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Documents authored by Braverman, Mark

Document
DOI: 10.4230/LIPIcs.ITCS.2023.25
Improved Monotonicity Testers via Hypercube Embeddings

Authors: Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We show improved monotonicity testers for the Boolean hypercube under the p-biased measure, as well as over the hypergrid [m]ⁿ. Our results are: 1) For any p ∈ (0,1), for the p-biased hypercube we show a non-adaptive tester that makes Õ(√n/ε²) queries, accepts monotone functions with probability 1 and rejects functions that are ε-far from monotone with probability at least 2/3. 2) For all m ∈ ℕ, we show an Õ(√nm³/ε²) query monotonicity tester over [m]ⁿ. We also establish corresponding directed isoperimetric inequalities in these domains, analogous to the isoperimetric inequality in [Subhash Khot et al., 2018]. Previously, the best known tester due to Black, Chakrabarty and Seshadhri [Hadley Black et al., 2018] had Ω(n^{5/6}) query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on m. Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension n into a function over a Boolean cube of a larger dimension n', while preserving its distance from being monotone; an embedding is considered efficient if n' is not much larger than n, and we show how to construct efficient embeddings in the above mentioned settings.
Cite as

Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved Monotonicity Testers via Hypercube Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 25:1-25:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2023.25,
  author =	{Braverman, Mark and Khot, Subhash and Kindler, Guy and Minzer, Dor},
  title =	{{Improved Monotonicity Testers via Hypercube Embeddings}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{25:1--25:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.25},
  URN =		{urn:nbn:de:0030-drops-175285},
  doi =		{10.4230/LIPIcs.ITCS.2023.25},
  annote =	{Keywords: Property Testing, Monotonicity Testing, Isoperimetric Inequalities}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.26
Rounding via Low Dimensional Embeddings

Authors: Mark Braverman and Dor Minzer

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
A regular graph G = (V,E) is an (ε,γ) small-set expander if for any set of vertices of fractional size at most ε, at least γ of the edges that are adjacent to it go outside. In this paper, we give a unified approach to several known complexity-theoretic results on small-set expanders. In particular, we show: 1) Max-Cut: we show that if a regular graph G = (V,E) is an (ε,γ) small-set expander that contains a cut of fractional size at least 1-δ, then one can find in G a cut of fractional size at least 1-O(δ/(εγ⁶)) in polynomial time. 2) Improved spectral partitioning, Cheeger’s inequality and the parallel repetition theorem over small-set expanders. The general form of each one of these results involves square-root loss that comes from certain rounding procedure, and we show how this can be avoided over small set expanders. Our main idea is to project a high dimensional vector solution into a low-dimensional space while roughly maintaining 𝓁₂² distances, and then perform a pre-processing step using low-dimensional geometry and the properties of 𝓁₂² distances over it. This pre-processing leverages the small-set expansion property of the graph to transform a vector valued solution to a different vector valued solution with additional structural properties, which give rise to more efficient integral-solution rounding schemes.
Cite as

Mark Braverman and Dor Minzer. Rounding via Low Dimensional Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 26:1-26:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2023.26,
  author =	{Braverman, Mark and Minzer, Dor},
  title =	{{Rounding via Low Dimensional Embeddings}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{26:1--26:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.26},
  URN =		{urn:nbn:de:0030-drops-175291},
  doi =		{10.4230/LIPIcs.ITCS.2023.26},
  annote =	{Keywords: Parallel Repetition, Small Set Expanders, Semi-Definite Programs}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.ITCS.2022
LIPIcs, Volume 215, ITCS 2022, Complete Volume

Authors: Mark Braverman

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
LIPIcs, Volume 215, ITCS 2022, Complete Volume
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13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 1-2410, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{braverman:LIPIcs.ITCS.2022,
  title =	{{LIPIcs, Volume 215, ITCS 2022, Complete Volume}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{1--2410},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022},
  URN =		{urn:nbn:de:0030-drops-155957},
  doi =		{10.4230/LIPIcs.ITCS.2022},
  annote =	{Keywords: LIPIcs, Volume 215, ITCS 2022, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.ITCS.2022.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Mark Braverman

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
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13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 0:i-0:xxiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{braverman:LIPIcs.ITCS.2022.0,
  author =	{Braverman, Mark},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{0:i--0:xxiv},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.0},
  URN =		{urn:nbn:de:0030-drops-155967},
  doi =		{10.4230/LIPIcs.ITCS.2022.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.5
Optimal Tiling of the Euclidean Space Using Permutation-Symmetric Bodies

