1,543 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
Effective Banach spaces
This thesis addresses Pour-El and Richards' fourth question from their book
"Computability in analysis and physics", concerning the relation between higher
order recursion theory and computability in analysis.
Among other things it is shown that there is a computability structure that
is uncountable. The example given is a structure on the Banach space of bounded
linear operators on the set of almost periodic functions.Comment: Master's thesis, University of Oslo, 1997. Adviser: Dag Normann.
Translated from Norwegian. Original title: "Effektive Banach-rom
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
A pseudorandom point in an ergodic dynamical system over a computable metric
space is a point which is computable but its dynamics has the same statistical
behavior as a typical point of the system.
It was proved in [Avigad et al. 2010, Local stability of ergodic averages]
that in a system whose dynamics is computable the ergodic averages of
computable observables converge effectively. We give an alternative, simpler
proof of this result.
This implies that if also the invariant measure is computable then the
pseudorandom points are a set which is dense (hence nonempty) on the support of
the invariant measure
Point degree spectra of represented spaces
We introduce the point degree spectrum of a represented space as a
substructure of the Medvedev degrees, which integrates the notion of Turing
degrees, enumeration degrees, continuous degrees, and so on. The notion of
point degree spectrum creates a connection among various areas of mathematics
including computability theory, descriptive set theory, infinite dimensional
topology and Banach space theory. Through this new connection, for instance, we
construct a family of continuum many infinite dimensional Cantor manifolds with
property whose Borel structures at an arbitrary finite rank are mutually
non-isomorphic. This provides new examples of Banach algebras of real valued
Baire class two functions on metrizable compacta, and strengthen various
theorems in infinite dimensional topology such as Pol's solution to
Alexandrov's old problem
Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
We consider the dynamical behavior of Martin-L\"of random points in dynamical
systems over metric spaces with a computable dynamics and a computable
invariant measure. We use computable partitions to define a sort of effective
symbolic model for the dynamics. Through this construction we prove that such
points have typical statistical behavior (the behavior which is typical in the
Birkhoff ergodic theorem) and are recurrent. We introduce and compare some
notions of complexity for orbits in dynamical systems and prove: (i) that the
complexity of the orbits of random points equals the Kolmogorov-Sina\"i entropy
of the system, (ii) that the supremum of the complexity of orbits equals the
topological entropy
Computation with Advice
Computation with advice is suggested as generalization of both computation
with discrete advice and Type-2 Nondeterminism. Several embodiments of the
generic concept are discussed, and the close connection to Weihrauch
reducibility is pointed out. As a novel concept, computability with random
advice is studied; which corresponds to correct solutions being guessable with
positive probability. In the framework of computation with advice, it is
possible to define computational complexity for certain concepts of
hypercomputation. Finally, some examples are given which illuminate the
interplay of uniform and non-uniform techniques in order to investigate both
computability with advice and the Weihrauch lattice
Computability on quasi-Polish spaces
We investigate the effectivizations of several equivalent definitions of
quasi-Polish spaces and study which characterizations hold effectively. Being a
computable effectively open image of the Baire space is a robust notion that
admits several characterizations. We show that some natural effectivizations of
quasi-metric spaces are strictly stronger
Dugundji systems and a retract characterization of effective zero-dimensionality
In this paper (as in [Ken15]), we consider an effective version of the
characterization of (separable metric) spaces as zero-dimensional iff every
nonempty closed subset is a retract of the space (actually, we have proved a
relative result for closed (zero-dimensional) subspaces of a fixed space). This
uses (in the converse direction) local compactness & bilocated sets as in
[Ken15], but in the forward direction the newer version has a simpler proof and
no compactness assumption. Furthermore, the proof of the forward implication
relates to so-called Dugundji systems: we elaborate both a general construction
of such systems for a proper nonempty closed subspace (using a computable form
of countable paracompactness), and modifications to make the sets pairwise
disjoint if the subspace is zero-dimensional, or to avoid the restriction to
proper subspaces. In a different direction, a second theorem applies in
-adic analysis the ideas of the first theorem to compute a more general form
of retraction, given a Dugundji system (possibly without disjointness).
Finally, we complement the mentioned effective retract characterization of
zero-dimensional subspaces by improving to equivalence the implications (or
Weihrauch reductions in some cases), for closed at-most-zero-dimensional
subsets with some negative information, among separate conditions of
computability of operations introduced in [Ken15,\S 4] and
corresponding to vanishing large inductive dimension, vanishing small inductive
dimension, existence of a countable basis of relatively clopen sets, and the
reduction principle for sequences of open sets. Thus, similarly to the robust
notion of effective zero-dimensionality of computable metric spaces in [Ken15],
there is a robust notion of `uniform effective zero-dimensionality' for a
represented pointclass consisting of at-most-zero-dimensional closed subsets.Comment: 33 pages, major revised version, intended for postproceedings of CCC
201
Logic Blog 2015f
The 2015 Logic Blog contains a large variety of results connected to logic,
some of them unlikely to be submitted to a journal. For the first time there is
a group theory part. There are results in higher randomness, and in computable
ergodic theory
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