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Documents authored by Brattka, Vasco

Document
DOI: 10.4230/DagRep.8.9.1
Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis (Dagstuhl Seminar 18361)

Authors: Vasco Brattka, Damir D. Dzhafarov, Alberto Marcone, and Arno Pauly

Published in: Dagstuhl Reports, Volume 8, Issue 9 (2019)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 18361 "Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis". It includes abstracts of talks presented during the seminar and open problems that were discussed, as well as a bibliography on Weihrauch complexity that was started during the previous meeting (Dagstuhl seminar 15392) and that saw some significant growth in the meantime. The session "Solved problems" is dedicated to the solutions to some of the open questions raised in the previous meeting (Dagstuhl seminar 15392).
Cite as

Vasco Brattka, Damir D. Dzhafarov, Alberto Marcone, and Arno Pauly. Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis (Dagstuhl Seminar 18361). In Dagstuhl Reports, Volume 8, Issue 9, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{brattka_et_al:DagRep.8.9.1,
  author =	{Brattka, Vasco and Dzhafarov, Damir D. and Marcone, Alberto and Pauly, Arno},
  title =	{{Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis (Dagstuhl Seminar 18361)}},
  pages =	{1--28},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{8},
  number =	{9},
  editor =	{Brattka, Vasco and Dzhafarov, Damir D. and Marcone, Alberto and Pauly, Arno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.9.1},
  URN =		{urn:nbn:de:0030-drops-103270},
  doi =		{10.4230/DagRep.8.9.1},
  annote =	{Keywords: Computability and complexity in analysis, computations on real numbers, reducibilities, descriptive complexity, computational complexity, reverse and}
}
Document
DOI: 10.4230/DagRep.7.2.89
Computability Theory (Dagstuhl Seminar 17081)

Authors: Klaus Ambos-Spies, Vasco Brattka, Rodney Downey, and Steffen Lempp

Published in: Dagstuhl Reports, Volume 7, Issue 2 (2017)

Abstract
Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Gödel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.
Cite as

Klaus Ambos-Spies, Vasco Brattka, Rodney Downey, and Steffen Lempp. Computability Theory (Dagstuhl Seminar 17081). In Dagstuhl Reports, Volume 7, Issue 2, pp. 89-101, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{ambosspies_et_al:DagRep.7.2.89,
  author =	{Ambos-Spies, Klaus and Brattka, Vasco and Downey, Rodney and Lempp, Steffen},
  title =	{{Computability Theory (Dagstuhl Seminar 17081)}},
  pages =	{89--101},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2017},
  volume =	{7},
  number =	{2},
  editor =	{Ambos-Spies, Klaus and Brattka, Vasco and Downey, Rodney and Lempp, Steffen},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.2.89},
  URN =		{urn:nbn:de:0030-drops-73540},
  doi =		{10.4230/DagRep.7.2.89},
  annote =	{Keywords: algorithmic randomness, computability theory, computable algebra, computable analysis, generic case complexity, proof mining}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.17
Monte Carlo Computability

Authors: Vasco Brattka, Rupert Hölzl, and Rutger Kuyper

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
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 functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable functions, and that of Monte Carlo computable functions. We give natural examples of computational problems witnessing these separations. As a specific problem which is Monte Carlo computable but neither Las Vegas computable nor non-deterministically computable, we study the problem of sorting infinite sequences that was recently introduced by Neumann and Pauly. Their results allow us to draw conclusions about the relation between algebraic models and Monte Carlo computability.
Cite as

Vasco Brattka, Rupert Hölzl, and Rutger Kuyper. Monte Carlo Computability. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brattka_et_al:LIPIcs.STACS.2017.17,
  author =	{Brattka, Vasco and H\"{o}lzl, Rupert and Kuyper, Rutger},
  title =	{{Monte Carlo Computability}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.17},
  URN =		{urn:nbn:de:0030-drops-70164},
  doi =		{10.4230/LIPIcs.STACS.2017.17},
  annote =	{Keywords: Weihrauch degrees, Weak Weak Konig's Lemma, Monte Carlo computability, algorithmic randomness, sorting}
}
Document
DOI: 10.4230/DagRep.5.9.77
Measuring the Complexity of Computational Content (Dagstuhl Seminar 15392)

