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Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

2016
*
Computational Complexity
*

Shpilka and Wigderson [22] had posed the problem of proving exponential

doi:10.1007/s00037-016-0132-0
fatcat:s6mnfmxlizhzve5aq4xwit2aum
*lower**bounds**for*(nonhomogeneous)*depth*three*arithmetic**circuits*with*bounded*bottom fanin over*a*field F of characteristic zero ... Over fields of characteristic zero, we show*a**lower**bound*of N Ω( √ d)*for*homogeneous*depth*five*circuits*(resp. also*for**depth*three*circuits*) with bottom fanin at most N µ ,*for*any fixed µ < 1. ... In particular, Ramprasad pointed out to us that*a*lemma in [7] can be improved quantitatively and that the ΣΠΣ*circuits*which come out of the*depth*reduction in [7] in fact have small bottom fanin. ...##
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Approaching the Chasm at Depth Four

2014
*
Journal of the ACM
*

having fanin

doi:10.1145/2629541
fatcat:de53fxiayzgvvggpwukfk7z5fa
*bounded*by √ n translates to*super*-polynomial*lower**bound**for*general*arithmetic**circuits*computing the permanent. ...*Lower**bounds*have been obtained earlier*for**depth*three*arithmetic**circuits*(with some restrictions) and constant*depth*multilinear*circuits*. ... the*lower**bound*estimate of Section 5 remains essentially valid under*a*random restriction. ...##
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Approaching the Chasm at Depth Four

2013
*
2013 IEEE Conference on Computational Complexity
*

having fanin

doi:10.1109/ccc.2013.16
dblp:conf/coco/0001KKS13
fatcat:udy6adogijecxciazfmr47ojce
*bounded*by √ n translates to*super*-polynomial*lower**bound**for*general*arithmetic**circuits*computing the permanent. ...*Lower**bounds*have been obtained earlier*for**depth*three*arithmetic**circuits*(with some restrictions) and constant*depth*multilinear*circuits*. ... the*lower**bound*estimate of Section 5 remains essentially valid under*a*random restriction. ...##
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Elusive functions and lower bounds for arithmetic circuits

2008
*
Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
*

In this paper, we show that problems of this type are closely related to proving

doi:10.1145/1374376.1374479
dblp:conf/stoc/Raz08
fatcat:ccqjajxfjrfivem6k7sor45w7e
*lower**bounds**for*the size of general*arithmetic**circuits*. ... In particular,*for*any constant r, this gives*a*constant degree polynomial such that any*depth*r*arithmetic**circuit**for*this polynomial is of size ≥ n 1+Ω(1) . ... Acknowledgement I am grateful to Zeev Dvir, Yael Tauman Kalai, Toni Pitassi, Omer Reingold, Amir Shpilka, Avi Wigderson and Amir Yehudayoff,*for*very helpful conversations and comments at different stages ...##
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Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

2010
*
SIAM journal on computing (Print)
*

In this paper we show that

doi:10.1137/080735850
fatcat:gonpk6xthzgdzldbrgsdqowlha
*lower**bounds**for**bounded**depth**arithmetic**circuits*imply derandomization of polynomial identity testing*for**bounded**depth**arithmetic**circuits*. ... To the best of our knowledge this is the first hardness-randomness tradeoff*for**bounded**depth**arithmetic**circuits*. ... Acknowledgement We would like to thank Prahladh Harsha, Chris Umans and Gil Alon*for*helpful discussions. ...##
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Hardness-randomness tradeoffs for bounded depth arithmetic circuits

2008
*
Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
*

In this paper we show that

doi:10.1145/1374376.1374482
dblp:conf/stoc/DvirSY08
fatcat:2meotxjq4zff7bbpqbqkqvekuq
*lower**bounds**for**bounded**depth**arithmetic**circuits*imply derandomization of polynomial identity testing*for**bounded**depth**arithmetic**circuits*. ... To the best of our knowledge this is the first hardness-randomness tradeoff*for**bounded**depth**arithmetic**circuits*. ... Acknowledgement We would like to thank Prahladh Harsha, Chris Umans and Gil Alon*for*helpful discussions. ...##
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Arithmetic Circuits: A survey of recent results and open questions

2009
*
Foundations and Trends® in Theoretical Computer Science
*

As examples we mention the connection between polynomial identity testing and

doi:10.1561/0400000039
fatcat:vejtujygx5ddjkm2crbxh2udcq
*lower**bounds*of Kabanets and Impagliazzo, the*lower**bounds*of Raz*for*multilinear formulas, and two new approaches*for*proving ... Being*a*more structured model than Boolean*circuits*, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve*for**arithmetic**circuits*... Since multilinear ΣΠΣ(k)*circuits*are actually*a*sum of k ROFs, this gives the currently best PIT algorithm*for*them 12 . ...##
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Functional lower bounds for restricted arithmetic circuits of depth four
[article]

