On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces

Quantitative Coding and Complexity Theory of Compact Metric Spaces

Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say); but concerning continuous data, already real numbers naturally suggest various encodings with very different computational properties. With respect to qualitative computability, Kreitz and Weihrauch (1985) had identified ADMISSIBILITY as crucial property for 'reasonable' encodings over the Cantor space of infinite binary sequences, so-called representations [doi:10.1007/11780342_48]: For (precisely) these does the sometimes so-called MAIN THEOREM apply, characterizing continuity of functions in terms of continuous realizers. We rephrase qualitative admissibility as continuity of both the representation and its multivalued inverse, adopting from [doi:10.4115/jla.2013.5.7] a notion of sequential continuity for multifunctions. This suggests its quantitative refinement as criterion for representations suitable for complexity investigations. Higher-type complexity is captured by replacing Cantor's as ground space with Baire or any other (compact) ULTRAmetric space: a quantitative counterpart to equilogical spaces in computability [doi:10.1016/j.tcs.2003.11.012]

Topological Domain Theory

This thesis presents Topological Domain Theory as a powerful and flexible framework for denotational semantics. Topological Domain Theory models a wide range of type constructions and can interpret many computational features. Furthermore, it has close connections to established frameworks for denotational semantics, as well as to well-studied mathematical theories, such as topology and computable analysis.We begin by describing the categories of Topological Domain Theory, and their categorical structure. In particular, we recover the basic constructions of domain theory, such as products, function spaces, fixed points and recursive types, in the context of Topological Domain Theory.As a central contribution, we give a detailed account of how computational effects can be modelled in Topological Domain Theory. Following recent work of Plotkin and Power, who proposed to construct effect monads via free algebra functors, this is done by showing that free algebras for a large class of parametrised equational theories exist in Topological Domain Theory. These parametrised equational theories are expressive enough to generate most of the standard examples of effect monads. Moreover, the free algebras in Topological Domain Theory are obtained by an explicit inductive construction, using only basic topological and set-theoretical principles.We also give a comparison of Topological and Classical Domain Theory. The category of omega-continuous dcpos embeds into Topological Domain Theory, and we prove that this embedding preserves the basic domain-theoretic constructions in most cases. We show that the classical powerdomain constructions on omega-continuous dcpos, including the probabilistic powerdomain, can be recovered in Topological Domain Theory.Finally, we give a synthetic account of Topological Domain Theory. We show that Topological Domain Theory is a specific model of Synthetic Domain Theory in the realizability topos over Scott's graph model. We give internal characterisations of the categories of Topological Domain Theory in this realizability topos, and prove the corresponding categories to be internally complete and weakly small. This enables us to show that Topological Domain Theory can model the polymorphic lambda-calculus, and to obtain a richer collection of free algebras than those constructed earlier.In summary, this thesis shows that Topological Domain Theory supports a wide range of semantic constructions, including the standard domain-theoretic constructions, computational effects and polymorphism, all within a single setting