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Secondorder unification and type inference for Churchstyle polymorphism
 In Conference Record of POPL 98: The 25TH ACM SIGPLANSIGACT Symposium on Principles of Programming Languages
, 1998
"... We present a proof of the undecidability of type inference for the Churchstyle system F  an abstraction of polymorphism. A natural reduction from the secondorder unification problem to type inference leads to strong restriction on instances  arguments of variables cannot contain variables. T ..."
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We present a proof of the undecidability of type inference for the Churchstyle system F  an abstraction of polymorphism. A natural reduction from the secondorder unification problem to type inference leads to strong restriction on instances  arguments of variables cannot contain variables. This requires another proof of the undecidability of the secondorder unification since known results use variables in arguments of other variables. Moreover, our proof uses elementary techniques, which is important from the methodological point of view, because Goldfarb's proof [Gol81] highly relies on the undecidability of the tenth Hilbert's problem. 1 1 Introduction The Churchstyle system F was independently introduced by Girard [Gir72] and Reynolds [Rey74] as an extension of the simplytyped calculus a type system introduced of H. B. Curry [Cur69]. As usual for type systems, the decidability of so called sequent decision problems was considered. A sequent decision problem in some ty...
Type Reconstruction with FirstClass Polymorphic Values
, 1989
"... We present the first type reconstruction system which combines the implicit typing of ML with the full power of the explicitly typed secondorder polymorphic lambda calculus. The system will accept MLstyle programs, explicitly typed programs, and programs that use explicit types for all firstclass ..."
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We present the first type reconstruction system which combines the implicit typing of ML with the full power of the explicitly typed secondorder polymorphic lambda calculus. The system will accept MLstyle programs, explicitly typed programs, and programs that use explicit types for all firstclass polymorphic values. We accomplish this flexibility by providing both generic and explicitlyquantified polymorphic types, as well as operators which convert between these two forms of polymorphism. This type reconstruction system is an integral part of the FX89 programming language. We present a type reconstruction algorithm for the system. The type reconstruction algorithm is proven sound and complete with respect to the formal typing rules.
Raising ML to the Power of System F
 In ICFP ’03: Proceedings of the eighth ACM SIGPLAN international conference on Functional programming
, 2003
"... We propose a type system MLF that generalizes ML with firstclass polymorphism as in System F. We perform partial type reconstruction. As in ML and in opposition to System F, each typable expression admits a principal type, which can be inferred. Furthermore, all expressions of ML are welltyped, wi ..."
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We propose a type system MLF that generalizes ML with firstclass polymorphism as in System F. We perform partial type reconstruction. As in ML and in opposition to System F, each typable expression admits a principal type, which can be inferred. Furthermore, all expressions of ML are welltyped, with a possibly more general type than in ML, without any need for type annotation. Only arguments of functions that are used polymorphically must be annotated, which allows to type all expressions of System F as well.
Towards a Practical Programming Language Based on the Polymorphic Lambda Calculus
, 1989
"... The value of polymorphism in programming languages has been demonstrated by languages such as ML [19]. Recent e cient implementations of ML have shown that a language with implicit polymorphism can be practical [1]. The core of ML's type system is limited, however, by the fact that only instances of ..."
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The value of polymorphism in programming languages has been demonstrated by languages such as ML [19]. Recent e cient implementations of ML have shown that a language with implicit polymorphism can be practical [1]. The core of ML's type system is limited, however, by the fact that only instances of polymorphic functions may be passed as arguments to other functions, but
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"... The quest for type inference with firstclasspolymorphic types Programming languages considerably benefit from static typechecking. In practice however, types may sometimes trammel programmers, for two opposite reasons. On the one hand, type annotations may quickly become a burden to write; while ..."
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The quest for type inference with firstclasspolymorphic types Programming languages considerably benefit from static typechecking. In practice however, types may sometimes trammel programmers, for two opposite reasons. On the one hand, type annotations may quickly become a burden to write; while they usefully serve as documentation for toplevel functions, they also obfuscatethe code when every local function must be decorated. On the other hand, since types are only approximations, any type system willreject programs that are perfectly wellbehaved and that could be accepted by another more expressive one; hence, sharp programmers may be irritated in such situations.
TypeReconstruction wit FirstClass Polymorphic Values
"... We present the first type reconstruction system which combines the implicit typing of ML with the full power of the explicitly typed secondorder polymorphic lambda calculus. The system will accept MLstyle programs, explicitly typed programs, and programs that use explicit types for all firstclass ..."
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We present the first type reconstruction system which combines the implicit typing of ML with the full power of the explicitly typed secondorder polymorphic lambda calculus. The system will accept MLstyle programs, explicitly typed programs, and programs that use explicit types for all firstclass polymorphic values. We accomplish this exibility by providing both generic and explicitlyquanti ed polymorphic types, as well as operators which convert between these two forms of polymorphism. This type reconstruction system is an integral part of the FX89 programming language. We present atype reconstruction algorithm for the system. The type reconstruction algorithm is proven sound and complete with respect to the formal typing rules.
EXISTENTIAL TYPE SYSTEMS BETWEEN CHURCH AND CURRY STYLE
"... Abstract. We study type checking, typability, and type inference problems for typefree style and Curry style secondorder existential systems where the typefree style differs from the Curry style in that the terms of the former contain information on where the existential quantifier elimination an ..."
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Abstract. We study type checking, typability, and type inference problems for typefree style and Curry style secondorder existential systems where the typefree style differs from the Curry style in that the terms of the former contain information on where the existential quantifier elimination and introduction take place but omit the information on which types are involved. We show that all the problems are undecidable employing reduction of secondorder unification in case of the typefree system and semiunification in case of the Curry style system. This provides a fine border between problems yielding to a reduction of secondorder unification problem and the semiunification problem. In addition, we investigate the subject reduction property of the system in the Currystyle. 5 10
THE UNDECIDABILITY OF TYPE RELATED PROBLEMS IN TYPEFREE STYLE SYSTEM F
"... Abstract. We consider here a number of variations on the System F, that are predicative secondorder systems whose terms are intermediate between the Curry style and Church style. The terms here contain the information on where the universal quantifier elimination and introduction in the type infere ..."
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Abstract. We consider here a number of variations on the System F, that are predicative secondorder systems whose terms are intermediate between the Curry style and Church style. The terms here contain the information on where the universal quantifier elimination and introduction in the type inference process must take place, which is similar to Church forms. However, they omit the information on which types are involved in the rules, which is similar to Curry forms. In this paper we prove the undecidability of the typechecking, type inference and typability problems for the system. Moreover, the proof works for the predicative version of the system with finitely stratified polymorphic types. The result includes the bounds on the Leivant’s level numbers for types used in the instances leading to the undecidability. 1.