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Lower bounds on threshold and related circuits via communication complexity

1994
*
IEEE Transactions on Information Theory
*

Each of the bounds derived is shown to be tight for some functions and some applications t o

doi:10.1109/18.312169
fatcat:a4ydrluivbhepmjpm4rtolxaqi
*threshold*-*circuit*complexity are indicated. ... Communication-complexity definitions and arguments are used to derive linear (Q(n)) and almost-linear (Q(n/ log n)) lower bounds on the size of*circuits*implementing certain functions. ...*depth*-3*threshold*-*circuit*complexity. ...##
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Satisfiability and Derandomization for Small Polynomial Threshold Circuits

2018
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International Workshop on Approximation Algorithms for Combinatorial Optimization
*

This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for

doi:10.4230/lipics.approx-random.2018.46
dblp:conf/approx/KabanetsL18
fatcat:ch4gewg3ybelzjjwutioqml6yq
*constant*-*depth**circuits*of super-linear wire complexity with linear*threshold*function (LTF) gates ... We study the problems of exact and (promise) approximate counting for PTF*circuits*of*constant**depth*. Satisfiability (#SAT ). ...*Threshold**circuits*. It is well known that the class of*constant*-*depth*polynomial-size TC 0*circuits*is equivalent to the class of*constant*-*depth*polynomial-size*circuits*with LTF gates [7] . ...##
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Page 3865 of Mathematical Reviews Vol. , Issue 98F
[page]

1998
*
Mathematical Reviews
*

While obtaining lower bounds for

*constant*-*depth**circuits*with*threshold*gates is an extremely important and much-studied prob- lem, it has proved to be extremely difficult to get very far with the general ... Secondly, any*depth*-2 AC“)*circuit*can be computed by polynomial-size*threshold*-MOD?’*circuits*. ...##
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Luby--Veličković--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits
[article]

2018
*
arXiv
*
pre-print

ε-

arXiv:1803.04553v1
fatcat:ynglyw7775fs5ho4mlu5rbrnwy
*fools*these*circuits*. ... The above PRG is actually a special case of a more general PRG which we establish for*constant*-*depth**circuits*containing multiple SYM or THR gates, including as a special case {SYM,THR}∘AC^0*circuits*. ... For some sufficiently small absolute*constant*c > 0, there is a PRG with seed length 2 O( √ log S) + polylog(1/ε) that ε-*fools*the class of size-S*constant*-*depth**circuits*that contain 2 √ c log S many ...##
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Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits

2018
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International Workshop on Approximation Algorithms for Combinatorial Optimization
*

ε-

doi:10.4230/lipics.approx-random.2018.56
dblp:conf/approx/ServedioT18
fatcat:knjpbzpf3zg2zpel4wrheeaqe4
*fools*these*circuits*. ... The above PRG is actually a special case of a more general PRG which we establish for*constant*-*depth**circuits*containing multiple SYM or THR gates, including as a special case {SYM, THR} • AC 0*circuits*... For some sufficiently small absolute*constant*c > 0, there is a PRG with seed length 2 O( √ log S) + polylog(1/ε) that ε-*fools*the class of size-S*constant*-*depth**circuits*that contain 2 √ c log S many ...##
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Bounded Indistinguishability and the Complexity of Recovering Secrets
[chapter]

2016
*
Lecture Notes in Computer Science
*

*circuits*of size poly(n). ... parties so that any set of fewer than σn parties can learn nothing about the secret, any set of at least ρn parties can reconstruct the secret, and where both the sharing and the reconstruction are done by

*constant*-

*depth*... Emanuele thanks Daniel Wichs for asking whether bounded indistinguishability

*fools*AND, and Mark Bun and Justin Thaler for many discussions about the approximate degree literature. ...

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Some Limitations of the Sum of Small-Bias Distributions

2017
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Theory of Computing
*

We present two approaches to constructing ε-biased distributions D on n bits and functions f : {0, 1} n → {0, 1} such that the XOR of two independent copies (D + D) does not

doi:10.4086/toc.2017.v013a016
dblp:journals/toc/LeeV17
fatcat:cqpqfsv4ezb67mdb6gegnksy3q
*fool*f . ... Suppose for every δ > 0 there exists a*constant*d such that NC 2 has AC 0*circuits*of size 2 n δ and*depth*d. ...*Depth*3*circuits*, DNF formulas and AC 0*circuits*Proof of Theorem 1.5. ...##
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On derandomizing algorithms that err extremely rarely

2014
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Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
*

We also obtain results for other classes of computational devices including log-space algorithms and Arithmetic

doi:10.1145/2591796.2591808
dblp:conf/stoc/GoldreichW14
fatcat:g25g7hytyrfidh4cjipix6owfy
*circuits*. ... In relation to the above question, we put forward the following quantified derandomization challenge: For a class of*circuits*C (e.g., P/poly or AC 0 , AC 0 [2]) and a bounding function B : N → N (e.g. ...*Constant*-*depth**circuits*(AC 0 ). ...##
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Pseudorandom Generators for Read-Once ACC^0

