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Deterministic approximate counting of depth-2 circuits

M. Luby, B. Velickovic, A. Wigderson
<i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rl24zg6tg5g2hdek7nnlgicgja" style="color: black;">[1993] The 2nd Israel Symposium on Theory and Computing Systems</a> </i> &nbsp;
We describe deterministic algorithms which for a given depth-2 circuit F approximate the probability that on a random input F outputs a speci c value .  ...  Our approach gives an algorithm which for a given GF 2] multivariate polynomial p and given > 0 approximates the number of zeros of p within a multiplicative factor 1+ .  ...  Introduction This paper deals with the problem of approximating the accepting probability of general depth-2 boolean circuits.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/istcs.1993.253488">doi:10.1109/istcs.1993.253488</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/istcs/LubyVW93.html">dblp:conf/istcs/LubyVW93</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qyjj6g727ngvhl7esl7cdzg2yy">fatcat:qyjj6g727ngvhl7esl7cdzg2yy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190220204501/http://pdfs.semanticscholar.org/2952/2ebdfa1b33d090aa26e39d86d5af15cf7707.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/29/52/29522ebdfa1b33d090aa26e39d86d5af15cf7707.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/istcs.1993.253488"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Deterministic simulation of probabilistic constant depth circuits

Miklos Ajtai, Avi Wigderson
<span title="">1985</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v4uclqixrrhlnk5pbw4wii6cfq" style="color: black;">26th Annual Symposium on Foundations of Computer Science (sfcs 1985)</a> </i> &nbsp;
Moreover, the functions $f_{n,\varepsilon}$ themselves can be computed by uniform polynomial size, constant depth circuits. Some (interrelated) consequences of this result are given below.  ...  property: for a random seed, the pseudorandom output "looks random" to any polynomial size, constant depth, unbounded fan-in circuit.  ...  We also present two results for the special case of depth 2 circuits. They deal, respectively, with finding an assignment and approximately counting the number of assignments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/sfcs.1985.19">doi:10.1109/sfcs.1985.19</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/focs/AjtaiW85.html">dblp:conf/focs/AjtaiW85</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5qb2lcvhabhznmf7msgi262rga">fatcat:5qb2lcvhabhznmf7msgi262rga</a> </span>
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Floating point representations in quantum circuit synthesis

Nathan Wiebe, Vadym Kliuchnikov
<span title="2013-09-27">2013</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bt7vbjky6zbevclechrlkkahae" style="color: black;">New Journal of Physics</a> </i> &nbsp;
We provide a non-deterministic quantum protocol that approximates the single qubit rotations R_x(2a^2 b^2) using R_x(2a) and R_x(2b) and a constant number of Clifford and T operations.  ...  We also discuss the T-depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths.  ...  The first component is a non-deterministic circuit that can combine two rotations R x (α) and R x (β) to approximate R x (α 2 β 2 ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1367-2630/15/9/093041">doi:10.1088/1367-2630/15/9/093041</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cu3txe5vjzdlvgxnvtotdjtwzq">fatcat:cu3txe5vjzdlvgxnvtotdjtwzq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810222535/http://www.imsc.res.in/~aqis13/extended/shorttalks/Nathan_Wiebe_89.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/52/97/52971e52c8d54a4502b1862e5dcc2adbfeff8d26.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1367-2630/15/9/093041"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> iop.org </button> </a>

Self-reducible with easy decision version counting problems admit additive error approximation. Connections to counting complexity, exponential time complexity, and circuit lower bounds [article]

Eleni Bakali
<span title="2016-11-05">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
error ϵ· 2^n', ∀ϵ>0, which for many of those problems (when n'=n+constant) implies additive approximation to the fraction f(x)/2^n.  ...  of all circuits for which the problems of counting either satisfying or unsatisfying assignments belong to TotP (which is the Karp-closure of self reducible problems with easy decision version).  ...  Acknowledgements I want to thank Manolis Zampetakis for mentioning to me the relationship between these results and circuit lower bounds.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1611.01706v1">arXiv:1611.01706v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6oqbq3mtaza2ndwjjow6uhy6v4">fatcat:6oqbq3mtaza2ndwjjow6uhy6v4</a> </span>
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Randomness Buys Depth for Approximate Counting

Emanuele Viola
<span title="2014-01-08">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t46w5lpc3ngnfmzil33zdjjrp4" style="color: black;">Computational Complexity</a> </i> &nbsp;
In this work we focus on the trade-off between the approximation parameter and the depth of the polynomial-size circuits that count approximately.  ...  fan-in) poly(n)-size depth-d circuits with error ≤ 1/3, but cannot be solved by deterministic poly(n)-size depth-(d + 1) circuits, for every d ≥ 2; and the depth of both is tight.  ...  We thank Kord Eickmeyer for helpful initial discussions on this problem, and Joshua Brody and Elad Verbin for sending us a preliminary version of [BV10] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-013-0076-6">doi:10.1007/s00037-013-0076-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mwxfkfjhhjdxbkb3stn6acc6su">fatcat:mwxfkfjhhjdxbkb3stn6acc6su</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140325040832/http://www.ccs.neu.edu/home/viola/papers/rbd.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d6/52/d652e6e6abf895ff71cb09768627f290b293f824.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-013-0076-6"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Randomness Buys Depth for Approximate Counting

