3 Search Results for "McGuire, Sam"

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DOI: 10.4230/LIPIcs.CCC.2024.9

Lifting Dichotomies

Authors: Yaroslav Alekseev, Yuval Filmus, and Alexander Smal

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].

Cite as

Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
  title =	{{Lifting Dichotomies}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.9},
  URN =		{urn:nbn:de:0030-drops-204051},
  doi =		{10.4230/LIPIcs.CCC.2024.9},
  annote =	{Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.82

Refuting Approaches to the Log-Rank Conjecture for XOR Functions

Authors: Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang, Wong, Xie, and Zhang (FOCS'13), and the second is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Cite as

Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 82:1-82:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hatami_et_al:LIPIcs.ICALP.2024.82,
  author =	{Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar and Ostuni, Anthony},
  title =	{{Refuting Approaches to the Log-Rank Conjecture for XOR Functions}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{82:1--82:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.82},
  URN =		{urn:nbn:de:0030-drops-202252},
  doi =		{10.4230/LIPIcs.ICALP.2024.82},
  annote =	{Keywords: Communication complexity, log-rank conjecture, XOR functions, additive structure}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.2

Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)

Authors: Russell Impagliazzo and Sam McGuire

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

Abstract

Computational pseudorandomness studies the extent to which a random variable Z looks like the uniform distribution according to a class of tests ℱ. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a high entropy distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class ℱ is closed under taking majorities. This equivalence constitutes (essentially) the so-called dense model theorem of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao’s proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where ℱ is not closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is false.

Cite as

Russell Impagliazzo and Sam McGuire. Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{impagliazzo_et_al:LIPIcs.ITCS.2021.2,
  author =	{Impagliazzo, Russell and McGuire, Sam},
  title =	{{Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{2:1--2:20},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.2},
  URN =		{urn:nbn:de:0030-drops-135417},
  doi =		{10.4230/LIPIcs.ITCS.2021.2},
  annote =	{Keywords: Computational entropy, dense model theorem, coin problem}
}

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1 Alekseev, Yaroslav
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2 Theory of computation → Communication complexity
1 Theory of computation → Oracles and decision trees
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