arXiv subjects

Vasco Brattka

Publications and source records attributed to Vasco Brattka.

At least 19 recordsLinked to original sources

arXiv2026

Loops, Inverse Limits and Non-Determinism

We introduce an operator on problems in Weihrauch complexity, which we call the infinite loop or inverse limit, and which corresponds to an infinite compositional product. This operation arises naturally whenever one implements algorithms that produce a sequence of results in an infinite loop, using some fixed subroutine. We prove that the corresponding operator is monotone with respect to (strong) Weihrauch reducibility but that it is not a closure operator. One of our findings is that weak Kőnig's lemma is closed under infinite loops, which implies that the class of non-deterministically computable problems is also closed under this operation. Consequently, this class allows for a high degree of flexibility in programming. As our main technical tools, we present an injective version of the recursion theorem and an infinitary version of the so-called independent choice theorem. We also show that, in general, the infinite loop operator is more powerful than the composition of the diamond operator followed by the parallelization operator. However, in many practical scenarios, these compositions yield a result, which coincides with the application of the infinite loop operator. Finally, we discuss the special situation of loops for single-valued problems and for problems on Turing degrees.

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arXiv2026

Computable Bases

In computable analysis typically topological spaces with countable bases are considered. The Theorem of Kreitz-Weihrauch implies that the subbase representation of a second-countable $T_0$ space is admissible with respect to the topology that the subbase generates. We consider generalizations of this setting to bases that are representable, but not necessarily countable. We introduce the notions of a computable presubbase and a computable prebase. We prove a generalization of the Theorem of Kreitz-Weihrauch for the presubbase representation that shows that any such representation is admissible with respect to the topology generated by compact intersections of the presubbase elements. For computable prebases we obtain representations that are admissible with respect to the topology that they generate. These concepts provide a natural way to investigate many topological spaces that have been studied in computable analysis. The benefit of this approach is that topologies can be described by their usual subbases and standard constructions for such subbases can be applied. Finally we discuss a Galois connection between presubbases and representations of $T_0$ spaces that indicates that presubbases and representations offer particular views on the same mathematical structure from different perspectives.

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arXiv2026

Effective bases and notions of effective second countability in computable analysis

We investigate different notions of "computable topological base" for represented spaces. We show that several non-equivalent notions of bases become equivalent when we consider computably enumerable bases. This indicates the existence of a robust notion of computably second countable represented space. These spaces are precisely those introduced by Grubba and Weihrauch under the name "computable topological spaces". The present work thus clarifies the articulation between Schröder's approach to computable topology based on the Sierpinski representation and other approaches based on notions of computable bases. These other approaches turn out to be compatible with the Sierpinski representation approach, but also strictly less general. We revisit Schröder's Effective Metrization Theorem, by showing that it characterizes those represented spaces that embed into computable metric spaces: those are the computably second countable strongly computably regular represented spaces. Finally, we study different forms of open choice problems. We show that having a computable open choice is equivalent to being computably separable, but that the "non-total open choice problem", i.e., open choice restricted to open sets that have non-empty complement, interacts with effective second countability in a satisfying way.

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arXiv2026

Computability of the Hahn-Banach Theorem Revisited

Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak Kőnig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience.

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arXiv2026

Uniform Computability of PAC Learning

We study uniform computability properties of PAC learning using Weihrauch complexity. We focus on closed concept classes, which are either represented by positive, by negative or by full information. Among other results, we prove that proper PAC learning from positive information is equivalent to the limit operation on Baire space, whereas improper PAC learning from positive information is closely related to Weak Kőnig's Lemma and even equivalent to it, when we have some negative information about the admissible hypotheses. If arbitrary hypotheses are allowed, then improper PAC learning from positive information is still in a finitary DNC range, which implies that it is non-deterministically computable, but does not allow for probabilistic algorithms. These results can also be seen as a classification of the degree of constructivity of the Fundamental Theorem of Statistical Learning. All the aforementioned results hold if an upper bound of the VC dimension is provided as an additional input information. We also study the question of how these results are affected if the VC dimension is not given, but only promised to be finite or if concept classes are represented by negative or full information. Finally, we also classify the complexity of the VC dimension operation itself, which is a problem that is of independent interest. For positive or full information it turns out to be equivalent to the binary sorting problem, for negative information it is equivalent to the jump of sorting. This classification allows also conclusions regarding the Borel complexity of PAC learnability.