Authors: Mark Braverman and Dor Minzer

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
What is the least surface area of a permutation-symmetric body B whose ℤⁿ translations tile ℝⁿ? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least Ω(√n). Remarkably, Kindler et al. showed that for general bodies B this is tight, i.e. that there is a tiling body of ℝⁿ whose surface area is O(√n). In theoretical computer science, the tiling problem is intimately related to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a "strong version" of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of ℝⁿ. In this paper, motivated by the study of a symmetric parallel repetition, we consider the permutation-symmetric variant of the tiling problem in ℝⁿ. We show that any permutation-symmetric body that tiles ℝⁿ must have surface area at least Ω(n/√{log n}), and that this bound is tight, i.e. that there is a permutation-symmetric tiling body of ℝⁿ with surface area O(n/√{log n}). We also give matching bounds for the value of the symmetric parallel repetition of Raz’s odd cycle game. Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies.
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Mark Braverman and Dor Minzer. Optimal Tiling of the Euclidean Space Using Permutation-Symmetric Bodies. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 5:1-5:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{braverman_et_al:LIPIcs.CCC.2021.5,
  author =	{Braverman, Mark and Minzer, Dor},
  title =	{{Optimal Tiling of the Euclidean Space Using Permutation-Symmetric Bodies}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{5:1--5:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.5},
  URN =		{urn:nbn:de:0030-drops-142796},
  doi =		{10.4230/LIPIcs.CCC.2021.5},
  annote =	{Keywords: PCP, Parallel Repetition, Tilings}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.27
On Rich 2-to-1 Games

Authors: Mark Braverman, Subhash Khot, and Dor Minzer

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We propose a variant of the 2-to-1 Games Conjecture that we call the Rich 2-to-1 Games Conjecture and show that it is equivalent to the Unique Games Conjecture. We are motivated by two considerations. Firstly, in light of the recent proof of the 2-to-1 Games Conjecture [Subhash Khot et al., 2017; Irit Dinur et al., 2018; Irit Dinur et al., 2018; Subhash Khot et al., 2018], we hope to understand how one might make further progress towards a proof of the Unique Games Conjecture. Secondly, the new variant along with perfect completeness in addition, might imply hardness of approximation results that necessarily require perfect completeness and (hence) are not implied by the Unique Games Conjecture.
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Mark Braverman, Subhash Khot, and Dor Minzer. On Rich 2-to-1 Games. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2021.27,
  author =	{Braverman, Mark and Khot, Subhash and Minzer, Dor},
  title =	{{On Rich 2-to-1 Games}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{27:1--27:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.27},
  URN =		{urn:nbn:de:0030-drops-135666},
  doi =		{10.4230/LIPIcs.ITCS.2021.27},
  annote =	{Keywords: PCP, Unique-Games, Perfect Completeness}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.46
Tiered Random Matching Markets: Rank Is Proportional to Popularity

Authors: Itai Ashlagi, Mark Braverman, Amin Saberi, Clayton Thomas, and Geng Zhao

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study the stable marriage problem in two-sided markets with randomly generated preferences. Agents on each side of the market are divided into a constant number of "soft" tiers, which capture agents' qualities. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the man-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes the results by Pittel [Pittel, 1989], which analyzed markets with uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market.
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Itai Ashlagi, Mark Braverman, Amin Saberi, Clayton Thomas, and Geng Zhao. Tiered Random Matching Markets: Rank Is Proportional to Popularity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ashlagi_et_al:LIPIcs.ITCS.2021.46,
  author =	{Ashlagi, Itai and Braverman, Mark and Saberi, Amin and Thomas, Clayton and Zhao, Geng},
  title =	{{Tiered Random Matching Markets: Rank Is Proportional to Popularity}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{46:1--46:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.46},
  URN =		{urn:nbn:de:0030-drops-135851},
  doi =		{10.4230/LIPIcs.ITCS.2021.46},
  annote =	{Keywords: Stable matching, stable marriage problem, tiered random markets, deferred acceptance}
}
Document
DOI: 10.4230/LIPIcs.FORC.2020.9
The Role of Randomness and Noise in Strategic Classification

Authors: Mark Braverman and Sumegha Garg

Published in: LIPIcs, Volume 156, 1st Symposium on Foundations of Responsible Computing (FORC 2020)

