4 Search Results for "Kötzing, Timo"


Document
The Minimization of Random Hypergraphs

Authors: Thomas Bläsius, Tobias Friedrich, and Martin Schirneck

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
We investigate the maximum-entropy model B_{n,m,p} for random n-vertex, m-edge multi-hypergraphs with expected edge size pn. We show that the expected size of the minimization min(B_{n,m,p}), i.e., the number of inclusion-wise minimal edges of B_{n,m,p}, undergoes a phase transition with respect to m. If m is at most 1/(1-p)^{(1-p)n}, then E[|min(B_{n,m,p})|] is of order Θ(m), while for m ≥ 1/(1-p)^{(1-p+ε)n} for any ε > 0, it is Θ(2^{(H(α) + (1-α) log₂ p) n}/√n). Here, H denotes the binary entropy function and α = - (log_{1-p} m)/n. The result implies that the maximum expected number of minimal edges over all m is Θ((1+p)ⁿ/√n). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff-Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable Y ∼ Bin(n,p) and any 0 < x < p, it holds that P[Y ≤ xn] = Θ(2^{-D(x‖p) n}/√n), where D is the binary Kullback-Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all n. Secondly, we establish the fact that the probability of a set of cardinality i being minimal after m i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at i^* = n + log_{1-p} m.

Cite as

Thomas Bläsius, Tobias Friedrich, and Martin Schirneck. The Minimization of Random Hypergraphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{blasius_et_al:LIPIcs.ESA.2020.21,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Schirneck, Martin},
  title =	{{The Minimization of Random Hypergraphs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.21},
  URN =		{urn:nbn:de:0030-drops-128871},
  doi =		{10.4230/LIPIcs.ESA.2020.21},
  annote =	{Keywords: Chernoff-Hoeffding theorem, maximum entropy, maximization, minimization, phase transition, random hypergraphs}
}
Document
Random Subgroups of Rationals

Authors: Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence beta such that the word problem for G with respect to beta is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence beta for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

Cite as

Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan. Random Subgroups of Rationals. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{gao_et_al:LIPIcs.MFCS.2019.25,
  author =	{Gao, Ziyuan and Jain, Sanjay and Khoussainov, Bakhadyr and Li, Wei and Melnikov, Alexander and Seidel, Karen and Stephan, Frank},
  title =	{{Random Subgroups of Rationals}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.25},
  URN =		{urn:nbn:de:0030-drops-109693},
  doi =		{10.4230/LIPIcs.MFCS.2019.25},
  annote =	{Keywords: Martin-L\"{o}f randomness, subgroups of rationals, finitely generated subgroups of rationals, learning in the limit, behaviourally correct learning}
}
Document
Towards an Atlas of Computational Learning Theory

Authors: Timo Kötzing and Martin Schirneck

Published in: LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)


Abstract
A major part of our knowledge about Computational Learning stems from comparisons of the learning power of different learning criteria. These comparisons inform about trade-offs between learning restrictions and, more generally, learning settings; furthermore, they inform about what restrictions can be observed without losing learning power. With this paper we propose that one main focus of future research in Computational Learning should be on a structured approach to determine the relations of different learning criteria. In particular, we propose that, for small sets of learning criteria, all pairwise relations should be determined; these relations can then be easily depicted as a map, a diagram detailing the relations. Once we have maps for many relevant sets of learning criteria, the collection of these maps is an Atlas of Computational Learning Theory, informing at a glance about the landscape of computational learning just as a geographical atlas informs about the earth. In this paper we work toward this goal by providing three example maps, one pertaining to partially set-driven learning, and two pertaining to strongly monotone learning. These maps can serve as blueprints for future maps of similar base structure.

Cite as

Timo Kötzing and Martin Schirneck. Towards an Atlas of Computational Learning Theory. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{kotzing_et_al:LIPIcs.STACS.2016.47,
  author =	{K\"{o}tzing, Timo and Schirneck, Martin},
  title =	{{Towards an Atlas of Computational Learning Theory}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{47:1--47:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Ollinger, Nicolas and Vollmer, Heribert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.47},
  URN =		{urn:nbn:de:0030-drops-57483},
  doi =		{10.4230/LIPIcs.STACS.2016.47},
  annote =	{Keywords: computational learning, language learning, partially set-driven learning, strongly monotone learning}
}
Document
A Solution to Wiehagen's Thesis

Authors: Timo Kötzing

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)


Abstract
Wiehagen's Thesis in Inductive Inference (1991) essentially states that, for each learning criterion, learning can be done in a normalized, enumerative way. The thesis was not a formal statement and thus did not allow for a formal proof, but support was given by examples of a number of different learning criteria that can be learned enumeratively. Building on recent formalizations of learning criteria, we are now able to formalize Wiehagen's Thesis. We prove the thesis for a wide range of learning criteria, including many popular criteria from the literature. We also show the limitations of the thesis by giving four learning criteria for which the thesis does not hold (and, in two cases, was probably not meant to hold). Beyond the original formulation of the thesis, we also prove stronger versions which allow for many corollaries relating to strongly decisive and conservative learning.

Cite as

Timo Kötzing. A Solution to Wiehagen's Thesis. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 494-505, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Copy BibTex To Clipboard

@InProceedings{kotzing:LIPIcs.STACS.2014.494,
  author =	{K\"{o}tzing, Timo},
  title =	{{A Solution to Wiehagen's Thesis}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{494--505},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.494},
  URN =		{urn:nbn:de:0030-drops-44823},
  doi =		{10.4230/LIPIcs.STACS.2014.494},
  annote =	{Keywords: Algorithmic Learning Theory, Wiehagen's Thesis, Enumeration Learning}
}
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