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16
Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
The Rho Cube
 In Proc. of FOSSACS, volume 2030 of LNCS
, 2001
"... www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. The rewriting calculus, or Rho Calculus (ρCal), is a simple calculus that uniformly integrates abstraction on patterns and nondeterminism. Therefore, it fully integrates rewriting and λcalculus. The original presentation of the calculus was unty ..."
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Cited by 32 (16 self)
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www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. The rewriting calculus, or Rho Calculus (ρCal), is a simple calculus that uniformly integrates abstraction on patterns and nondeterminism. Therefore, it fully integrates rewriting and λcalculus. The original presentation of the calculus was untyped. In this paper we present a uniform way to decorate the terms of the calculus with types. This gives raise to a new presentation à la Church, together with nine (8+1) type systems which can be placed in a ρcube that extends the λcube of Barendregt. Due to the matching capabilities of the calculus, the type systems use only one abstraction mechanism and therefore gives an original answer to the identification of the standard “λ ” and “Π” abstractors. As a consequence, this brings matching and rewriting as the first class concepts of the Rhoversions of the Logical Framework (LF) of Harper
Rewriting calculus with(out) types
 Proceedings of the fourth workshop on rewriting logic and applications
, 2002
"... The last few years have seen the development of a new calculus which can be considered as an outcome of the last decade of various researches on (higher order) term rewriting systems, and lambda calculi. In the Rewriting Calculus (or Rho Calculus, ρCal), algebraic rules are considered as sophisticat ..."
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Cited by 22 (13 self)
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The last few years have seen the development of a new calculus which can be considered as an outcome of the last decade of various researches on (higher order) term rewriting systems, and lambda calculi. In the Rewriting Calculus (or Rho Calculus, ρCal), algebraic rules are considered as sophisticated forms of “lambda terms with patterns”, and rule applications as lambda applications with pattern matching facilities. The calculus can be customized to work modulo sophisticated theories, like commutativity, associativity, associativitycommutativity, etc. This allows us to encode complex structures such as list, sets, and more generally objects. The calculus can either be presented “à la Curry ” or “à la Church ” without sacrificing readability and without complicating too much the metatheory. Many static type systems can be easily pluggedin on top of the calculus in the spirit of the rich typeoriented literature. The Rewriting Calculus could represent a lingua franca to encode many paradigms of computations together with a formal basis used to build powerful theorem provers based on lambda calculus and efficient rewriting, and a step towards new proof engines based on the CurryHoward isomorphism. 1
A Combinatory Algebra for Sequential Functionals of Finite Type
 University of Utrecht
, 1997
"... It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic ..."
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Cited by 21 (2 self)
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It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of "dIdomains with coherence", or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined in Plotkin's paper [16]. It is a simply typed calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reduction relation attached to it which ensures (by Plotkin's "Activity Lemma") that all PCF definable highertype functionals have a sequential, i.e. nonparal...
The HOL logic extended with quantification over type variables
 Formal Methods in System Design, 3(12):724
, 1993
"... Abstract. The HOL system is an LCFstyle mechanized proofassistant for conducting proofs in higher order logic. This paper discusses a proposal to extend the primitive basis of the logic underlying the HOL system with a very simple form of quantification over types. It is shown how certain practica ..."
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Cited by 19 (0 self)
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Abstract. The HOL system is an LCFstyle mechanized proofassistant for conducting proofs in higher order logic. This paper discusses a proposal to extend the primitive basis of the logic underlying the HOL system with a very simple form of quantification over types. It is shown how certain practical problems with using the definitional mechanisms of HOL would be solved by the additional expressive power gained by making this extension.
On functors expressible in the polymorphic typed lambda calculus
 Logical Foundations of Functional Programming
, 1990
"... This is a preprint of a paper that has been submitted to Information and Computation. ..."
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Cited by 16 (1 self)
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This is a preprint of a paper that has been submitted to Information and Computation.
Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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Cited by 12 (4 self)
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Linear Logic, Totality and Full Completeness
 In Proceedings of LiCS `94
, 1994
"... I give a `totality space' model for linear logic [4], derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and gametheoretic models [1, 5], but is based upon a notion of total object. Using this model, I prove a full comple ..."
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Cited by 12 (2 self)
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I give a `totality space' model for linear logic [4], derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and gametheoretic models [1, 5], but is based upon a notion of total object. Using this model, I prove a full completeness result, along the lines of the results for game theoretic models in [1] and [5]. In other words, I show that the mapping of proofs to their interpretations (here collections of total objects uniform for a given functor) in the model is a surjection. 1 Introduction We shall give a model of linear logic by formalising a particular view of what an abstract datatype is. Consider a datatype A. There are objects s of type A, and programs t that accept an argument of type A. Taking any such s and t, we may execute the program on the data, and obtain a particular computationthe trace of the execution of the program. We shall consider only this facet of datatypes. For a given data (or pro...
Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations
, 1994
"... Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. ..."
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Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...
Comparing Cubes
"... We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subje ..."
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Cited by 5 (3 self)
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We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the relationship between the two cubes, which leads to some unexpected results in the eld of systems with dependent types.