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Type systems
 The Computer Science and Engineering Handbook
, 1997
"... This paper presents an overview of the programming language Modula3, and a more detailed description of its type system. 1 ..."
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Cited by 202 (1 self)
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This paper presents an overview of the programming language Modula3, and a more detailed description of its type system. 1
Formal Parametric Polymorphism
 THEORETICAL COMPUTER SCIENCE
, 1993
"... A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this appro ..."
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Cited by 124 (6 self)
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A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds's work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this approach, called system R . Girard's system F deals with terms and types; R is an extension of F that deals also with relations between types. In R , it is possible to derive theorems about functions from their types, or "theorems for free", as Wadler calls them. An easy "theorem for free" asserts that the type "(X)XBool contains only constant functions; this is not provable in F. There are many harder and more substantial examples. Various metatheorems can also be obtained, such as a syntactic version of Reynolds's abstraction theorem.
A theory of primitive objects: secondorder systems
 Proc. ESOP’94  European Symposium on Programming
"... We describe a secondorder calculus of objects. The calculus supports object subsumption, method override, and the type Self. It is constructed as an extension of System F with subtyping, recursion, and firstorder object types. 1. ..."
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Cited by 54 (8 self)
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We describe a secondorder calculus of objects. The calculus supports object subsumption, method override, and the type Self. It is constructed as an extension of System F with subtyping, recursion, and firstorder object types. 1.
Inductively Defined Types in the Calculus of Constructions
 IN: PROCEEDINGS OF THE FIFTH CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS. SPRINGER VERLAG LNCS
, 1989
"... We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm ..."
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Cited by 43 (2 self)
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We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm & Berarducci on synthesis of functions on term algebras in the secondorder polymorphiccalculus (F2). We give several applications of this generalization, including a representation of F2programs in F3, along with a definition of functions reify, reflect, and eval for F2 in F3. We also show how to define induction over inductively defined types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, F!. This is because a proof by induction can be realized by primitive recursion, which is already de nable in F!.
Universes for Generic Programs and Proofs in Dependent Type Theory
 Nordic Journal of Computing
, 2003
"... We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, el ..."
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Cited by 42 (2 self)
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We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductiverecursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.
Extensible records in a pure calculus of subtyping
 In Theoretical Aspects of ObjectOriented Programming
, 1994
"... Extensible records were introduced by Mitchell Wand while studying type inference in a polymorphic λcalculus with record types. This paper describes a calculus with extensible records, F <:ρ, that can be translated into a simpler calculus, F <: , lacking any record primitives. Given independent axi ..."
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Cited by 38 (9 self)
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Extensible records were introduced by Mitchell Wand while studying type inference in a polymorphic λcalculus with record types. This paper describes a calculus with extensible records, F <:ρ, that can be translated into a simpler calculus, F <: , lacking any record primitives. Given independent axiomatizations of F <:ρ and F <: (the former being an extension of the latter) we show that the translation preserves typing, subtyping, and equality. F <:ρ can then be used as an expressive calculus of extensible records, either directly or to give meaning to yet other languages. We show that F <:ρ can express many of the standard benchmark examples that appear in the literature. Like other record calculi that have been proposed, F <:ρ has a rather complex set of rules but, unlike those other calculi, its rules are justified by a translation to a very simple calculus. We argue that thinking in terms of translations may help in simplifying and organizing the various record calculi that have been proposed, as well as in generating new ones.
Efficient Interpretation by Transforming Data Types and Patterns to Functions
 In Trends in Functional Programming, volume 7. Intellect
, 2007
"... In this paper we present the stepwise construction of an efficient interpreter for lazy functional programming languages like Haskell and Clean. The interpreter is realized by first transforming the source language to the intermediate language SAPL (Simple Application Programming Language) consistin ..."
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Cited by 11 (4 self)
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In this paper we present the stepwise construction of an efficient interpreter for lazy functional programming languages like Haskell and Clean. The interpreter is realized by first transforming the source language to the intermediate language SAPL (Simple Application Programming Language) consisting of pure functions only. During this transformation algebraic data types and patternbased function definitions are mapped to functions. This eliminates the need for constructs for Algebraic Data Types and Pattern Matching in SAPL. For SAPL a simple and elegant interpreter is constructed using straightforward graph reduction techniques. This interpreter can be considered as a prototype implementation of lazy functional programming languages. Using abstract interpretation techniques the interpreter is optimised. The performance of the resulting interpreter turns out to be very competitive in a comparison with other interpreters like Hugs, Helium, GHCi and Amanda for a number benchmarks. For some benchmarks the interpreter even rivals the speed of the GHC compiler. Due to its simplicity and the stepwise construction this implementation is an ideal subject for introduction courses on implementation aspects of lazy functional programming languages. 1
The GirardReynolds isomorphism
 Proc. of 4th Int. Symp. on Theoretical Aspects of Computer Science, TACS 2001
, 2001
"... Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented ..."
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Cited by 8 (1 self)
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Abstract. The secondorder polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in secondorder intuitionistic propositional logic, P2, can be represented in F2. Reynolds additionally proved an abstraction theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. The Girard projection discards all firstorder quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embedding should also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions corresponding to inductive definitions of naturals, products, sums, and fixpoint types. 1