Authors: Vasco Brattka, Akitoshi Kawamura, Alberto Marcone, and Arno Pauly

Published in: Dagstuhl Reports, Volume 5, Issue 9 (2016)

Abstract
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 on the seminar topic that we compiled during the seminar.
Cite as

Vasco Brattka, Akitoshi Kawamura, Alberto Marcone, and Arno Pauly. Measuring the Complexity of Computational Content (Dagstuhl Seminar 15392). In Dagstuhl Reports, Volume 5, Issue 9, pp. 77-104, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Article{brattka_et_al:DagRep.5.9.77,
  author =	{Brattka, Vasco and Kawamura, Akitoshi and Marcone, Alberto and Pauly, Arno},
  title =	{{Measuring the Complexity of Computational Content (Dagstuhl Seminar 15392)}},
  pages =	{77--104},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2016},
  volume =	{5},
  number =	{9},
  editor =	{Brattka, Vasco and Kawamura, Akitoshi and Marcone, Alberto and Pauly, Arno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.9.77},
  URN =		{urn:nbn:de:0030-drops-56861},
  doi =		{10.4230/DagRep.5.9.77},
  annote =	{Keywords: Computability and complexity in analysis, computations on real numbers, reducibilities, descriptive complexity, computational complexity, reverse and constructive mathematics}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.130
Las Vegas Computability and Algorithmic Randomness

Authors: Vasco Brattka, Guido Gherardi, and Rupert Hölzl

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
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 powerful than mere oracle access to a Martin-Löf random object. As a concrete problem that is Las Vegas computable but not computable with access to a Martin-Löf random oracle we study the problem of finding Nash equilibria.
Cite as

Vasco Brattka, Guido Gherardi, and Rupert Hölzl. Las Vegas Computability and Algorithmic Randomness. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 130-142, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{brattka_et_al:LIPIcs.STACS.2015.130,
  author =	{Brattka, Vasco and Gherardi, Guido and H\"{o}lzl, Rupert},
  title =	{{Las Vegas Computability and Algorithmic Randomness}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{130--142},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.130},
  URN =		{urn:nbn:de:0030-drops-49093},
  doi =		{10.4230/LIPIcs.STACS.2015.130},
  annote =	{Keywords: Weihrauch degrees, weak weak K\"{o}nig's lemma, Las Vegas computability, algorithmic randomness, Nash equilibria}
}
Document
DOI: 10.4230/DagRep.1.10.14
Computing with Infinite Data: Topological and Logical Foundations (Dagstuhl Seminar 11411)

Authors: Ulrich Berger, Vasco Brattka, Victor Selivanov, Dieter Spreen, and Hideki Tsuiki

Published in: Dagstuhl Reports, Volume 1, Issue 10 (2012)

Abstract
There is a large gap between mathematical structures and the structures computer implementations are based on. To stimulate research to overcome this---especially for infinitary structures---highly non-trivial problem the Dagstuhl Seminar 11411 ``Computing with Infinite Data: Topological and Logical Foundations'' was held. This report collects the ideas that were presented and discussed during the course of the seminar.
Cite as

Ulrich Berger, Vasco Brattka, Victor Selivanov, Dieter Spreen, and Hideki Tsuiki. Computing with Infinite Data: Topological and Logical Foundations (Dagstuhl Seminar 11411). In Dagstuhl Reports, Volume 1, Issue 10, pp. 14-36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@Article{berger_et_al:DagRep.1.10.14,
  author =	{Berger, Ulrich and Brattka, Vasco and Selivanov, Victor and Spreen, Dieter and Tsuiki, Hideki},
  title =	{{Computing with Infinite Data: Topological and Logical Foundations (Dagstuhl Seminar 11411)}},
  pages =	{14--36},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2012},
  volume =	{1},
  number =	{10},
  editor =	{Berger, Ulrich and Brattka, Vasco and Selivanov, Victor and Spreen, Dieter and Tsuiki, Hideki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.10.14},
  URN =		{urn:nbn:de:0030-drops-33721},
  doi =		{10.4230/DagRep.1.10.14},
  annote =	{Keywords: Exact real number computation, Stream computation, Infinite computations, Computability in analysis, Hierarchies, Reducibility, Topological complexity}
}
Document
DOI: 10.4230/OASIcs.CCA.2009.2261
Weihrauch Degrees, Omniscience Principles and Weak Computability

