5 Search Results for "Venkitesh, S."


Document
RANDOM
On the Probabilistic Degree of an n-Variate Boolean Function

Authors: Srikanth Srinivasan and S. Venkitesh

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}ⁿ → {0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x₁,…,x_n), has degree at least log n - O(log log n). This was improved to a tight (log n - O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)^{1/2-o(1)} and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)^c, then the minimum possible probabilistic degree of such an f is at least (log n)^{c/(c+1)-o(1)}, and we show this is tight up to (log n)^{o(1)} factors.

Cite as

Srikanth Srinivasan and S. Venkitesh. On the Probabilistic Degree of an n-Variate Boolean Function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 42:1-42:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{srinivasan_et_al:LIPIcs.APPROX/RANDOM.2021.42,
  author =	{Srinivasan, Srikanth and Venkitesh, S.},
  title =	{{On the Probabilistic Degree of an n-Variate Boolean Function}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{42:1--42:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.42},
  URN =		{urn:nbn:de:0030-drops-147356},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.42},
  annote =	{Keywords: truly n-variate, Boolean function, probabilistic polynomial, probabilistic degree}
}
Document
More on AC^0[oplus] and Variants of the Majority Function

Authors: Nutan Limaye, Srikanth Srinivasan, and Utkarsh Tripathi

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)


Abstract
In this paper we prove two results about AC^0[oplus] circuits. (1) We show that for d(N) = o(sqrt(log N/log log N)) and N <= s(N) <= 2^(dN^(1/4d^2)) there is an explicit family of functions {f_N:{0,1}^N - > {0,1}} such that - f_N has uniform AC^0 formulas of depth d and size at most s; - f_N does not have AC^0[oplus] formulas of depth d and size s^epsilon, where epsilon is a fixed absolute constant. This gives a quantitative improvement on the recent result of Limaye, Srinivasan, Sreenivasaiah, Tripathi, and Venkitesh, (STOC, 2019), which proved a similar Fixed-Depth Size-Hierarchy theorem but for d << log log N and s << exp(N^(1/2^Omega(d))). As in the previous result, we use the Coin Problem to prove our hierarchy theorem. Our main technical result is the construction of uniform size-optimal formulas for solving the coin problem with improved sample complexity (1/delta)^O(d) (down from (1/delta)^(2^O(d)) in the previous result). (2) In our second result, we show that randomness buys depth in the AC^0[oplus] setting. Formally, we show that for any fixed constant d >= 2, there is a family of Boolean functions that has polynomial-sized randomized uniform AC^0 circuits of depth d but no polynomial-sized (deterministic) AC^0[oplus] circuits of depth d. Previously Viola (Computational Complexity, 2014) showed that an increase in depth (by at least 2) is essential to avoid superpolynomial blow-up while derandomizing randomized AC^0 circuits. We show that an increase in depth (by at least 1) is essential even for AC^0[oplus]. As in Viola’s result, the separating examples are promise variants of the Majority function on N inputs that accept inputs of weight at least N/2 + N/(log N)^(d-1) and reject inputs of weight at most N/2 - N/(log N)^(d-1).

Cite as

Nutan Limaye, Srikanth Srinivasan, and Utkarsh Tripathi. More on AC^0[oplus] and Variants of the Majority Function. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 22:1-22:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{limaye_et_al:LIPIcs.FSTTCS.2019.22,
  author =	{Limaye, Nutan and Srinivasan, Srikanth and Tripathi, Utkarsh},
  title =	{{More on AC^0\lbrackoplus\rbrack and Variants of the Majority Function}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.22},
  URN =		{urn:nbn:de:0030-drops-115844},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.22},
  annote =	{Keywords: AC^0\lbrackoplus\rbrack, Coin Problem, Promise Majority}
}
Document
On the Probabilistic Degrees of Symmetric Boolean Functions

Authors: Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)


Abstract
The probabilistic degree of a Boolean function f:{0,1}^n -> {0,1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions - specifically symmetric Boolean functions - have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems. In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

Cite as

Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. On the Probabilistic Degrees of Symmetric Boolean Functions. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 28:1-28:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{srinivasan_et_al:LIPIcs.FSTTCS.2019.28,
  author =	{Srinivasan, Srikanth and Tripathi, Utkarsh and Venkitesh, S.},
  title =	{{On the Probabilistic Degrees of Symmetric Boolean Functions}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{28:1--28:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.28},
  URN =		{urn:nbn:de:0030-drops-115908},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.28},
  annote =	{Keywords: Symmetric Boolean function, probabilistic degree}
}
Document
Parity Helps to Compute Majority

Authors: Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan

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


Abstract
We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades. Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.

Cite as

Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity Helps to Compute Majority. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.23,
  author =	{Oliveira, Igor Carboni and Santhanam, Rahul and Srinivasan, Srikanth},
  title =	{{Parity Helps to Compute Majority}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{23:1--23:17},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.23},
  URN =		{urn:nbn:de:0030-drops-108453},
  doi =		{10.4230/LIPIcs.CCC.2019.23},
  annote =	{Keywords: Computational Complexity, Boolean Circuits, Lower Bounds, Parity, Majority}
}
Document
Fourier Bounds and Pseudorandom Generators for Product Tests

Authors: Chin Ho Lee

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


Abstract
We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. We also extend our results to functions f_i whose range is [-1,1].

Cite as

Chin Ho Lee. Fourier Bounds and Pseudorandom Generators for Product Tests. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{lee:LIPIcs.CCC.2019.7,
  author =	{Lee, Chin Ho},
  title =	{{Fourier Bounds and Pseudorandom Generators for Product Tests}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{7:1--7:25},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.7},
  URN =		{urn:nbn:de:0030-drops-108296},
  doi =		{10.4230/LIPIcs.CCC.2019.7},
  annote =	{Keywords: bounded independence plus noise, Fourier spectrum, product test, pseudorandom generators}
}
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