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arXiv2024

Computability of Initial Value Problems

We demonstrate that techniques of Weihrauch complexity can be used to get easy and elegant proofs of known and new results on initial value problems. Our main result is that solving continuous initial value problems is Weihrauch equivalent to weak Kőnig's lemma, even if only solutions with maximal domains of existence are considered. This result simultaneously generalizes negative and positive results by Aberth and by Collins and Graça, respectively. It can also be seen as a uniform version of a Theorem of Simpson. Beyond known techniques we exploit for the proof that weak Kőnig's lemma is closed under infinite loops. One corollary of our main result is that solutions with maximal domain of existence of continuous initial value problems can be computed non-deterministically, and for computable instances there are always solutions that are low as points in the function space. Another corollary is that in the case that there is a fixed finite number of solutions, these solutions are all computable for computable instances and they can be found uniformly in a finite mind-change computation.

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arXiv2023

On the Complexity of Computing Gödel Numbers

Given a computable sequence of natural numbers, it is a natural task to find a Gödel 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 this problem is at least learnable in the limit. Here we study the problem on all computable sequences and we classify the Weihrauch complexity of it. For this purpose we can, among other methods, utilize the amalgamation technique known from learning theory. As a benchmark for the classification we use closed and compact choice problems and their jumps on natural numbers, and we argue that these problems correspond to induction and boundedness principles, as they are known from the Kirby-Paris hierarchy in reverse mathematics. We provide a topological as well as a computability-theoretic classification, which reveal some significant differences.

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arXiv2022

Weihrauch Complexity and the Hagen School of Computable Analysis

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 mathematics. Here we present a historical account of the early developments of Weihrauch complexity by the Hagen school of computable analysis that started more than thirty years ago, and we indicate how this has influenced, informed, and anticipated more recent developments of the subject.

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arXiv2021

Stashing And Parallelization Pentagons

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 introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".

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arXiv2020

The Discontinuity Problem

Matthias Schröder 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 suitable way. However, in which sense this answers Schröder's question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo-Fraenkel set theory with dependent choice and the axiom of determinacy AD. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy AD, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

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arXiv2020

Completion of Choice

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 problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete.

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arXiv2020

Weihrauch goes Brouwerian

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 the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it generates a total version of Weihrauch reducibility that is defined like the usual version of Weihrauch reducibility, but in terms of total realizers. From a logical perspective completion can be seen as a way to make problems independent of their premises. Alongside with the completion operator and total Weihrauch reducibility we need to study precomplete representations that are required to describe these concepts. In order to show that the parallelized total Weihrauch lattice forms a Brouwer algebra, we introduce a new multiplicative version of an implication. While the parallelized total Weihrauch lattice forms a Brouwer algebra with this implication, the total Weihrauch lattice fails to be a model of intuitionistic linear logic in two different ways. In order to pinpoint the algebraic reasons for this failure, we introduce the concept of a Weihrauch algebra that allows us to formulate the failure in precise and neat terms. Finally, we show that the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which also implies that the theory of our Brouwer algebra is Jankov logic.

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arXiv2018

Weihrauch Complexity in Computable Analysis

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.

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arXiv2018

Connected Choice and the Brouwer Fixed Point Theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. 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. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak Kőnig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

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arXiv2018

On the algebraic structure of Weihrauch degrees

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 an ideal with respect to the compositional product, we can consider suitable quotients of the Weihrauch degrees. We also prove some specific characterizations using the implication. In order to introduce and study compositional products and implications, we introduce and study a function space of multi-valued continuous functions. This space turns out to be particularly well-behaved for effectively traceable spaces that are closely related to admissibly represented spaces.

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arXiv2018

A Galois connection between Turing jumps and limits

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. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis.

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arXiv2017

On the Uniform Computational Content of Computability Theory

We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

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arXiv2017

On the Uniform Computational Content of Ramsey's Theorem

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 cardinality $n\geq1$ and an arbitrary finite number of colors $k\geq2$ is equivalent to the $n$--th jump of weak Kőnig's lemma. In particular, Ramsey's theorem for cardinality $n\geq1$ is $Σ^0_{n+2}$--measurable in the effective Borel hierarchy, but not $Σ^0_{n+1}$--measurable. Secondly, we obtain interesting lower bounds, for instance the $n$--th jump of weak Kőnig's lemma is Weihrauch reducible to (the stable version of) Ramsey's theorem of cardinality $n+2$ for $n\geq2$. We prove that with strictly increasing numbers of colors Ramsey's theorem forms a strictly increasing chain in the Weihrauch lattice. Our study of jumps also shows that certain uniform variants of Ramsey's theorem that are indistinguishable from a non-uniform perspective play an important role. For instance, the colored version of Ramsey's theorem explicitly includes the color of the homogeneous set as output information, and the jump of this problem (but not the uncolored variant) is equivalent to the stable version of Ramsey's theorem of the next greater cardinality. Finally, we briefly discuss the particular case of Ramsey's theorem for pairs, and we provide some new separation techniques for problems that involve jumps in this context. In particular, we study uniform results regarding the relation of boundedness and induction problems to Ramsey's theorem, and we show that there are some significant differences with the non-uniform situation in reverse mathematics.

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