Vasco Brattka

Universität der Bundeswehr München ·  Institute for Theoretical Computer Science, Mathematics, and Operations Research

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 theory this problem is well studied from several perspectives and one question studied there is for which sequences...

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 important interface between more proof-theoretic and more computability-theoretic studies in the realm of reverse...

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we intro...

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable...

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we intro...

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 discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a s...

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of...

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal role in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable...

We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of...

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingl...

We demonstrate that the Weihrauch lattice can be used to study the uniform computational content of computability theoretic properties and theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete extensions of Peano arithmetic, 1-genericity, Martin-L\"of randomness and cohesiveness. Th...

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.

The purpose of this addendum is to close a gap in the proof of , which characterizes the computational content of the Bolzano-Weierstraß Theorem for arbitrary computable metric spaces.

We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility. We show that a more detailed picture emerges than what a related study by Giusto, Brown, and Simpson has revealed in the setting of reverse mathematics. In particular, different formulations of the Vitali Covering Theore...

The history of computability theory and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, Émil Borel discussed his ideas on computable real number functions in his introduction to measure theory. On the other hand, Alan Turing had computable real numbers in mind when he introduced his now fa...

We revisit the investigation of the computational content of the Brouwer Fixed Point Theorem in [7], and answer the two open questions from that work. First, we show that the computational hardness is independent of the dimension, as long as it is greater than 1 (in [7] this was only established for dimension greater than 2). Second, we show that r...

We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility. We show that a more detailed picture emerges than what a related study by Giusto, Brown, and Simpson has revealed in the setting of reverse mathematics. In particular, different formulations of the Vitali Covering Theore...

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 with similar structures such as residuated lattices and concurrent Kleene algebras. Introducing the notion of...

The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction to measure theory. On the other hand, Alan Turing had computable real numbers in mind when he introduced his...

This report documents the program and the outcomes of Dagstuhl Seminar 15392 "Measuring the Complexity of Computational Content: Weihrauch Reducibility and Reverse Analysis." It includes abstracts on most talks presented during the seminar, a list of open problems that were discussed and partially solved during the meeting as well as a bibliography...

We study the uniform computational content of different versions of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a complete (i.e., "large") metric space cannot be decomposed into countably many nowhere dense (i.e., "small") pieces. The Baire Category Theorem i...

We study the uniform computational content of Ramsey's Theorem in the Weihrauch lattice. Our central results provide information on how Ramsey's Theorem behaves under product, parallelization and jumps. From these results we can derive a number of important properties of Ramsey's Theorem. For one, the parallelization of Ramsey's Theorem for size $n...

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 randomness. In particular, we show that a natural concept of Las Vegas computability on infinite objects is more p...

This special issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Dagstuhl Seminar on Computing with Infinite Data: Topological and Logical Foundations. The workshop took place in the Schloss Dagstuhl - Leibniz Center for Informatics in the first half of October 2011.

Alan Turing was an inspirational figure who is now recognised as a genius of modern mathematics. In addition to leading the Allied forces' code-breaking effort at Bletchley Park in World War II, he proposed the theoretical foundations of modern computing and anticipated developments in areas from information theory to computer chess. His ideas have...

We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a probabilistic Turing machine that receives a random binary sequence as auxiliary input. The machine can take adv...

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Connected choice is the operation that finds a point...

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Connected choice is the operation that finds a point...

We discuss the legacy of Alan Turing and his impact on computability and analysis.

We classify the computational content of the Bolzano–Weierstraß Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a compu...

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) We show that a real number z in [0,1] is computably random if and only if every nondecreasing computable function [0,1]->R is differentiable at z. (2) A real number z in [0,1] is weakly 2-random if and only if every almost everywhere differen...

We classify the computational content of the Bolzano-Weierstrass Theorem and variants thereof in the Weihrauch lattice. For this purpose we first introduce the concept of a derivative or jump in this lattice and we show that it has some properties similar to the Turing jump. Using this concept we prove that the derivative of closed choice of a comp...

We discuss computability properties of the set PG(x) of elements of best approximation of some point x∈X by elements of G⊆X in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about PG(x) as a closed set. In the case that G is finite-dimensional, one ca...

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solution...

We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed...

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solution...

In this paper we study Weihrauch reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest lower bound operation. We prove that parallelizati...

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. Th...

We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations such as union and intersection, but also typical topological operations such as the closure of the complement, the closure of the interior, the boundary and the derivative of a set. Thes...

Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T−1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T−1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulati...

We show that given a computable Banach space X and a finite-dimensional subspace U of X the set of elements of best approximation of x ∈ X (by elements of U) can be computed as a compact set with negative information. If X is uniformly convex, we can even compute the (unique) element of best approximation. Furthermore, given a uniformly convex comp...

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and oth...

This special issue of the Journal of Universal Computer Science (J. UCS) contains a selection of articles presented at the Fourth International Conference on Computability and Complexity in Analysis (CCA), held in Siena, Italy, June 16-18, 2007. This conference is the fourteenth event in the series of CCA annual meetings; seventy-four participants...

The ordinary notion of algorithmic randomness of reals can be characterised as Martin-Löf randomness with respect to the Lebesgue measure or as Kolmogorov randomness with respect to the binary representation. In this paper we study the question of how the notion of algorithmic randomness induced by the signed-digit representation of the real number...

The Graph Theorem of classical recursion theory states that a total function on the natural numbers is computable, if and only if its graph is recursive. It is known that this result can be generalized to real number functions where it has an important practical interpretation: the total computable real number functions are precisely those which ca...