Abstract
We investigate the problem of designing optimal classifiers in the "strategic classification" setting, where the classification is part of a game in which players can modify their features to attain a favorable classification outcome (while incurring some cost). Previously, the problem has been considered from a learning-theoretic perspective and from the algorithmic fairness perspective. Our main contributions include - Showing that if the objective is to maximize the efficiency of the classification process (defined as the accuracy of the outcome minus the sunk cost of the qualified players manipulating their features to gain a better outcome), then using randomized classifiers (that is, ones where the probability of a given feature vector to be accepted by the classifier is strictly between 0 and 1) is necessary. - Showing that in many natural cases, the imposed optimal solution (in terms of efficiency) has the structure where players never change their feature vectors (and the randomized classifier is structured in a way, such that the gain in the probability of being classified as a "1" does not justify the expense of changing one’s features). - Observing that the randomized classification is not a stable best-response from the classifier’s viewpoint, and that the classifier doesn’t benefit from randomized classifiers without creating instability in the system. - Showing that in some cases, a noisier signal leads to better equilibria outcomes - improving both accuracy and fairness when more than one subpopulation with different feature adjustment costs are involved. This is particularly interesting from a policy perspective, since it is hard to force institutions to stick to a particular randomized classification strategy (especially in a context of a market with multiple classifiers), but it is possible to alter the information environment to make the feature signals inherently noisier.
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Mark Braverman and Sumegha Garg. The Role of Randomness and Noise in Strategic Classification. In 1st Symposium on Foundations of Responsible Computing (FORC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 156, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{braverman_et_al:LIPIcs.FORC.2020.9,
  author =	{Braverman, Mark and Garg, Sumegha},
  title =	{{The Role of Randomness and Noise in Strategic Classification}},
  booktitle =	{1st Symposium on Foundations of Responsible Computing (FORC 2020)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-142-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{156},
  editor =	{Roth, Aaron},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2020.9},
  URN =		{urn:nbn:de:0030-drops-120255},
  doi =		{10.4230/LIPIcs.FORC.2020.9},
  annote =	{Keywords: Strategic classification, noisy features, randomized classification, fairness}
}
Document
DOI: 10.4230/LIPIcs.DISC.2019.8
On the Computational Power of Radio Channels

Authors: Mark Braverman, Gillat Kol, Rotem Oshman, and Avishay Tal

Published in: LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

Abstract
Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes' inputs. Our techniques are general, and apply to many types of radio channels studied in the literature.
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Mark Braverman, Gillat Kol, Rotem Oshman, and Avishay Tal. On the Computational Power of Radio Channels. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{braverman_et_al:LIPIcs.DISC.2019.8,
  author =	{Braverman, Mark and Kol, Gillat and Oshman, Rotem and Tal, Avishay},
  title =	{{On the Computational Power of Radio Channels}},
  booktitle =	{33rd International Symposium on Distributed Computing (DISC 2019)},
  pages =	{8:1--8:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-126-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{146},
  editor =	{Suomela, Jukka},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.8},
  URN =		{urn:nbn:de:0030-drops-113152},
  doi =		{10.4230/LIPIcs.DISC.2019.8},
  annote =	{Keywords: radio channel, lower bounds, approximate majority}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.10
Optimal Short-Circuit Resilient Formulas

Authors: Mark Braverman, Klim Efremenko, Ran Gelles, and Michael A. Yitayew

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract
We consider fault-tolerant boolean formulas in which the output of a faulty gate is short-circuited to one of the gate’s inputs. A recent result by Kalai et al. [FOCS 2012] converts any boolean formula into a resilient formula of polynomial size that works correctly if less than a fraction 1/6 of the gates (on every input-to-output path) are faulty. We improve the result of Kalai et al., and show how to efficiently fortify any boolean formula against a fraction 1/5 of short-circuit gates per path, with only a polynomial blowup in size. We additionally show that it is impossible to obtain formulas with higher resilience and sub-exponential growth in size. Towards our results, we consider interactive coding schemes when noiseless feedback is present; these produce resilient boolean formulas via a Karchmer-Wigderson relation. We develop a coding scheme that resists up to a fraction 1/5 of corrupted transmissions in each direction of the interactive channel. We further show that such a level of noise is maximal for coding schemes with sub-exponential blowup in communication. Our coding scheme takes a surprising inspiration from Blockchain technology.
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Mark Braverman, Klim Efremenko, Ran Gelles, and Michael A. Yitayew. Optimal Short-Circuit Resilient Formulas. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{braverman_et_al:LIPIcs.CCC.2019.10,
  author =	{Braverman, Mark and Efremenko, Klim and Gelles, Ran and Yitayew, Michael A.},
  title =	{{Optimal Short-Circuit Resilient Formulas}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{10:1--10:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.10},
  URN =		{urn:nbn:de:0030-drops-108326},
  doi =		{10.4230/LIPIcs.CCC.2019.10},
  annote =	{Keywords: Circuit Complexity, Noise-Resilient Circuits, Interactive Coding, Coding Theory, Karchmer-Wigderson Games}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.6
Semi-Direct Sum Theorem and Nearest Neighbor under l_infty