Emanuele Viola
<span title="">2011</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v4uclqixrrhlnk5pbw4wii6cfq" style="color: black;">2011 IEEE 52nd Annual Symposium on Foundations of Computer Science</a> </i> &nbsp;
In this work we focus on the trade-off between the approximation parameter and the depth of the polynomial-size circuits that count approximately.  ...  fan-in) poly(n)-size depth-d circuits with error ≤ 1/3, but cannot be solved by deterministic poly(n)-size depth-(d + 1) circuits, for every d ≥ 2; and the depth of both is tight.  ...  We thank Kord Eickmeyer for helpful initial discussions on this problem, and Joshua Brody and Elad Verbin for sending us a preliminary version of [BV10] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/focs.2011.19">doi:10.1109/focs.2011.19</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/focs/Viola11a.html">dblp:conf/focs/Viola11a</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zfg7s5qtzbcx5i3cvb2vl273ma">fatcat:zfg7s5qtzbcx5i3cvb2vl273ma</a> </span>
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Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

Ninad Rajgopal, Rahul Santhanam, Srikanth Srinivasan, Michael Wagner
<span title="2018-08-20">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5b35w7jagjckfngb6pj76nbam4" style="color: black;">International Symposium on Mathematical Foundations of Computer Science</a> </i> &nbsp;
We give a deterministic algorithm for counting the number of satisfying assignments of any AC 0 [⊕] circuit C of size s and depth d over n variables in time 2 n−f (n,s,d) , where As a consequence, we get  ...  that for each d, there is a language in E NP that does not have AC 0 [⊕] circuits of size 2 o(n 1/(d+1) ) .  ...  The Razborov-Smolensky approximation method yields size lower bounds of the form 2 Ω(n 1/2(d−1) ) for Majority against AC 0 [⊕] circuits of depth d over n variables.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2018.78">doi:10.4230/lipics.mfcs.2018.78</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/mfcs/RajgopalS018.html">dblp:conf/mfcs/RajgopalS018</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ses4ukqknjhq7iwemh6dpfhnlu">fatcat:ses4ukqknjhq7iwemh6dpfhnlu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220205180538/https://drops.dagstuhl.de/opus/volltexte/2018/9660/pdf/LIPIcs-MFCS-2018-78.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/29/db/29dbde46be983924ef0a9372cf7589b40c4dfe69.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2018.78"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

On the complexity of counting in the polynomial hierarchy

Marek Piotrów
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
., On the complexity of counting in the polynomial hierarchy, Theoretical Computer Science 81 (1991) 77-95.  ...  We investigate the time complexity of the following counting problem: for a given set of words A, a length n and a number k, check whether A contains exactly k words of length n (in symbol, if cr,( n)  ...  It suffices to consider a depth k circuit as a depth k + 1 circuit of bottom fan-in 1 and choose D < D,, + , . Then for sufficiently large n we have 2 18" < 2 I:?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(91)90317-u">doi:10.1016/0304-3975(91)90317-u</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vlw7vlbfbzgerdeqx44hzwvdnq">fatcat:vlw7vlbfbzgerdeqx44hzwvdnq</a> </span>
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On the Computational Power of Radio Channels

Mark Braverman, Gillat Kol, Rotem Oshman, Avishay Tal, Michael Wagner
<span title="2019-10-11">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2pcluxp32rfojgwekdrryxoaqu" style="color: black;">International Symposium on Distributed Computing</a> </i> &nbsp;
Next, we use the technique of random restrictions, used to prove AC 0 lower bounds, to prove a tight lower bound of Ω(1/ 2 ) on computing a (1 ± )-approximation to the sum of the nodes' inputs.  ...  Using techniques from circuit complexity, we show that in many cases, the answer is "no".  ...  The circuit C P is given by C P = t∈T V t , a depth-3 circuit of size O(n • r • 2 r ) = 2 O(r+log n) , since |T | ≤ 2 r and the size of each sub-circuit V t is O(n • r).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.disc.2019.8">doi:10.4230/lipics.disc.2019.8</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/wdag/BravermanKOT19.html">dblp:conf/wdag/BravermanKOT19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ezmqszklgbculk6ywmvifyreh4">fatcat:ezmqszklgbculk6ywmvifyreh4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220307212132/https://drops.dagstuhl.de/opus/volltexte/2019/11315/pdf/LIPIcs-DISC-2019-8.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/71/56/71565037f586b5fd72efcfdf13d7af832cfdc1a5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.disc.2019.8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Pseudo-Derandomizing Learning and Approximation

