237 research outputs found
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
The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma
We classify the computational content of the Bolzano-Weierstrass Theorem and
variants thereof in the Weihrauch lattice. For this purpose we first introduce
the concept of a derivative or jump in this lattice and we show that it has
some properties similar to the Turing jump. Using this concept we prove that
the derivative of closed choice of a computable metric space is the cluster
point problem of that space. By specialization to sequences with a relatively
compact range we obtain a characterization of the Bolzano-Weierstrass Theorem
as the derivative of compact choice. In particular, this shows that the
Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's
Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the
jump of the lesser limited principle of omniscience LLPO and the
Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump
of the idempotent closure of LLPO. We also introduce the compositional product
of two Weihrauch degrees f and g as the supremum of the composition of any two
functions below f and g, respectively. We can express the main result such that
the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's
Lemma and the Monotone Convergence Theorem. We also study the class of weakly
limit computable functions, which are functions that can be obtained by
composition of weakly computable functions with limit computable functions. We
prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this
class. Likewise, the unique cluster point problem on real numbers is complete
for the class of functions that are limit computable with finitely many mind
changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers
and, more generally, the unbounded cluster point problem on real numbers is
uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias
Schr\"oder, and the authors of the original paper. The addendum closes a gap
in the proof of Theorem 11.2, which characterizes the computational content
of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
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
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
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Completion of Choice
We systematically study the completion of choice problems in the Weihrauch
lattice. Choice problems play a pivotal role in Weihrauch complexity. For one,
they can be used as landmarks that characterize important equivalences classes
in the Weihrauch lattice. On the other hand, choice problems also characterize
several natural classes of computable problems, such as finite mind change
computable problems, non-deterministically computable problems, Las Vegas
computable problems and effectively Borel measurable functions. The closure
operator of completion generates the concept of total Weihrauch reducibility,
which is a variant of Weihrauch reducibility with total realizers. Logically
speaking, the completion of a problem is a version of the problem that is
independent of its premise. Hence, studying the completion of choice problems
allows us to study simultaneously choice problems in the total Weihrauch
lattice, as well as the question which choice problems can be made independent
of their premises in the usual Weihrauch lattice. The outcome shows that many
important choice problems that are related to compact spaces are complete,
whereas choice problems for unbounded spaces or closed sets of positive measure
are typically not complete.Comment: 30 page
Weihrauch goes Brouwerian
We prove that the Weihrauch lattice can be transformed into a Brouwer algebra
by the consecutive application of two closure operators in the appropriate
order: first completion and then parallelization. The closure operator of
completion is a new closure operator that we introduce. It transforms any
problem into a total problem on the completion of the respective types, where
we allow any value outside of the original domain of the problem. This closure
operator is of interest by itself, as it generates a total version of Weihrauch
reducibility that is defined like the usual version of Weihrauch reducibility,
but in terms of total realizers. From a logical perspective completion can be
seen as a way to make problems independent of their premises. Alongside with
the completion operator and total Weihrauch reducibility we need to study
precomplete representations that are required to describe these concepts. In
order to show that the parallelized total Weihrauch lattice forms a Brouwer
algebra, we introduce a new multiplicative version of an implication. While the
parallelized total Weihrauch lattice forms a Brouwer algebra with this
implication, the total Weihrauch lattice fails to be a model of intuitionistic
linear logic in two different ways. In order to pinpoint the algebraic reasons
for this failure, we introduce the concept of a Weihrauch algebra that allows
us to formulate the failure in precise and neat terms. Finally, we show that
the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which
also implies that the theory of our Brouwer algebra is Jankov logic.Comment: 36 page
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