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30
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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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
Programming with inductive and coinductive types
, 1992
"... Abstract We look at programming with inductive and coinductive datatypes, which are inspired theoretically by initial algebras and final coalgebras, respectively. A predicative calculus which incorporates these datatypes as primitive constructs is presented. This calculus allows reduction sequence ..."
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Abstract We look at programming with inductive and coinductive datatypes, which are inspired theoretically by initial algebras and final coalgebras, respectively. A predicative calculus which incorporates these datatypes as primitive constructs is presented. This calculus allows reduction sequences which are significantly more efficient for two dual classes of common programs than do previous calculi using similar primitives. Several techniques for programming in this calculus are illustrated with numerous examples. A short survey of related work is also included.
A Cube of Proof Systems for the Intuitionistic Predicate mu,nuLogic
 Dept. of Informatics, Univ. of Oslo
, 1997
"... This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight pr ..."
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Cited by 6 (5 self)
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This paper is an attempt at a systematizing study of the proof theory of the intuitionistic predicate ¯; logic (conventional intuitionistic predicate logic extended with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers). We identify eight prooftheoretically interesting naturaldeduction calculi for this logic and propose a classification of these into a cube on the basis of the embeddibility relationships between these. 1 Introduction ¯,logics, i.e. logics with logical constants ¯ and for the least and greatest fixpoint operators on positive predicate transformers, have turned out to be a useful formalism in a number of computer science areas. The classical 1storder predicate ¯,logic can been used as a logic of (nondeterministic) imperative programs and as a database query language. It is also one of the relation description languages studied in descriptive complexity theory (finite model theory) (for a survey on this hi...
Realizability and parametricity in pure type systems
 In the Proceedings of FoSSaCS 2011 (Saarbruecken
, 2011
"... Abstract. We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is express ..."
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Abstract. We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is expressive enough to internalize both theories. Thanks to the PTS setting, we abstract most idiosyncrasies specific to particular type theories. This confers generality to the results, and reveals parallels between parametricity and realizability. 1
Monad Translating Inductive and Coinductive Types
 In Proc. Partial Evaluation and SemanticsBased Program Manipulation
, 2003
"... We show that the callbyname monad translation of simply typed lambda calculus extended with sum and product types extends to special and general inductive and coinductive types so that its crucial property of preserving typings and  and commuting reductions is maintained. Speci c similarpu ..."
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We show that the callbyname monad translation of simply typed lambda calculus extended with sum and product types extends to special and general inductive and coinductive types so that its crucial property of preserving typings and  and commuting reductions is maintained. Speci c similarpurpose translations such as CPS translations follow from the general monad translations by specialization for appropriate concrete monads.
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. 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. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.
Coding Recursion a la Mendler (Extended Abstract)
 Department of Computer Science, Utrecht University
, 2000
"... Abstract We advocate the Mendler style of coding terminating recursion schemes as combinators by showing on the example of two simple and much used schemes (courseofvalue iteration and simultaneous iteration) that choosing the Mendler style can sometimes lead to handier constructions than followin ..."
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Cited by 4 (1 self)
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Abstract We advocate the Mendler style of coding terminating recursion schemes as combinators by showing on the example of two simple and much used schemes (courseofvalue iteration and simultaneous iteration) that choosing the Mendler style can sometimes lead to handier constructions than following the construction style of cata and para like combinators. 1 Introduction This paper is intended as an advert for something we call the Mendler style. This is a not too widely known style of coding terminating recursion schemes by combinators that di ers from the construction style of the famous cata and para combinators (for iteration and primitiverecursion, respectively) [Mal90,Mee92], here called the conventional style. The paper ar...
Monotone FixedPoint Types and Strong Normalization
 In Proceedings of CSL 1998, Lecture Notes in Computer Science. Submitted
, 1998
"... Several systems of fixedpoint types (also called retract types or recursive types with explicit isomorphisms) extending system F are considered. The seemingly strongest systems have monotonicity witnesses and use them in the definition of beta reduction for those types. A more naive approach lea ..."
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Several systems of fixedpoint types (also called retract types or recursive types with explicit isomorphisms) extending system F are considered. The seemingly strongest systems have monotonicity witnesses and use them in the definition of beta reduction for those types. A more naive approach leads to nonnormalizing terms. All the other systems are strongly normalizing because they embed in a reductionpreserving way into the system of noninterleaved positive fixedpoint types which can be shown to be strongly normalizing by an easy extension of some proof of strong normalization for system F. Due to the presence of F's impredicativity it is also possible to embed monotone inductive types (with full primitive recursion) into the system of noninterleaved positive fixedpoint types. In the author's view this gives the easiest way to proving strong normalization for systems of inductive types. 1 Definition of the systems of monotone fixedpoint types We consider extensions o...
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