7 research outputs found
Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces
We analyze the reducibilities induced by, respectively, uniformly continuous,
Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces,
and determine whether under suitable set-theoretical assumptions the induced
degree-structures are well-behaved.Comment: 37 pages, 2 figures, revised version, accepted for publication in the
Festschrift that will be published on the occasion of Victor Selivanov's 60th
birthday by Ontos-Verlag. A mistake has been corrected in Section
Quantitative Coding and Complexity Theory of Compact Metric Spaces
Specifying a computational problem requires fixing encodings for input and
output: encoding graphs as adjacency matrices, characters as integers, integers
as bit strings, and vice versa. For such discrete data, the actual encoding is
usually straightforward and/or complexity-theoretically inessential (up to
polynomial time, say); but concerning continuous data, already real numbers
naturally suggest various encodings with very different computational
properties. With respect to qualitative computability, Kreitz and Weihrauch
(1985) had identified ADMISSIBILITY as crucial property for 'reasonable'
encodings over the Cantor space of infinite binary sequences, so-called
representations [doi:10.1007/11780342_48]: For (precisely) these does the
sometimes so-called MAIN THEOREM apply, characterizing continuity of functions
in terms of continuous realizers.
We rephrase qualitative admissibility as continuity of both the
representation and its multivalued inverse, adopting from
[doi:10.4115/jla.2013.5.7] a notion of sequential continuity for
multifunctions. This suggests its quantitative refinement as criterion for
representations suitable for complexity investigations. Higher-type complexity
is captured by replacing Cantor's as ground space with Baire or any other
(compact) ULTRAmetric space: a quantitative counterpart to equilogical spaces
in computability [doi:10.1016/j.tcs.2003.11.012]
Continuous reducibility and dimension of metric spaces
If is a Polish metric space of dimension , then by Wadge's lemma,
no more than two Borel subsets of can be incomparable with respect to
continuous reducibility. In contrast, our main result shows that for any metric
space of positive dimension, there are uncountably many Borel subsets
of that are pairwise incomparable with respect to continuous
reducibility.
The reducibility that is given by the collection of continuous functions on a
topological space is called the \emph{Wadge quasi-order} for
. We further show that this quasi-order, restricted to the Borel
subsets of a Polish space , is a \emph{well-quasiorder (wqo)} if and
only if has dimension , as an application of the main result.
Moreover, we give further examples of applications of the technique, which is
based on a construction of graph colorings
Regular tree languages in low levels of the Wadge Hierarchy
In this article we provide effective characterisations of regular languages
of infinite trees that belong to the low levels of the Wadge hierarchy. More
precisely we prove decidability for each of the finite levels of the hierarchy;
for the class of the Boolean combinations of open sets (i.e.
the union of the first levels); and for the Borel class
(i.e. for the union of the first levels)