54 research outputs found
A rich hierarchy of functionals of finite types
We are considering typed hierarchies of total, continuous functionals using
complete, separable metric spaces at the base types. We pay special attention
to the so called Urysohn space constructed by P. Urysohn. One of the properties
of the Urysohn space is that every other separable metric space can be
isometrically embedded into it. We discuss why the Urysohn space may be
considered as the universal model of possibly infinitary outputs of algorithms.
The main result is that all our typed hierarchies may be topologically
embedded, type by type, into the corresponding hierarchy over the Urysohn
space. As a preparation for this, we prove an effective density theorem that is
also of independent interest.Comment: 21 page
A Note on Closed Subsets in Quasi-zero-dimensional Qcb-spaces (Extended Abstract)
We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category of quasi-zero-dimensional qcb-spaces is cartesian closed. Prominent examples of spaces in are the spaces in the sequential hierarchy of the Kleene-Kreisel continuous functionals. Moreover, we characterise some types of closed subsets of -spaces in terms of their ability to allow extendability of continuous functions. These results are related to an open problem in Computable Analysis
06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality
From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Anomaly Cancelation in Field Theory and F-theory on a Circle
We study the manifestation of local gauge anomalies of four- and
six-dimensional field theories in the lower-dimensional Kaluza-Klein theory
obtained after circle compactification. We identify a convenient set of
transformations acting on the whole tower of massless and massive states and
investigate their action on the low-energy effective theories in the Coulomb
branch. The maps employ higher-dimensional large gauge transformations and
precisely yield the anomaly cancelation conditions when acting on the one-loop
induced Chern-Simons terms in the three- and five-dimensional effective theory.
The arising symmetries are argued to play a key role in the study of the
M-theory to F-theory limit on Calabi-Yau manifolds. For example, using the fact
that all fully resolved F-theory geometries inducing multiple Abelian gauge
groups or non-Abelian groups admit a certain set of symmetries, we are able to
generally show the cancelation of pure Abelian or pure non-Abelian anomalies in
these models.Comment: 48 pages, 2 figures; v2: typos corrected, comments on circle fluxes
adde
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
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
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