Igor Carboni Oliveira, Rahul Santhanam, Michael Wagner
<span title="2018-08-02">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gdc7gjvgv5hd7mr5o3uwxvngwu" style="color: black;">International Workshop on Approximation Algorithms for Combinatorial Optimization</a> </i> &nbsp;
Finally, we investigate the notion of approximate canonization of Boolean circuits.  ...  We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [7].  ...  If BPE requires circuits of size 2 n Ω(1) almost everywhere, then AC 0 [p] circuits can be approximately canonized in pseudo-deterministic quasi-polynomial time.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.approx-random.2018.55">doi:10.4230/lipics.approx-random.2018.55</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/approx/OliveiraS18.html">dblp:conf/approx/OliveiraS18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gjphuxvvubakxevkekyacqxryq">fatcat:gjphuxvvubakxevkekyacqxryq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220129195424/https://drops.dagstuhl.de/opus/volltexte/2018/9459/pdf/LIPIcs-APPROX-RANDOM-2018-55.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/16/cb/16cbe62a7d9b9b043f83e247a5aabdda6aedb46b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.approx-random.2018.55"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Algorithms for Circuits and Circuits for Algorithms

Ryan Williams
<span title="">2014</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fvcoh2vdcnbppmicpce3jet6r4" style="color: black;">2014 IEEE 29th Conference on Computational Complexity (CCC)</a> </i> &nbsp;
and proofs of circuit size lower bounds.  ...  Algorithms for circuits refers to the design of interesting algorithms which can perform non-trivial circuit analysis of some kind, on either a circuit or a Boolean function given as a truth table.  ...  Approximate Circuit Analysis A different form of circuit analysis is that of (additive) approximate counting; that is, approximating the fraction of satisfying assignments to a given circuit: Circuit Approximation  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ccc.2014.33">doi:10.1109/ccc.2014.33</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/coco/Williams14.html">dblp:conf/coco/Williams14</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t2irb2av2bcb5hzfi2dxkrsvqu">fatcat:t2irb2av2bcb5hzfi2dxkrsvqu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20150919045218/http://stanford.edu/~rrwill/ccc14-survey.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/10/56/10562a5dfdf466163493b34b990f62baa9e4b5b0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ccc.2014.33"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

On derandomizing algorithms that err extremely rarely

Oded Goldreich, Avi Widgerson
<span title="">2014</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC &#39;14</a> </i> &nbsp;
For the latter we present a deterministic polynomial-time hitting set generator for the class of n-variate polynomials of degree d over GF(2) that evaluate to 0 on at most an O(2 −d ) fraction of their  ...  , B(n) = n log n or B(n) = exp(n 0.99 ))), given an n-input circuit C from C that evaluates to 1 on all but at most B(n) of its inputs, find (in deterministic polynomial-time) an input x such that C(x)  ...  Specifically, the known deterministic algorithms for approximate counting for AC 0 run in quasipolynomial time.  ... 
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More on AC^0[oplus] and Variants of the Majority Function

Nutan Limaye, Srikanth Srinivasan, Utkarsh Tripathi, Michael Wagner
<span title="2019-12-10">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yjwtenvm3vc4tjdmag7doffpby" style="color: black;">Foundations of Software Technology and Theoretical Computer Science</a> </i> &nbsp;
Formally, we show that for any fixed constant d ≥ 2, there is a family of Boolean functions that has polynomialsized randomized uniform AC 0 circuits of depth d but no polynomial-sized (deterministic)  ...  AC 0 [⊕] circuits of depth d.  ...  Ajtai and Ben-Or [3] showed that for any randomized depth-d AC 0 circuit of size at most s, there is deterministic AC 0 circuit of depth d + 2 and size at most poly(s) that computes the same function  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.fsttcs.2019.22">doi:10.4230/lipics.fsttcs.2019.22</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/fsttcs/Limaye0T19.html">dblp:conf/fsttcs/Limaye0T19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u54vfutravhctcqt4rwdpe34pm">fatcat:u54vfutravhctcqt4rwdpe34pm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220307212038/https://drops.dagstuhl.de/opus/volltexte/2019/11584/pdf/LIPIcs-FSTTCS-2019-22.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/66/34/6634f699a674fbadf4c307f354cc6fd712528fd6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.fsttcs.2019.22"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

My favorite ten complexity theorems of the past decade [chapter]

Lance Fortnow
<span title="">1994</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We use each of the theorems as a springboard to discuss work done in various areas of complexity theory.  ...  Their proof uses a new idea of approximating bounded-depth circuits by low-degree polynomials over a nite eld.  ...  Instead of the restriction method used by H astad (Theorem 2), Razborov uses an approximation method. Razborov shows how to approximate each AND and OR with approximate AND and approximate OR.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-58715-2_130">doi:10.1007/3-540-58715-2_130</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oqj4nco6ozaxpclwtipoor2egi">fatcat:oqj4nco6ozaxpclwtipoor2egi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190307201121/http://pdfs.semanticscholar.org/e668/276631996f52624e7b65098308b579c153a4.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e6/68/e668276631996f52624e7b65098308b579c153a4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-58715-2_130"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 5582 of Mathematical Reviews Vol. , Issue 96i [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Every counting problem in #Zo is computable in deterministic polynomial time, and every prob- lem in #2; has a fully polynomial-time randomized approximation scheme.  ...  /A,\-circuits of depth 2d that cannot be computed by a small monotone perceptron of depth d.  ... 
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