands for “f is a function from α into β. ” If we interpret “: ” as “∈ ” we have the rule: f: α → β t: α f(t) : β This is one of the formation rules of typed lambda calculus, except that there we write ft instead of f(t). In λcalculus, λx.M represents the function f such that fx = M. This makes the following rule a natural one: [x: α] M: β λx.M: α → β We now set up the λterms and their types more formally.
The inhabitation problem for intersection types
, 2003
"... In the system λ ∧ of intersection types, without ω, the problem as to whether an arbitrary type has an inhabitant, has been shown to be undecidable by Urzyczyn in [10]. For one subsystem of λ∧, that lacks the ∧introduction rule, the inhabitation problem has been shown to be decidable in Kurata and ..."
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Cited by 1 (1 self)
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In the system λ ∧ of intersection types, without ω, the problem as to whether an arbitrary type has an inhabitant, has been shown to be undecidable by Urzyczyn in [10]. For one subsystem of λ∧, that lacks the ∧introduction rule, the inhabitation problem has been shown to be decidable in Kurata and Takahashi [9]. The natural question that arises is: What other subsystems of λ∧, have a decidable inhabitation problem? The work in [2], which classifies distinct and inhabitationdistinct subsystems of λ∧, leads to the extension of the undecidability result to λ ∧ without the (η) rule. By new methods, this paper shows, for the remaining six (two of them trivial) distinct subsystems of λ∧, that inhabitation is decidable. For the latter subsystems inhabitant finding algorithms are provided.
Notes on game semantics
, 2006
"... The subject called game semantics grew out as a coherent body of work from two seminal works of the early 1990’s, with forerunners in logic, recursion theory, and semantics. Game semantics allows to provide precise and also natural, interactive semantics to most of the classical features of programm ..."
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The subject called game semantics grew out as a coherent body of work from two seminal works of the early 1990’s, with forerunners in logic, recursion theory, and semantics. Game semantics allows to provide precise and also natural, interactive semantics to most of the classical features of programming such as functions, control, references. The precision is measured by definability and in some cases by full interpretation are being developed, which opens the way for connecting the uses of games in semantics and in verification. 1
www.elsevier.com/locate/apal Classical Fω, orthogonality and symmetric candidates
, 2008
"... We present a version of system Fω, called Fcω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We ..."
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We present a version of system Fω, called Fcω, in which the layer of type constructors is essentially the traditional one of Fω, whereas provability of types is classical. The proofterm calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λcalculus. We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the (classical) layer of terms, we use Barbanera and Berardi’s method based on a symmetric notion of reducibility candidate. We prove that orthogonality does not capture the fixpoint construction of symmetric candidates. We establish the consistency of Fcω, and relate the calculus to the traditional system Fω, also when the latter is extended with axioms for classical logic.