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Typability and Type Checking in System F Are Equivalent and Undecidable
 ANNALS OF PURE AND APPLIED LOGIC
, 1998
"... Girard and Reynolds independently invented System F (a.k.a. the secondorder polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions ..."
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Girard and Reynolds independently invented System F (a.k.a. the secondorder polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lowerbounds have been determined for typability in F, but this report is the first to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semiunification, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to type checking, the two problems are equivalent. The reduction from type checking to typability uses a novel method to construct lambda terms that simulate arbitrarily chosen type environments. All the results also hold for the lambdaIotacalculus.
Typability and Type Checking in the SecondOrder lambdaCalculus Are Equivalent and Undecidable
, 1993
"... We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considere ..."
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We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lowerbound complexity results for System F have been achieved, but they have remained "embarrassing open problems" 3 for System F itself. We first prove that type checking in System F is undecidable by a reduction from semiunification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructingterms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require that specif subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment maybe simulated. We develop this method, which we call \constants for free", for both the K and I calculi.
Recognizability in the Simply Typed LambdaCalculus, in "16th Workshop on Logic, Language, Information and Computation
 Lecture Notes in Artificial Intelligence, vol. 5514, Springer, 2009, p. 48–60, http://hal.inria.fr/inria00412654/en/. Scientific Books (or Scientific Book chapters
"... Abstract. We define a notion of recognizable sets of simply typed λterms that extends the notion of recognizable sets of strings or trees. This definition is based on finite models. Using intersection types, we generalize the notions of automata for strings and trees so as to grasp recognizability ..."
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Abstract. We define a notion of recognizable sets of simply typed λterms that extends the notion of recognizable sets of strings or trees. This definition is based on finite models. Using intersection types, we generalize the notions of automata for strings and trees so as to grasp recognizability for λterms. We then expose the closure properties of this notion and present some of its applications. 1
H.: Loader and Urzyczyn are logically related
 ICALP 2012, Part II. LNCS
, 2012
"... Abstract. In simply typed λcalculus with one ground type the following theorem due to Loader holds. (i) Given the full model F over a finite set, the question whether some element f ∈ F is λdefinable is undecidable. In the λcalculus with intersection types based on countably many atoms, the fol ..."
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Abstract. In simply typed λcalculus with one ground type the following theorem due to Loader holds. (i) Given the full model F over a finite set, the question whether some element f ∈ F is λdefinable is undecidable. In the λcalculus with intersection types based on countably many atoms, the following is proved by Urzyczyn. (ii) It is undecidable whether a type is inhabited. Both statements are major results presented in [3]. We show that (i) and (ii) follow from each other in a natural way, by interpreting intersection types as continuous functions logically related to elements of F. From this, and a result by Joly on λdefinability, we get that Urzyczyn’s theorem already holds for intersection types with at most two atoms.
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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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.
Preciseness of Subtyping on Intersection and Union Types?
"... Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two dif ..."
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Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type which is a mathematical object that describes the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected. We propose a technique for formalising and proving operational preciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types. The key feature is the link between typings and the operational semantics. We then prove soundness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness. 1
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.
DRAFT
, 2008
"... This book about typed lambda terms comes in two volumes: the present one about lambda terms typed using simple, recursive and intersection types and a planned second volume about higher order, dependent and inductive types. In some sense this book is a sequel to Barendregt [1984]. That book is about ..."
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This book about typed lambda terms comes in two volumes: the present one about lambda terms typed using simple, recursive and intersection types and a planned second volume about higher order, dependent and inductive types. In some sense this book is a sequel to Barendregt [1984]. That book is about untyped lambda calculus. Types give the untyped terms more structure: function applications are allowed only in some cases. In this way one can single out untyped terms having special properties. But there is more to it. The extra structure makes the theory of typed terms quite different from the untyped ones. The emphasis of the book is on syntax. Models are introduced only in so far they give useful information about terms and types or if the theory can be applied to them. The writing of the book has been different from that about the untyped lambda calculus. First of all, since many researchers are working on typed lambda calculus, we were aiming at a moving target. Also there was a wealth of material to work with. For these reasons the book has been written by several