2012
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2012 IEEE 27th Conference on Computational Complexity
*

We give an explicit construction of a pseudorandom generator for the class of read-once

doi:10.1109/ccc.2012.37
dblp:conf/coco/GavinskyLS12
fatcat:67a6f2di4ncybh3sxhzamp57gy
*constant**depth**circuits*with unbounded fan-in AND, OR, NOT and generalized modulo m gates, where m is an arbitrary ... fixed*constant*. ... This amounts to*constant**depth**circuits*with arbitrary symmetric gates; which is known to also be equivalent to TC 0 , where arbitrary*threshold*gates are allowed [6] , [7] . ...##
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Polynomial Threshold Functions: Structure, Approximation and Pseudorandomness
[article]

2009
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arXiv
*
pre-print

Our second result relates to constructing pseudorandom generators

arXiv:0911.3473v3
fatcat:7iegsdeyo5ey5nwbx5g6jwjyr4
*fooling*low-degree polynomial*threshold*functions. ... We prove that any low-degree polynomial*threshold*function, which can be represented as a function of a small number of linear*threshold*functions, can also be*fooled*by k-wise independence. ... Later, Braverman [10] proved that polylogarithmic-wise independence*fools*small*constant**depth**circuits*, settling a conjecture of Linial and Nisan [20] . ...##
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Shielding circuits with groups

2013
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Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13
*

We show how to efficiently compile any given

doi:10.1145/2488608.2488640
dblp:conf/stoc/MilesV13
fatcat:zohseoo4yzdyxgksrycfbxeca4
*circuit*C into a leakage-resilient*circuit*C such that any function on the wires of C that leaks information during a computation C(x) yields advantage in computing ... This includes communication protocols, and AC 0*circuits*augmented with few arbitrary symmetric gates. ... AC 0 with symmetric gates Recall that in §3.4, it was shown that (A 5 ) t t −Ω(log t) -*fools*the class of unbounded-fan-in*constant*-*depth**circuits*that contain t O(log t) And/Or/Not gates and O(log 2 t ...##
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Nonuniform Kolmogorov extractors
[article]

2012
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arXiv
*
pre-print

precisely, as instantiations of more general results, we show that while O(1) amount of advice regarding the source is not enough for extracting a string with randomness rate 1 from a source string with

arXiv:1204.6696v1
fatcat:swfpfse3ungrbi57u7vnshyh5a
*constant*... Pseudo-random generator*fooling*bounded-size*constant*-*depth**circuits*The derandomization in the proof of Theorem 4.6 is done using the Nisan-Wigderson pseudo-random generator that "*fools*"*constant*-*depth*... Therefore, we can use the Nisan-Wigderson ( [NW94] ) pseudorandom generator NW-gen that*fools*bounded-size*constant*-*depth**circuits*and has seeds of size polylogaritmic in the size of the output. ...##
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Fine-Grained Cryptography
[chapter]

2016
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Lecture Notes in Computer Science
*

secure against adversaries with an a-priori bounded polynomial amount of resources (time, space or parallel-time), where the honest algorithms use less resources than the adversaries they are designed to

doi:10.1007/978-3-662-53015-3_19
fatcat:ueuecgreavgr3koy6pdlwfwps4
*fool*... Previously, one-way permutations and pseudo-random generators (with linear stretch) computable in AC 0 and secure against AC 0*circuits*were known from the works of Håstad and Braverman. ... of polylog*thresholds*by*constant*-*depth**circuits*. ...##
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Pseudo-Derandomizing Learning and Approximation

2018
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International Workshop on Approximation Algorithms for Combinatorial Optimization
*

BPE) requires large Boolean

doi:10.4230/lipics.approx-random.2018.55
dblp:conf/approx/OliveiraS18
fatcat:gjphuxvvubakxevkekyacqxryq
*circuits*. ... Finally, we investigate the notion of approximate canonization of Boolean*circuits*. ... If there is a family of quick pseudorandom generators G m : {0, 1} (m) → {0, 1} m that ε-*fool**depth*-e size-m C-*circuits*, for C f has*depth*≤ d and size ≤ s(n), and D n is an oracle*circuit*of*depth*≤ ...##
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Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates
[article]

2016
*
arXiv
*
pre-print

For

arXiv:1610.02686v1
fatcat:74xhlc3drncynhzp4v7nnitiry
*depth*3*circuits*we show that a*circuit*with k= O(n^2/3) can compute MAJ_n correctly on all inputs. ... We study the following computational problem: for which values of k, the majority of n bits MAJ_n can be computed with a*depth*two formula whose each gate computes a majority function of at most k bits ... Acknowledgments We would like to thank the participants of Low-*Depth*Complexity Workshop (St. Petersburg, Russia, May 21-25, 2016) for many helpful discussions. ...
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