Matthias Schröder

A Convenient Category of Domains

We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual @w-continuous 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.

Spaces allowing Type-2 Complexity Theory revisited

The basic concept of Type-2 Theory of Effectivity (TTE) to define computability on topological spaces (X, τ ) or limit spaces (X,) are representations, i. e. surjection functions from the Baire space onto X. Representations having the topological property of admissibility are known to provide a reasonable computability theory. In this article, we investigate several additional properties of representations which guarantee that such representations induce a reasonable Type-2 Complexity Theory on the represented spaces. For each of these properties, we give a nice characterization of the class of spaces that are equipped with a representation having the respective property. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Compactly generated domain theory

We propose compactly generated monotone convergence spaces as a well-behaved topological generalisation of directed-complete partial orders (dcpos). The category of such spaces enjoys the usual properties of categories of ‘predomains’ in denotational semantics. Moreover, such properties are retained if one restricts to spaces with a countable pseudobase in the sense of E. Michael, a fact that permits connections to be made with computability theory, realizability semantics and recent work on the closure properties of topological quotients of countably based spaces (qcb spaces). We compare the standard domain-theoretic constructions of products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases, though not in general.

Admissible Representations of Probability Measures

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type-2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. This canonical representation turns out to be admissible with respect to the weak topology on M(X). Moreover, we prove that for countably based topological spaces X the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology on M(X).

Two preservation results for countable products of sequential spaces

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.

Some hierarchies of QCB 0 -spaces

We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain P\omega, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.

Computing with sequences, weak topologies and the axiom of choice

We study computability on sequence spaces, as they are used in functional analysis. It is known that non-separable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice non-separable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology is one which makes all bounded linear functionals continuous). Surprisingly, it turns out that when one replaces the Axiom of Choice by the Axiom of Dependent Choice and the Baire Property, then some non-separable normed spaces can be represented admissibly on Turing machines with respect to the weak topology (which is just the weakest compatible topology). Thus the ability to adequately handle sequence spaces on Turing machines sensitively relies on the underlying axiomatic setting.

Admissible representations in computable analysis

Computable Analysis investigates computability on real numbers and related spaces. One approach to Computable Analysis is Type Two Theory of Effectivity (TTE). TTE provides a computational framework for non-discrete spaces with cardinality of the continuum. Its basic tool are representations. A representation equips the objects of a given space with “names”, which are infinite words. Computations are performed on these names. We discuss the property of admissibility as a well-behavedness criterion for representations. Moreover we investigate and characterise the class of spaces which have such an admissible representation. This category turns out to have a remarkably rich structure.

Observationally-induced Effect Monads: Upper and Lower Powerspace Constructions

Alex Simpson has suggested to use an observationally-induced approach towards modelling computational effects in denotational semantics. The principal idea is that a single observation algebra is used for defining the computational type structure. He advocates that besides giving algebraic structure this approach also allows the characterisation of the monadic types concretely. We show that free observationally-induced algebras exist in the category of continuous maps between topological spaces for arbitrary pre-chosen observation algebras. Moreover, we use this approach to give a lower and an upper powerdomain construction on general topological spaces, both of which generalise the classical characterisations on continuous dcpos. Our lower powerdomain construction is for all topological spaces given by the space of non-empty closed subsets with the lower Vietoris topology. Dually, our upper powerdomain construction is for a wide class of topological spaces given by the space of proper open filters of its topology with the upper Vietoris topology. We also give a counterexample showing that this characterisation does not hold for all topological spaces.

Some Examples of Non-Metrizable Spaces Allowing a Simple Type-2 Complexity Theory

Representations of spaces are the key device in Type-2 Theory of Effectivity (TTE) for defining computability on non-countable spaces. Almost-compact representations permit a simple measurement of the time complexity of functions using discrete parameters, namely the desired output precision together with “size” information about the argument, rather than continuous ones.We present some interesting examples of non-metrizable topological vector spaces that have almost-compact admissible representations, including spaces of real polynomial functions and of distributions with compact support.