Computable analysis
View article
Elementary Computable Topology Open
We revise and extend the foundation of computable topology in the frame- work of Type-2 theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to a…
View article
The Computable Multi-Functions on Multi-represented Sets are Closed under Programming Open
In the representation approach to computable analysis (TTE) (Grz55, KW85, Wei00), abstract data like rational numbers, real numbers, compact sets or continuous real functions are represented by finite or infinite sequences (Σ ∗ ,Σ ω )o f s…
View article
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees Open
We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakl…
View article
Polynomial Time corresponds to Solutions of Polynomial Ordinary\n Differential Equations of Polynomial Length Open
We provide an implicit characterization of polynomial time computation in\nterms of ordinary differential equations: we characterize the class\n$\\operatorname{PTIME}$ of languages computable in polynomial time in terms of\ndifferential eq…
View article
A comparison of concepts from computable analysis and effective descriptive set theory Open
Computable analysis and effective descriptive set theory are both concerned with complete metric spaces, functions between them and subsets thereof in an effective setting. The precise relationship of the various definitions used in the tw…
View article
Notions of Probabilistic Computability on Represented Spaces Open
We define and compare several probabilistically weakened notions of computability for mappings from represented spaces (that are equipped with a measure or outer measure) into effective metric spaces. We thereby generalize definitions by K…
View article
COMPUTABLY COMPACT METRIC SPACES Open
We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in c…
View article
On computability and disintegration Open
We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim.…
View article
ON THE STRUCTURE OF COMPUTABLE REDUCIBILITY ON EQUIVALENCE RELATIONS OF NATURAL NUMBERS Open
We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficient…
View article
Interactions between computability theory and set theory Open
In this thesis, we explore connections between computability theory and set theory. We investigate an extension of reverse mathematics to a higher-order context, focusing in particular on determinacy principles, and an extension of computa…
View article
Some Notes on Fine Computability Open
A metric defined by Fine induces a topology on the unit interval which is strictly stronger than the ordinary Euclidean topology and which has some interesting applications in Walsh analysis. We investigate computability properties of a co…
View article
Continuous models of computation: from computability to complexity Open
In 1941, Claude Shannon introduced a continuous-time analog model of computation,
namely the General Purpose Analog Computer (GPAC).
The GPAC is a physically feasible model in the sense that it can be
implemented in practice through the…
View article
Unconstrained Church-Turing thesis cannot possibly be true Open
The Church-Turing thesis asserts that if a partial strings-to-strings function is effectively computable then it is computable by a Turing machine. In the 1930s, when Church and Turing worked on their versions of the thesis, there was a ro…
View article
The unreasonable effectiveness of Nonstandard Analysis Open
As suggested by the title, the aim of this paper is to uncover the vast computational content of classical Nonstandard Analysis. To this end, we formulate a template ${\mathfrak{C}\mathfrak{I}}$ which converts a theorem of ‘pure’ Nonstanda…
View article
Dividing by Zero - How Bad Is It, Really? Open
In computable analysis testing a real number for being zero is a fundamental example of a non-computable task. This causes problems for division: We cannot ensure that the number we want to divide by is not zero. In many cases, any real nu…
View article
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations Open
The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynom…
View article
Weihrauch Complexity in Computable Analysis Open
We provide a self-contained introduction into Weihrauch complexity and its applications to computable analysis. This includes a survey on some classification results and a discussion of the relation to other approaches.
View article
Poly-time computability of the Feigenbaum Julia set Open
We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.
View article
Las Vegas Computability and Algorithmic Randomness Open
In this article we try to formalize the question "What can be computed with access to randomness?" We propose the very fine-grained Weihrauch lattice as an approach to differentiate between different types of computation with access to ran…
View article
Monte Carlo Computability Open
We introduce Monte Carlo computability as a probabilistic concept of computability on infinite objects and prove that Monte Carlo computable functions are closed under composition. We then mutually separate the following classes of functio…
View article
On the Subrecursive Computability of Several Famous Constants Open
For any class F of total functions in the set N of the natural numbers, we define the notion of F-computable real number. A real number α is called F-computable if there exist one-argument functions f, g and h in F such that for any n in N…
View article
Polynomial Running Times for Polynomial-Time Oracle Machines Open
This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider functions on the natural numbers as running t…
View article
Dividing by zero - how bad is it, really? Open
In computable analysis testing a real number for being zero is a fundamental example of a non-computable task. This causes problems for division: We cannot ensure that the number we want to divide by is not zero. In many cases, any real nu…
View article
Normal Numbers and Limit Computable Cantor Series Open
Given any oracle, $A$ , we construct a basic sequence $Q$ , computable in the jump of $A$ , such that no $A$ -computable real is $Q$ -distribution-normal. A corollary to this is that there is a $\\Delta^{0}_{n+1}$ basic sequence with respe…
View article
Computable randomness and betting for computable probability spaces Open
Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and fu…
View article
All Sampling Methods Produce Outliers Open
Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P.
View article
Weihrauch-completeness for layerwise computability Open
We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time o…
View article
M. Levin’s construction of absolutely normal numbers with very low discrepancy Open
Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational com…
View article
Iterative forcing and hyperimmunity in reverse mathematics Open
The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcin…
View article
Computable analysis with applications to dynamic systems Open
Numerical computation is traditionally performed using floating-point arithmetic and truncated forms of infinite series, a methodology which allows for efficient computation at the cost of some accuracy. For most applications, these errors…