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 $QZ$ of quasi-zero-dimensional qcb$_0$-spaces is cartesian closed. Prominent examples of spaces in $QZ$ are the spaces in the sequential hierarchy of the Kleene-Kreisel continuous functionals. Moreover, we characterise some types of closed subsets of $QZ$-spaces in terms of their ability to allow extendability of continuous functions. These results are related to an open problem in Computable Analysis

Relativizations of the P =? DNP Question for the BSS Model

We consider the uniform BSS model of computation where the machines can perform additions, multiplications, and tests of the form $xgeq 0$. The oracle machines can also check whether a tuple of real numbers belongs to a given oracle set ${cal O}$ or not. We construct oracles such that the classes P and DNP relative to these oracles are equal or not equal

Towards the Complexity of Riemann Mappings (Extended Abstract)

We show that under reasonable assumptions there exist Riemann mappings which are as hard as tally $sharp$-P even in the non-uniform case. More precisely, we show that under a widely accepted conjecture from numerical mathematics there exist single domains with simple, i.e. polynomial time computable, smooth boundary whose Riemann mapping is polynomial time computable if and only if tally $sharp$-P equals P. Additionally, we give similar results without any assumptions using tally $UP$ instead of $sharp$-P and show that Riemann mappings of domains with polynomial time computable analytic boundaries are polynomial time computable

Computability of Homology for Compact Absolute Neighbourhood Retracts

In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specify both a function which defines a retraction from a neighbourhood of the space in the Hilbert cube to the space itself, and a sufficiently fine over-approximation of the set. However, neither the retraction itself, nor a description of an approximation of the set in the Hausdorff metric, is sufficient to compute the homology groups. We express the conditions in the language of computable analysis, which is a powerful framework for studying computability in topology and geometry, and use cubical homology to perform the computations

Computable Separation in Topology, from T_0 to T_3

This article continues the study of computable elementary topology started in (Weihrauch, Grubba 2009). We introduce a number of computable versions of the topological $T_0$ to $T_3$ separation axioms and solve their logical relation completely. In particular, it turns out that computable $T_1$ is equivalent to computable $T_2$. The strongest axiom $SCT_3$ is used in (Grubba, Schroeder, Weihrauch 2007) to construct a computable metric

Effective Dispersion in Computable Metric Spaces

We investigate the relationship between computable metric spaces $(X,d,alpha )$ and $(X,d,beta ),$ where $(X,d)$ is a given metric space. In the case of Euclidean space, $alpha$ and $beta$ are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space. This notion is essential in the proof of the main result of this paper: $(X,d,alpha )$ is effectively totally bounded if and only if $(X,d,beta )$ is effectively totally bounded, i.e. the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space

Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem $f:Ato B$, additional structural information about the input $xin A$ (that is any kind of promise that $x$ belongs to a certain subset $A\u27subseteq A$) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problem show much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation $Acdotvec x=0$ for a given singular real $ntimes n$-matrix $A$ is possible when knowing $rank(A)in{0,1,ldots,n-1}$; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric $ntimes n$-matrix $A$ is possible when knowing the number of distinct eigenvalues: an integer between $1$ and $n$ (the latter corresponding to the nondegenerate case). And again we show that $n$--fold (i.e. roughly $log n$ bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding emph{some single} eigenvector of $A$ requires and suffices with $Theta(log n)$--fold advice

Canonical Effective Subalgebras of Classical Algebras as Constructive Metric Completions

We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the results are applicable in various established schools of computability, such as type 1 and type 2 effectivity, domain representations, equilogical spaces, and others

Separations of Non-monotonic Randomness Notions

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-L"of randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of Martin-L"of randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr\u27s model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent ``non-monotonic\u27\u27 notion of randomness, now called Kolmogorov-Loveland-randomness, has been shown to be quite close to Martin-L"of randomness, but whether these two classes coincide remains a fundamental open question. In order to get a better understanding of non-monotonic randomness notions, Miller and Nies introduced some interesting intermediate concepts, where one only allows non-adaptive strategies, i.e., strategies that can still bet non-monotonically, but such that the sequence of betting positions is known in advance (and computable). Recently, these notions were shown by Kastermans and Lempp to differ from Martin-L"of randomness. We continue the study of the non-monotonic randomness notions introduced by Miller and Nies and obtain results about the Kolmogorov complexities of initial segments that may and may not occur for such sequences, where these results then imply a complete classification of these randomness notions by order of strength

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 on closed sets 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, the Closed Graph Theorem and the Uniform Boundedness Theorem. Well-known omniscience principles from constructive mathematics such as $LPO$ and $LLPO$ can naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. 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