Authors: Mark Braverman and Young Kun Ko

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
We introduce semi-direct sum theorem as a framework for proving asymmetric communication lower bounds for the functions of the form V_{i=1}^n f(x,y_i). Utilizing tools developed in proving direct sum theorem for information complexity, we show that if the function is of the form V_{i=1}^n f(x,y_i) where Alice is given x and Bob is given y_i's, it suffices to prove a lower bound for a single f(x,y_i). This opens a new avenue of attack other than the conventional combinatorial technique (i.e. "richness lemma" from [Miltersen et al., 1995]) for proving randomized lower bounds for asymmetric communication for functions of such form. As the main technical result and an application of semi-direct sum framework, we prove an information lower bound on c-approximate Nearest Neighbor (ANN) under l_infty which implies that the algorithm of [Indyk, 2001] for c-approximate Nearest Neighbor under l_infty is optimal even under randomization for both decision tree and cell probe data structure model (under certain parameter assumption for the latter). In particular, this shows that randomization cannot improve [Indyk, 2001] under decision tree model. Previously only a deterministic lower bound was known by [Andoni et al., 2008] and randomized lower bound for cell probe model by [Kapralov and Panigrahy, 2012]. We suspect further applications of our framework in exhibiting randomized asymmetric communication lower bounds for big data applications.
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Mark Braverman and Young Kun Ko. Semi-Direct Sum Theorem and Nearest Neighbor under l_infty. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2018.6,
  author =	{Braverman, Mark and Ko, Young Kun},
  title =	{{Semi-Direct Sum Theorem and Nearest Neighbor under l\underlineinfty}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.6},
  URN =		{urn:nbn:de:0030-drops-94101},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.6},
  annote =	{Keywords: Asymmetric Communication Lower Bound, Data Structure Lower Bound, Nearest Neighbor Search}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.11
A Candidate for a Strong Separation of Information and Communication

Authors: Mark Braverman, Anat Ganor, Gillat Kol, and Ran Raz

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
The weak interactive compression conjecture asserts that any two-party communication protocol with communication complexity C and information complexity I can be compressed to a protocol with communication complexity poly(I)polylog(C). We describe a communication problem that is a candidate for refuting that conjecture. Specifically, while we show that the problem can be solved by a protocol with communication complexity C and information complexity I=polylog(C), the problem seems to be hard for protocols with communication complexity poly(I)polylog(C)=polylog(C).
Cite as

Mark Braverman, Anat Ganor, Gillat Kol, and Ran Raz. A Candidate for a Strong Separation of Information and Communication. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2018.11,
  author =	{Braverman, Mark and Ganor, Anat and Kol, Gillat and Raz, Ran},
  title =	{{A Candidate for a Strong Separation of Information and Communication}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{11:1--11:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.11},
  URN =		{urn:nbn:de:0030-drops-83322},
  doi =		{10.4230/LIPIcs.ITCS.2018.11},
  annote =	{Keywords: communication complexity, amortized communication complexity, communication compression, direct sum, information complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.12
Information Value of Two-Prover Games

Authors: Mark Braverman and Young Kun Ko

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1 - \epsilon)-probability by communicating at most \epsilon bits of information, then one can win n copies with probability at least 2^{-O(\epsilon n)}. This gives an intuitive explanation why Raz's counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length $m$ by communicating O(m^{-2})-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1-\epsilon)^n, then one can win one copy with at least (1 - O(\epsilon))-probability by communicating O(\epsilon) bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker's Inequality.
Cite as

Mark Braverman and Young Kun Ko. Information Value of Two-Prover Games. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2018.12,
  author =	{Braverman, Mark and Ko, Young Kun},
  title =	{{Information Value of Two-Prover Games}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.12},
  URN =		{urn:nbn:de:0030-drops-83466},
  doi =		{10.4230/LIPIcs.ITCS.2018.12},
  annote =	{Keywords: Two Prover Game, Parallel Repetition, Odd-Cycle Game, Amortized Value of the Game}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.18
Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All

