Related papers: The computational content of multidimensional disc…
We explore the low levels of the structure of the continuous Weihrauch degrees of first-order problems. In particular, we show that there exists a minimal discontinuous first-order degree, namely that of $\accn$, without any determinacy…
The Weihrauch degrees and strong Weihrauch degrees are partially ordered structures representing degrees of unsolvability of various mathematical problems. Their study has been widely applied in computable analysis, complexity theory, and…
There is a way of assigning a realizability notion to each degree of incomputability. In our setting, we make use of Weihrauch degrees (degrees of incomputability/discontinuity of partial multi-valued functions) to obtain Lifschitz-like…
Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify…
Weihrauch complexity is now an established and active part of mathematical logic. It can be seen as a computability-theoretic approach to classifying the uniform computational content of mathematical problems. This theory has become an…
We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…
In this work we investigate the Weihrauch degree of the problem $\mathsf{DS}$ of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem $\mathsf{BS}$ of finding a bad sequence…
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…
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary…
Weihrauch reducibility is a notion of reducibility between computational problems that is useful to calibrate the uniform computational strength of a multivalued function. It complements the analysis of mathematical theorems done in reverse…
Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning…
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.
We consider the problem of numerically integrating functions with hyperplane discontinuities over the entire Euclidean space in many dimensions. We describe a simple process through which the Euclidean space is partitioned into simplices on…
Matthias Schr\"oder has asked the question whether there is a weakest discontinuous problem in the continuous version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the…
We answer a question by Vasco Brattka and Guido Gherardi by proving that the Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further…
We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison…
We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this,…
Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS…
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…
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…