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A Characterization Of Lambda Definability In Categorical Models Of Implicit Polymorphism
 Theoretical Computer Science
, 1995
"... . Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A categorytheoretic framework known as glueing or sconing is used to extend the JungTiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categor ..."
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. Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A categorytheoretic framework known as glueing or sconing is used to extend the JungTiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a wellestablished tool for studying the semantics of various typed lambda calculi. The main lines of research are focused in two areas, the first of which strives for an understanding of Strachey's notion of parametric polymorphism. The main idea is that a parametricly polymorphic function acts independently from the types to which its type variables are instantiated, and that this uniformity may be captured by imposing a relational structure on the types [OHT93, MSd93, MaR91, Wad89, Rey83, Str67]. The other line of research concerns lambda definability and the full abstraction problem for various models of languag...
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
On the Role of Category Theory in the Area of Algebraic Specifications
 In LNCS , Proc. WADT11
, 1996
"... . The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing pa ..."
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. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...
A Model for Formal Parametric Polymorphism: A PER Interpretation for System R
, 1995
"... System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In system R, considerably more lambdaterms can be proved equal than in system F: for example, the encoded weak products of F are strong products in R. Also, many "theorems for free" à la Wadler can be ..."
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System R is an extension of system F that formalizes Reynolds' notion of relational parametricity. In system R, considerably more lambdaterms can be proved equal than in system F: for example, the encoded weak products of F are strong products in R. Also, many "theorems for free" à la Wadler can be proved formally in R. In this paper we describe a semantics for system R. As a first step, we give a precise and general reconstruction of the per model of system F of Bainbridge et al., presenting it as a categorical model in the sense of Seely. Then we interpret system R in this model.
Relational Databases and Indexed Categories
 In Proceedings of the International Category Theory Meeting 1991, CMS Conference Proceedings 13, 391407, American Mathematical Society
, 1992
"... . A description of relational databases in categorical terminology given here has as intended application the study of database dynamics, in particular we view (i) updates as database objects in a suitable category indexed by a topos; (ii) Lfuzzy databases as database objects in sheaves. Indexed ca ..."
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. A description of relational databases in categorical terminology given here has as intended application the study of database dynamics, in particular we view (i) updates as database objects in a suitable category indexed by a topos; (ii) Lfuzzy databases as database objects in sheaves. Indexed categories are constructed to model the databases on a fixed family of domains and also all databases for a varying family of domains. Further, we show that the process of constructing the relational completion of a relational database is a monad in a 2category of functors. Introduction We use the term relation for a subobject of a finite product of objects in a category. Following the relational database literature, we use the term domain for an object of the ambient category (and warn readers that these are not the ordered objects which go by the name "domain" elsewhere in theoretical Computer Science.) A relational database, as defined by E. F. Codd [3], is first of all a family of rela...
Typed Applicative Structures and Normalization by Evaluation for System F ω
"... Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the alg ..."
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Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.
An Introduction to Polymorphic Lambda Calculus
 Logical Foundations of Functional Programming
, 1994
"... Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that ..."
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Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that essentially the same programming language was formulated independently by the two of us, especially since we were led to the language by entirely different motivations. In my own case, I was seeking to extend conventional typed programming languages to permit the definition of "polymorphic" procedures that could accept arguments of a variety of types. I started with the ordinary typed lambda calculus and added the ability to pass types as parameters (an idea that was "in the air" at the time, e.g. [4]). For example, as in the ordinary typed lambda calculus one can write f int!int : x int : f(f (x)) to denote the "doubling" function for the type int, which accepts a function from integers
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ii COPYRIGHT ..."
Polymorphic Linear Logic and Topos Models
 Math. Reports, Academy of Science (Canada) XII
, 1990
"... We give a definition of a "linear fibration", which is a hyperdoctrine model of polymorphic linear logic, and show how to internalise the fibration, generating topos models. This gives a constructive set theoretical context for the logic of Petri nets, as recently developed by N. Mart'iOliet and J. ..."
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We give a definition of a "linear fibration", which is a hyperdoctrine model of polymorphic linear logic, and show how to internalise the fibration, generating topos models. This gives a constructive set theoretical context for the logic of Petri nets, as recently developed by N. Mart'iOliet and J. Meseguer. Also, we sketch how this can be further extended to include the exponential operator ! . In this context, the topos model we construct can be embedded in the model constructed by A.M. Pitts. 0 Introduction In [4], it is shown how to enrich the logic of Petri nets with gedanken states and processes, by embedding it into linear logic. Recently, Mart'iOliet and Meseguer have asked how to extend this even further to include polymorphism. It turns out that the process is fairly straightforward, and for maximal impact (so as to include constructive set theory), can be internalised to give topos models of polymorphic linear logic. Here we give the necessary definitions, and sketch the ...
On PlotkinAbadi Logic for Parametric Polymorphism  Towards a Categorical Understanding
"... . The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] ..."
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. The idea of parametric polymorphism is that of a single operator that can be used for different data types and whose behaviour is somehow uniform for each type. Reynolds [Reynolds, 1983] uses binary relations to define a uniformity condition for parametric polymorphism. In [Plotkin & Abadi, 1993] the authors proposed a second order logic for second order lambdacalculus; this logic is able to handle parametric polymorphism in the binary relational sense of Reynolds. In this paper we examine a categorical framework for this logic. This framework is based on the notion of categorical model of second order lambdacalculus as given, for example, in [Pitts, 1987, Seely, 1987, Robinson, 1992, Jacobs, 1991]. Going through the categorical constructions of the model, an unexpected property of quantification over type variables appears. A simple categorical calculation indicates what is the appropriate way to obtain the right adjoint to weakening that models universal quantification. The resul...