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11
A Uniform Approach to Domain Theory in Realizability Models
 Mathematical Structures in Computer Science
, 1996
"... this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability&quo ..."
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Cited by 20 (6 self)
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this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domaintheoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesianclosure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...
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 definiti ..."
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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...
Algebras for the partial map classifier monad
 Lecture Notes in Mathematics
, 1991
"... Dedicated to the always helpful Max Kelly ..."
Equational lifting monads
 Proceedings CTCS '99, Electronic Notes in Computer Science
, 1999
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of ..."
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Cited by 3 (2 self)
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of the Kleisli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1
Commutation Structures
, 2005
"... structure on an object A is just a map X⊗A → A⊗X. We study aspects of such structures in case A has a dual object. We consider a monoidal category V, ⊗, I; for simplicity we let it be strict (the application we have in mind is anyway a category of endofuncors on a ..."
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structure on an object A is just a map X⊗A → A⊗X. We study aspects of such structures in case A has a dual object. We consider a monoidal category V, ⊗, I; for simplicity we let it be strict (the application we have in mind is anyway a category of endofuncors on a
1 Physics, Topology, Logic and Computation:
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning
The connected Vietoris powerlocale
, 2008
"... The Vietoris powerlocale V X is a pointfree analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V c X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the cate ..."
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The Vietoris powerlocale V X is a pointfree analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V c X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the category of locales. The strength gives rise to a product map × : V c X × V c Y → V c (X × Y), showing that the product of two of these connected sublocales is again connected. If X is locally connected then V c X is overt. In the case where X is the localic completion Y of a generalized metric space Y, the points of V c Y are characterized as certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric. The results are applied to the particular case of the pointfree real line R, giving a choicefree constructive version of the Intermediate Value Theorem and Rolle’s Theorem. The work is constructive in the sense of toposvalidity with natural numbers object. Its geometric aspects (preserved under inverse image functors) are stressed, and exploited to give a pointwise development of the pointfree locale theory. The connected Vietoris powerlocale itself is a geometric construction. 1
A Presentation Of The Initial LiftAlgebra
 Journal of Pure and Applied Algebra
, 1997
"... The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the success ..."
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The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.
From algebras to objects: Generation and composition
 Journal of Universal Computer Science
, 2005
"... Abstract: This paper addresses objectification, a formal specification technique which inspects the potential for objectorientation of a declarative model and brings the ’implicit objects ’ explicit. Criteria for such objectification are formalized and implemented in a runnable prototype tool which ..."
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Abstract: This paper addresses objectification, a formal specification technique which inspects the potential for objectorientation of a declarative model and brings the ’implicit objects ’ explicit. Criteria for such objectification are formalized and implemented in a runnable prototype tool which embeds Vdmsl into Vdm++. The paper also includes a quick presentation of a (coinductive) calculus of such generated objects, framed as generalised Moore machines.
Generic Process Algebra: A Programming Challenge
"... Abstract: Emerging interaction paradigms, such as serviceoriented computing, and new technological challenges, such as exogenous component coordination, suggest new roles and application areas for process algebras. This, however, entails the need for more generic and adaptable approaches to their d ..."
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Abstract: Emerging interaction paradigms, such as serviceoriented computing, and new technological challenges, such as exogenous component coordination, suggest new roles and application areas for process algebras. This, however, entails the need for more generic and adaptable approaches to their design. For example, some applications may require similar programming constructs coexisting with different interaction disciplines. In such a context, this paper pursues a research programme on a coinductive rephrasal of classic process algebra, proposing a clear separation between structural aspects and interaction disciplines. A particular emphasis is put on the study of interruption combinators defined by natural corecursion. The paper also illustrates the verification of their properties in an equational and pointfree reasoning style as well as their direct encoding in Haskell.