Authors: Mark Braverman, Sumegha Garg, and Ariel Schvartzman

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured that the answer is no. In this paper we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap. We prove that any undirected network with k source-sink pairs that exhibits a (1+epsilon) gap between its MCF rate and its network coding rate can be used to construct a family of graphs G' whose gap is log(|G'|)^c for some constant c < 1. The resulting gap is close to the best currently known upper bound, log(|G'|), which follows from the connection between MCF and sparsest cuts. Our construction relies on a gap-amplifying graph tensor product that, given two graphs G1,G2 with small gaps, creates another graph G with a gap that is equal to the product of the previous two, at the cost of increasing the size of the graph. We iterate this process to obtain a gap of log(|G'|)^c from any initial gap.
Cite as

Mark Braverman, Sumegha Garg, and Ariel Schvartzman. Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2017.18,
  author =	{Braverman, Mark and Garg, Sumegha and Schvartzman, Ariel},
  title =	{{Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.18},
  URN =		{urn:nbn:de:0030-drops-81599},
  doi =		{10.4230/LIPIcs.ITCS.2017.18},
  annote =	{Keywords: Network coding, Gap Amplification, Multicommodity flows}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.61
Coding for Interactive Communication Correcting Insertions and Deletions

Authors: Mark Braverman, Ran Gelles, Jieming Mao, and Rafail Ostrovsky

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol. In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and tolerates noise rates of up to 1/18 - epsilon. To this end we develop a novel primitive we name edit distance tree code. The edit distance tree code is designed to replace the Hamming distance constraints in Schulman's tree codes (STOC 93), with a stronger edit distance requirement. However, the straightforward generalization of tree codes to edit distance does not seem to yield a primitive that suffices for communication in the presence of synchronization problems. Giving the "right" definition of edit distance tree codes is a main conceptual contribution of this work.
Cite as

Mark Braverman, Ran Gelles, Jieming Mao, and Rafail Ostrovsky. Coding for Interactive Communication Correcting Insertions and Deletions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 61:1-61:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{braverman_et_al:LIPIcs.ICALP.2016.61,
  author =	{Braverman, Mark and Gelles, Ran and Mao, Jieming and Ostrovsky, Rafail},
  title =	{{Coding for Interactive Communication Correcting Insertions and Deletions}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{61:1--61:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.61},
  URN =		{urn:nbn:de:0030-drops-61981},
  doi =		{10.4230/LIPIcs.ICALP.2016.61},
  annote =	{Keywords: Interactive communication, coding, edit distance}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.87
Information Complexity Is Computable

Authors: Mark Braverman and Jon Schneider

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
The information complexity of a function f is the minimum amount of information Alice and Bob need to exchange to compute the function f. In this paper we provide an algorithm for approximating the information complexity of an arbitrary function f to within any additive error epsilon > 0, thus resolving an open question as to whether information complexity is computable. In the process, we give the first explicit upper bound on the rate of convergence of the information complexity of f when restricted to b-bit protocols to the (unrestricted) information complexity of f.
Cite as

Mark Braverman and Jon Schneider. Information Complexity Is Computable. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 87:1-87:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{braverman_et_al:LIPIcs.ICALP.2016.87,
  author =	{Braverman, Mark and Schneider, Jon},
  title =	{{Information Complexity Is Computable}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{87:1--87:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.87},
  URN =		{urn:nbn:de:0030-drops-62203},
  doi =		{10.4230/LIPIcs.ICALP.2016.87},
  annote =	{Keywords: Communication complexity, convergence rate, information complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.610
Search using queries on indistinguishable items

Authors: Mark Braverman and Gal Oshri

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
We investigate the problem of determining a set S of k indistinguishable integers in the range [1, n]. The algorithm is allowed to query an integer q \in [1,n], and receive a response comparing this integer to an integer randomly chosen from S. The algorithm has no control over which element of S the query q is compared to. We show tight bounds for this problem. In particular, we show that in the natural regime where k <= n, the optimal number of queries to attain n^{-Omega(1)} error probability is Theta(k^3 log n). In the regime where k > n, the optimal number of queries is Theta(n^2 k log n). Our main technical tools include the use of information theory to derive the lower bounds, and the application of noisy binary search in the spirit of Feige, Raghavan, Peleg, and Upfal (1994). In particular, our lower bound technique is likely to be applicable in other situations that involve search under uncertainty.
Cite as

