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A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
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 Probabilistic Powerdomains in Topological Domain Theory
"... We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given b ..."
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We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ωcontinuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domaintheoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countablybased spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.
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.