Authors: Vasco Brattka and Guido Gherardi

Published in: OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)

Abstract
In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this 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 show that parallelization is a closure operator for this semi-lattice and the parallelized Weihrauch degrees even form a lattice with the product of multi-valued functions as greatest lower bound operation. We show that the Medvedev lattice and hence the Turing upper semi-lattice can both be embedded into the parallelized Weihrauch lattice in a natural way. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. This allows a new purely topological or computational approach to metamathematics that sheds new light on the nature of theorems. As crucial corner points of this classification scheme we study the limited principle of omniscience $\LPO$, the lesser limited principle of omniscience $\LLPO$ and their parallelizations. We show that parallelized $\LLPO$ is equivalent to Weak König's Lemma and hence to the Hahn-Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized $\LLPO$ and we present a new proof that the class of weakly computable operations is closed under composition. This proof is based on a computational version of Kleene's ternary logic. Moreover, we characterize weakly computable operations on computable metric spaces as operations that admit upper semi-computable compact-valued selectors and we show that any single-valued weakly computable operation is already computable in the ordinary sense.
Cite as

Vasco Brattka and Guido Gherardi. Weihrauch Degrees, Omniscience Principles and Weak Computability. In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 83-94, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brattka_et_al:OASIcs.CCA.2009.2261,
  author =	{Brattka, Vasco and Gherardi, Guido},
  title =	{{Weihrauch Degrees, Omniscience Principles and Weak Computability}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{83--94},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Bauer, Andrej and Hertling, Peter and Ko, Ker-I},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2261},
  URN =		{urn:nbn:de:0030-drops-22617},
  doi =		{10.4230/OASIcs.CCA.2009.2261},
  annote =	{Keywords: Computable analysis, constructive analysis, reverse mathematics, effective descriptive set theory}
}
Document
DOI: 10.4230/OASIcs.CCA.2009.2262
Effective Choice and Boundedness Principles in Computable Analysis

Authors: Vasco Brattka and Guido Gherardi

Published in: OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)

Abstract
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.
Cite as

Vasco Brattka and Guido Gherardi. Effective Choice and Boundedness Principles in Computable Analysis. In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 95-106, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brattka_et_al:OASIcs.CCA.2009.2262,
  author =	{Brattka, Vasco and Gherardi, Guido},
  title =	{{Effective Choice and Boundedness Principles in Computable Analysis}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{95--106},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Bauer, Andrej and Hertling, Peter and Ko, Ker-I},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2262},
  URN =		{urn:nbn:de:0030-drops-22629},
  doi =		{10.4230/OASIcs.CCA.2009.2262},
  annote =	{Keywords: Computable analysis, constructive analysis, reverse mathematics, effective descriptive set theory}
}
Document
DOI: 10.4230/DagSemRep.327
Computability and Complexity in Analysis (Dagstuhl Seminar 01461)

Authors: Vasco Brattka, Peter Hertling, Mariko Yasugi, and Ning Zhong

Published in: Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Abstract
Cite as

Vasco Brattka, Peter Hertling, Mariko Yasugi, and Ning Zhong. Computability and Complexity in Analysis (Dagstuhl Seminar 01461). Dagstuhl Seminar Report 327, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2002)

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@TechReport{brattka_et_al:DagSemRep.327,
  author =	{Brattka, Vasco and Hertling, Peter and Yasugi, Mariko and Zhong, Ning},
  title =	{{Computability and Complexity in Analysis (Dagstuhl Seminar 01461)}},
  pages =	{1--24},
  ISSN =	{1619-0203},
  year =	{2002},
  type = 	{Dagstuhl Seminar Report},
  number =	{327},
  institution =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.327},
  URN =		{urn:nbn:de:0030-drops-152103},
  doi =		{10.4230/DagSemRep.327},
}
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