Mark Braverman and Gal Oshri. Search using queries on indistinguishable items. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 610-621, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{braverman_et_al:LIPIcs.STACS.2013.610,
  author =	{Braverman, Mark and Oshri, Gal},
  title =	{{Search using queries on indistinguishable items}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{610--621},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.610},
  URN =		{urn:nbn:de:0030-drops-39696},
  doi =		{10.4230/LIPIcs.STACS.2013.610},
  annote =	{Keywords: Search, Noisy Search, Information Theory, Query Complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2009.2311
Fractional Pebbling and Thrifty Branching Programs

Authors: Mark Braverman, Stephen Cook, Pierre McKenzie, Rahul Santhanam, and Dustin Wehr

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
We study the branching program complexity of the {\em tree evaluation problem}, introduced in \cite{BrCoMcSaWe09} as a candidate for separating \nl\ from\logcfl. The input to the problem is a rooted, balanced $d$-ary tree of height$h$, whose internal nodes are labelled with $d$-ary functions on$[k]=\{1,\ldots,k\}$, and whose leaves are labelled with elements of $[k]$.Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root. Deterministic $k$-way branching programs as related to black pebbling algorithms have been studied in \cite{BrCoMcSaWe09}. Here we introduce the notion of {\em fractional pebbling} of graphs to study non-deterministicbranching program size. We prove that this yields non-deterministic branching programs with $\Theta(k^{h/2+1})$ states solving the Boolean problem ``determine whether the root has value 1'' for binary trees - this isasymptotically better than the branching program size corresponding toblack-white pebbling. We prove upper and lower bounds on the fractionalpebbling number of $d$-ary trees, as well as a general result relating thefractional pebbling number of a graph to the black-white pebbling number. We introduce a simple semantic restriction called {\em thrifty} on $k$-way branching programs solving tree evaluation problems and show that the branchingprogram size bound of $\Theta(k^h)$ is tight (up to a constant factor) for all $h\ge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are thrifty as well, and that the bound of $\Theta(k^{h/2+1})$ is tight for non-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty branching programs are optimal among $k$-way branching programs solving the tree evaluation problem - proving this for deterministic programs would separate \lspace\ from \logcfl\, and proving it for non-deterministic programs would separate \nl\ from \logcfl.
Cite as

Mark Braverman, Stephen Cook, Pierre McKenzie, Rahul Santhanam, and Dustin Wehr. Fractional Pebbling and Thrifty Branching Programs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 109-120, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{braverman_et_al:LIPIcs.FSTTCS.2009.2311,
  author =	{Braverman, Mark and Cook, Stephen and McKenzie, Pierre and Santhanam, Rahul and Wehr, Dustin},
  title =	{{Fractional Pebbling and Thrifty Branching Programs}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{109--120},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2311},
  URN =		{urn:nbn:de:0030-drops-23111},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2311},
  annote =	{Keywords: Branching programs, space complexity, tree evaluation, pebbling}
}
Document
Invited Talk
DOI: 10.4230/OASIcs.CCA.2009.2250
Computability and Complexity of Julia Sets (Invited Talk)

Authors: Mark Braverman

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

Abstract
Studying dynamical systems is key to understanding a wide range of phenomena ranging from planetary movement to climate patterns to market dynamics. Various numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite. However, the theory of computation behind these problems appears to be very difficult to develop. In fact, little is known about computability of even the most natural problems arising from dynamical systems. In this talk I will survey the recent study of the computational properties of dynamical systems that arise from iterating quadratic polynomials on the complex plane. These give rise to the amazing variety of fractals known as Julia sets, and are closely connected to the Mandelbrot set. Julia sets are perhaps the most drawn objects in Mathematics due to their fascinating fractal structure. The theory behind them is even more fascinating, and the dynamical systems generating them are in many ways archetypal. I will present both positive and negative results on the computability and complexity of Julia sets. In conclusion of the talk I will discuss possible future directions and challenges in the study of the computability and complexity of dynamical systems.
Cite as

Mark Braverman. Computability and Complexity of Julia Sets (Invited Talk). In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, p. 3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{braverman:OASIcs.CCA.2009.2250,
  author =	{Braverman, Mark},
  title =	{{Computability and Complexity of Julia Sets}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{3--3},
  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-dev.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2250},
  URN =		{urn:nbn:de:0030-drops-22508},
  doi =		{10.4230/OASIcs.CCA.2009.2250},
  annote =	{Keywords: Computability, computable analysis, dynamical systems, complex dynamics, Julia sets Computability, computable analysis, dynamical systems, complex dynamics, Julia sets}
}
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