174 research outputs found
Computably Based Locally Compact Spaces
ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in
which the topology on a space is treated, not as an infinitary lattice, but as
an exponential object of the same category as the original space, with an
associated lambda-calculus. In this paper, this is shown to be equivalent to a
notion of computable basis for locally compact sober spaces or locales,
involving a family of open subspaces and accompanying family of compact ones.
This generalises Smyth's effectively given domains and Jung's strong proximity
lattices. Part of the data for a basis is the inclusion relation of compact
subspaces within open ones, which is formulated in locale theory as the
way-below relation on a continuous lattice. The finitary properties of this
relation are characterised here, including the Wilker condition for the cover
of a compact space by two open ones. The real line is used as a running
example, being closely related to Scott's domain of intervals. ASD does not use
the category of sets, but the full subcategory of overt discrete objects plays
this role; it is an arithmetic universe (pretopos with lists). In particular,
we use this subcategory to translate computable bases for classical spaces into
objects in the ASD calculus.Comment: 70pp, LaTeX2e, uses diagrams.sty; Accepted for "Logical Methods in
Computer Science" LMCS-2004-19; see http://www.cs.man.ac.uk/~pt/ASD for
related papers. ACM-class: F.4.
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
Closed Choice and a Uniform Low Basis Theorem
We study closed choice principles for different spaces. Given information
about what does not constitute a solution, closed choice determines a solution.
We show that with closed choice one can characterize several models of
hypercomputation in a uniform framework using Weihrauch reducibility. The
classes of functions which are reducible to closed choice of the singleton
space, of the natural numbers, of Cantor space and of Baire space correspond to
the class of computable functions, of functions computable with finitely many
mind changes, of weakly computable functions and of effectively Borel
measurable functions, respectively. We also prove that all these classes
correspond to classes of non-deterministically computable functions with the
respective spaces as advice spaces. Moreover, we prove that closed choice on
Euclidean space can be considered as "locally compact choice" and it is
obtained as product of closed choice on the natural numbers and on Cantor
space. We also prove a Quotient Theorem for compact choice which shows that
single-valued functions can be "divided" by compact choice in a certain sense.
Another result is the Independent Choice Theorem, which provides a uniform
proof that many choice principles are closed under composition. Finally, we
also study the related class of low computable functions, which contains the
class of weakly computable functions as well as the class of functions
computable with finitely many mind changes. As one main result we prove a
uniform version of the Low Basis Theorem that states that closed choice on
Cantor space (and the Euclidean space) is low computable. We close with some
related observations on the Turing jump operation and its initial topology
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems
according to their computational content. Basically, we are asking the question
which theorems can be continuously or computably transferred into each other?
For this purpose theorems are considered via their realizers which are
operations with certain input and output data. The technical tool to express
continuous or computable relations between such operations is Weihrauch
reducibility and the partially ordered degree structure induced by it. We have
identified certain choice principles which are cornerstones among Weihrauch
degrees and it turns out that certain core theorems in analysis can be
classified naturally in this structure. In particular, we study theorems such
as the Intermediate Value Theorem, the Baire Category Theorem, the Banach
Inverse Mapping Theorem and others. We also explore how existing
classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into
this picture. We compare the results of our classification with existing
classifications in constructive and reverse mathematics and we claim that in a
certain sense our classification is finer and sheds some new light on the
computational content of the respective theorems. We develop a number of
separation techniques based on a new parallelization principle, on certain
invariance properties of Weihrauch reducibility, on the Low Basis Theorem of
Jockusch and Soare and based on the Baire Category Theorem. Finally, we present
a number of metatheorems that allow to derive upper bounds for the
classification of the Weihrauch degree of many theorems and we discuss the
Brouwer Fixed Point Theorem as an example
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