Michael B. Smyth

Quasi Uniformities: Reconciling Domains with Metric Spaces

We show that quasi-metric or quasi-uniform spaces provide, inter alia, a common generalization of cpos and metric spaces as used in denotational semantics. To accommodate the examples suggested by computer science, a reworking of basic notions involving limits and completeness is found to be necessary. Specific results include general fixed point theorem and a sequential completion construction.

Semi-metrics, closure spaces and digital topology

Abstract We investigate certain generalized topological structures, with the aim of showing that they can provide a suitable framework within which to compare various approaches to digital topology (including tolerance geometry) with “ordinary” topology. Within an appropriate category of these structures, ordinary spaces arise as inverse limits of digital spaces. In the first instance, the structures can be taken to be (a slight generalization of) Cech closure spaces. The structures of this type which seem to be the most useful are those which can be realized as topological graphs (see Section 3). Domain equations for the (real) unit interval provide our main detailed application (Section 4). Another application area which seems promising is that of modal semantics.

I-categories as a framework for solving domain equations

Abstract An abstract notion of category of information systems or I-category is introduced as a generalisation of Scotts well-known category of information systems. As in the theory of partial orders, I-categories can be complete or ω-algebraic, and it is shown that ω-algebraic I-categories can be obtained from a certain completion of countable I-categories. The proposed axioms for a complete I-category introduce a global partial order on the morphisms of the category, making them a cpo. An initial algebra theorem for a class of functors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations; an effective version of these results for ω-algebraic I-categories is also provided. Some basic examples of I-categories representing the categories of sets, Boolean algebras, Scott domains and continuous Scott domains are constructed.

Topology and tolerance

Abstract How may the structures employed in ordinary analysis, such as the reals, be accommodated in the digital universe? Tolerance space theory, which was introduced by Poincare and (independently) Zeeman, as a kind of alternative topology, is a promising candidate. Here we argue that the way forward lies in viewing topology and tolerance as complementary rather than as alternative(s).

Categories of Information Systems

An abstract notion of “information category” (I-category) is introduced as a generalization of Scotts well-known category of information systems. The proposed axioms introduce a global partial order on the morphisms of the category, making them an ω-algebraic cpo. An initial algebra theorem for a class of endofunctors continuous on the cpo of morphisms is proved, thus giving canonical solution of domain equations. An effective version of these results, in the general setting, is also provided. Some basic examples of categories of information systems are dealt with.

AFPP vs FPP

The main theme of this paper is that almost fixed point properties of discrete structures and fixed point properties of (topological) spaces are interdeducible via a suitable category which contains both graphs and spaces as objects. To carry out the program, we have to consider (almost) fixed points of multifunctions, and for this we need a preliminary discussion of power structures for graphs and simplicial complexes. Specific applications developed are: a “digital convexity” (discrete) version of Kakutanis fixed point theorem for convex-valued multifunctions; and fixed point properties of dendrites in terms of those of finite discrete trees.

Compact metric information systems

We present information systems for compact metric spaces using the notions of diameter and strong inclusion of open sets. It is shown that the category of compact metric information systems and metric approximable mappings, dual to the category of compact metric spaces and non-expansive maps, is a partially complete I-category in which canonical solution of domain equations can be found by taking the union (least upper bound) of certain Cauchy chains. For the class of contracting functors, the domain equation has a unique solution. We present such a class which includes the product, the co-product and the hyperspace functor (with Hausdorff metric).

Fixed Points in Digital Topology (via Helly Posets)

Abstract We extend some of our previous results on fixed points of graph multifunctions to posets. The posets of most interest here are the (finite) Khalimsky spaces, in their specialization order. Retracts of Khalimsky spaces coincide with Helly posets. Notions of convexity can be defined in these spaces, providing the basis for certain “geometric” fixed point theorems.

The Constructive Maximal Point Space

Abstract We argue that constructive maximality (Martin-Lof [14]) can with advantage be employed in the study of maximal point spaces, and related questions in quantitative domain theory.

Matroids from Modules

Abstract The aim of this work is to show that (oriented) matroid methods can be applied to many discrete geometries, namely those based on modules over integral (ordered) domains. The trick is to emulate the structure of a vector space within the module, thereby allowing matroid methods to be used as if the module were a vector space. Only those submodules which are “closed under existing divisors”, and hence behave like vector subspaces, are used as subspaces of the matroid. It is also shown that Hubler‘s axiomatic discrete geometry can be characterised in terms of modules over the ring of integers.