26 research outputs found
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
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
Recommended from our members
Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Computability on the Countable Ordinals and the Hausdorff-Kuratowski Theorem (Extended Abstract)
While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of countable ordinals via a representation in the sense of computable analysis. The computability structure is characterized by the computability of four specific operations, and we prove further relevant operations to be computable. Some alternative approaches are discussed, too. As an application in effective descriptive set theory, we can then state and prove computable uniform versions of the Lusin separation theorem and the Hausdorff-Kuratowski theorem. Furthermore, we introduce an operator on the Weihrauch lattice corresponding to iteration of some principle over a countable ordinal
Extending the Reach of the Point-To-Set Principle
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled
the theory of computing to be used to answer open questions about fractal
geometry in Euclidean spaces . These are classical questions,
meaning that their statements do not involve computation or related aspects of
logic.
In this paper we extend the reach of the point-to-set principle from
Euclidean spaces to arbitrary separable metric spaces . We first extend two
fractal dimensions--computability-theoretic versions of classical Hausdorff and
packing dimensions that assign dimensions and to
individual points --to arbitrary separable metric spaces and to
arbitrary gauge families. Our first two main results then extend the
point-to-set principle to arbitrary separable metric spaces and to a large
class of gauge families.
We demonstrate the power of our extended point-to-set principle by using it
to prove new theorems about classical fractal dimensions in hyperspaces. (For a
concrete computational example, the stages used to
construct a self-similar fractal in the plane are elements of the
hyperspace of the plane, and they converge to in the hyperspace.) Our third
main result, proven via our extended point-to-set principle, states that, under
a wide variety of gauge families, the classical packing dimension agrees with
the classical upper Minkowski dimension on all hyperspaces of compact sets. We
use this theorem to give, for all sets that are analytic, i.e.,
, a tight bound on the packing dimension of the hyperspace
of in terms of the packing dimension of itself