This tutorial gives a brief introduction to computable analysis. The objective of this theory is to study algorithmic aspects of real numbers, real number functions, subsets of real numbers, and higher type operators over the real numbers. In this theory, the classical computability notions and complexity notions based on the Turing machine model a...

By means of different effectivities of the epigraphs and hypographs of real functions we introduce several effectivizations of the semi-continuous real functions. We call a real function f lower semi-computable of type one if its hypograph hypo(f): = (x, y): f(x) > y & x ∈ dom(f) is recursively enumerably open in dom(f) × IR; f is lower semi-comput...

and it is entitled “Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments”. Hilbert presented his question in terms of nomography (the theory of nomograms ,w hich are graphical presentations of continuous functions depending on several arguments for computational purposes), a field whic...

We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the...

We study the Borel complexity of topological operations on closed subsets of computable metric spaces. The investigated operations include set theoretic operations as union and intersection, but also typical topological operations such as the closure of the complement, the closure of the interior, the boundary and the derivative of a set. These ope...

We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the...

Special issue RAIRO - Theoretical Informatics and Applications 41(1)

We present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Ne...

We introduce the notion of a 'q-ideal in a commutative Banach algebra, and investigate the relation between q-ideals and maximal ideals constructively.

In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a di...

We present a modified real RAM model which is equipped with the usual discrete and real-valued arithmetic operations and with a finite precision test <k which allows comparisons of real numbers only up to a variable uncertainty 1/k+1 Furthermore our feasible RAM has an extended semantics which allows approximative computations. Using a logarithmic...

In computable analysis, a computable normed space turns out to be necessarily separable. In classical functional analysis, non-separable spaces occur naturally, for example as dual spaces of common separable spaces. The question arises whether the notion of a computable normed space can be generalized to the non-separable case in a meaningful way....

We study computability on sequence spaces, as they are used in functional analysis. It is known that non-separable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice non-separable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology...

The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and oth...

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computabil- ity for single-valued as well as for multi-valued functions by a direct effecti...

Self-adjoint operators and their spectra play a crucial role in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from...

Many problems in Linear Algebra can be solved by Gaussian Elimination. This famous algorithm applies to an algebraic model of real number computation where operations +,-,*,/ and tests like, e.g., < and == are presumed exact. Implementations of algebraic algorithms on actual digital computers often lead to numerical instabilities, thus revealing a...

Without Abstract

The notions “recursively enumerable” and “recursive” are the basic notions of effectivity in classical recursion theory. In computable analysis, these notions are generalized to closed subsets of Euclidean space using their metric distance functions. We study a further generalization of these concepts to subsets of computable metric spaces. It appe...

In computable analysis recursive metric spaces play an important role, since these are, roughly speaking, spaces with computable metric and limit operation. Unfortunately, the concept of a metric space is not powerful enough to capture all interesting phenomena which occur in computable analysis. Some computable objects are naturally considered as...

Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological s...

Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T -1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T -1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formula...

In formal analogy to separable metric spaces we introduce the concept of a generated quasi-metric space. In a corresponding way as each point of a separable metric space can be represented as the limit of a sequence in some countable dense subset, each point of a generated quasi-metric space can be considered as the infimum of a sequence in the gen...

Computable analysis is the Turing machine based theory of computabil-ity on the real numbers and other topological spaces. Similarly as Eršov's concept of numberings can be used to deal with discrete struc-tures, Kreitz and Weihrauch's concept of representations can be used to handle structures of continuum cardinality. In this context the choice o...

In the year 1900 in his famous lecture in Paris Hilbert formulated 23 challeng-ing problems which inspired many ground breaking mathematical investigations in the last century. Among these problems the 13th was concerned with the solution of higher order algebraic equations. Hilbert conjectured that such equations are not solvable by functions whic...

We prove three results about representations of real numbers (or elements of other topological spaces) by infinite strings. Such representations are useful for the description of real number computations performed by digital computers or by Turing machines. First, we show that the so-called admissible representations, a topologically natural class...

Generalizing the notion of a recursively enumerable (r.e.) set to sets of real numbers and other metric spaces is an important topic in computable analysis (which is the Turing machine based theory of computable real number functions). A closed subset of a computable metric space is called r. e. closed, if all open rational balls which intersect th...

We investigate the computable content of the Uniform Boundedness Theorem which states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the sequence is “given”. If it is just gi...

This part of the volume contains the papers accepted for presentation at the fifth workshop on Computability and Complexity in Analysis, CCA 2002, which took place on July 12-13, 2002 in Málaga, Spain. It was co-located with ICALP 2002, the 29th International Colloquium on Automata, Languages, and Programming. Both events were hosted by the Compute...

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 inter- esting applications in Walsh analysis. We investigate computability properties of a corresponding Fine representation of the real numbers and we construct a structure which characterizes this rep...

Do the solutions of linear equations depend computably on their co- ecients? Implicitly, this has been one of the central questions in linear algebra since the very beginning of the subject and the famous Gau algorithm is one of its numerical answers. Today there exists a tremen- dous number of algorithms which solve this problem for dierent types...

We investigate the computable content of the Uniform Bounded-ness Theorem and of the closely related Banach-Steinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It tu...

1Introduction Co nputability on the continuum has been investigated over the years, but we are interested in the mathematical approach such as the one by Pour-El and Richards [2]. In their approach, classical mathematics is accepted. The point is to see how computable objects and operators look like in ordinary mathematics. Co nputabilities of real...