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29
Relational Properties of Domains
 Information and Computation
, 1996
"... New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property o ..."
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Cited by 99 (5 self)
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New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with userdeclared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the coinduct...
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 15 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
A Categorical Model for Higher Order Imperative Programming
 Mathematical Structures in Computer Science
, 1993
"... This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers. ..."
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Cited by 14 (13 self)
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This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.
Container Types Categorically
, 2000
"... A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a ..."
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Cited by 12 (0 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...
Two Models of Synthetic Domain Theory
, 1997
"... This paper is concerned with models of SDT encompassing traditional categories of domains used in denotational semantics [7,18], showing that the synthetic approach generalises the standard theory of domains and suggests new problems to it. Consider a (locally small) category of domains D with a (sm ..."
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Cited by 11 (3 self)
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This paper is concerned with models of SDT encompassing traditional categories of domains used in denotational semantics [7,18], showing that the synthetic approach generalises the standard theory of domains and suggests new problems to it. Consider a (locally small) category of domains D with a (small) dense generator G equipped with a Grothendieck topology. Assume further that every cover in G is effective epimorphic in D. Then, by Yoneda, D embeds fully and faithfully in the topos of sheaves on G for the canonical topology, which thus provides a settheoretic universe for our original category of domains. In this paper we explore such a situation for two traditional categories of domains and, in particular, show that the Grothendieck toposes so arising yield models of SDT. In a subsequent paper we will investigate intrinsic characterizations, within our models, of these categories of domains. First, we present a model of SDT embedding the category !Cpo of posets with least upper bounds of countable chains (hence called !complete) and
Mathematics of generic specifications for model management
 Encyclopedia of Database Technologies and Applications
, 2005
"... This article (further referred to as MathI), and the next one (further referred to as MathII, see p. 359), form a mathematical companion to the article in this encyclopedia on Generic Model Management (further referred to as GenMMt, see p.258). Articles MathI and II present the basics of the arro ..."
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Cited by 9 (7 self)
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This article (further referred to as MathI), and the next one (further referred to as MathII, see p. 359), form a mathematical companion to the article in this encyclopedia on Generic Model Management (further referred to as GenMMt, see p.258). Articles MathI and II present the basics of the arrow diagram machinery that provides model management with truly generic specifications. Particularly, it allows us to build a generic pattern for heterogeneous data and schema transformation, which is presented in MathII for the first time in the literature.
Prototyping Relational Specifications Using HigherOrder Objects
, 1994
"... An approach is described for the generation of certain mathematical objects (like sets, correspondences, mappings) in terms of relations using relationalgebraic descriptions of higherorder objects. From nonconstructivecharacterizations executable relational specifications are obtained. We als ..."
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Cited by 7 (4 self)
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An approach is described for the generation of certain mathematical objects (like sets, correspondences, mappings) in terms of relations using relationalgebraic descriptions of higherorder objects. From nonconstructivecharacterizations executable relational specifications are obtained. We also showhowtodevelop more efficient algorithms from the frequently inefficient specifications within the calculus of binary relations.