2021
*
arXiv
*
pre-print

Thus they argued that

arXiv:2107.09703v1
fatcat:omis7sfwbje2jfpxnj6eamzzzy
*a*2^ω(log^O(1)n) "functional"*lower**bound**for*an explicit polynomial Q against Σ∧ΣΠ*circuits*would imply*a**lower**bound**for*the "corresponding Boolean function" of Q against non-uniform ... Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^O(1)-variate and degree d polynomial P_d∈ VNP such that if any*depth**four**circuit*C of*bounded*formal degree d ... Prior to that the best known*lower**bound**for**depth**four**circuits*was*super*-*quadratic*[GST20] (which improves upon*super*-linear*lower**bounds*due to Shoup and Smolensky [SS97] and Raz [Raz10] ). ...##
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Lower Bounds for Multilinear Order-Restricted ABPs

2019
*
International Symposium on Mathematical Foundations of Computer Science
*

As

doi:10.4230/lipics.mfcs.2019.52
dblp:conf/mfcs/RamyaR19
fatcat:tyxudljc5ba3dmeleu2nsrdq54
*a*corollary, we show that any size S syntactic multilinear ABP can be transformed into*a*size S O( √ n)*depth**four*syntactic multilinear ΣΠΣΠ*circuit*where the bottom Σ gates compute polynomials on ... Proving*super*-polynomial*lower**bounds*on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is*a*challenging problem in Algebraic Complexity Theory ... Agrawal and Vinay [1] showed that proving exponential*lower**bounds**for**depth**four**circuits*is sufficient to prove Valiant's hypothesis. ...##
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Lower bounds for multilinear bounded order ABPs
[article]

2019
*
arXiv
*
pre-print

Proving

arXiv:1901.04377v1
fatcat:radqvl3pczeudo3qqfqwyxcsjy
*super*-polynomial size*lower**bounds**for*syntactic multilinear Algebraic Branching Programs(smABPs) computing an explicit polynomial is*a*challenging problem in Algebraic Complexity Theory. ... We prove exponential*lower**bound**for*the size of*a*strict circular-interval ABP computing an explicit n-variate multilinear polynomial in VP. ...*super*-polynomial*lower**bounds**for*syntactic multilinear*circuits*. ...##
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An exponential lower bound for homogeneous depth-5 circuits over finite fields
[article]

2015
*
arXiv
*
pre-print

In this paper, we show exponential

arXiv:1507.00177v1
fatcat:s2fof3vribgr7grxz56hx34ece
*lower**bounds**for*the class of homogeneous*depth*-5*circuits*over all small finite fields. ... To the best of our knowledge, this is the first*super*-polynomial*lower**bound**for*this class*for*any field F_q ≠F_2. ... We would also like to thank Eric Allender*for*answering our questions about connections between boolean*circuits*and*arithmetic**circuits*over finite fields and pointing us to reference [AAD00] . ...##
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A Selection of Lower Bounds for Arithmetic Circuits
[chapter]

2014
*
Perspectives in Computational Complexity
*

We conclude with some recent progress on

doi:10.1007/978-3-319-05446-9_5
fatcat:6crjza2rtvhhbn3ez2ghj2l7uy
*lower**bounds**for*homogeneous*depth**four**circuits*. Remark. ...*A**quadratic**lower**bound**for**depth*three*circuits*by Shpilka and Wigderson [SW01] ,*for**bounded*occur*bounded**depth*formulas by Agrawal, Saha, Saptharishi and Saxena [ASSS12] and the n 1+Ω(1/r)*lower*...##
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On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
[article]

2013
*
arXiv
*
pre-print

The result [GK1998] can only rule out such

arXiv:1401.0189v1
fatcat:qeb2ot7mrraqnnoi2tcteslz6e
*a*possibility*for**depth*3*circuits*of size 2^o(n). ... An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger*circuit*size*lower**bounds*than Permanent in*a*... In*a*very recent work (and independent of ours), Kumar and Saraf have proved*super*polynomial*circuit*size*lower**bound**for*homogeneous*depth*4*circuits*(with no fan-in restriction) computing the NW n,ǫ ...##
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Lower Bounds for Tropical Circuits and Dynamic Programs

2014
*
Theory of Computing Systems
*

In this paper we present some

doi:10.1007/s00224-014-9574-4
fatcat:jswaye4o4jg7tdhz6a34a2j5ay
*lower**bounds*arguments*for*tropical*circuits*, and hence,*for*dynamic programs. ... The power of tropical*circuits*lies somewhere between that of monotone boolean*circuits*and monotone*arithmetic**circuits*. ... Acknowledgements I am thankful to Dima Grigoriev, Georg Schnitger and Igor Sergeev*for*interesting discussions. ...##
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Limitations of sum of products of Read-Once Polynomials
[article]

2015
*
arXiv
*
pre-print

We prove an exponential

arXiv:1512.03607v1
fatcat:s2ijgisiyvgl5dgsi7tilutsdu
*lower**bound**for*the size of the ΣΠ^[N^1/30]*arithmetic**circuits*built over syntactically multi-linear ΣΠΣ^[N^8/15]*arithmetic**circuits*computing*a*product of variable disjoint linear ... We show that the same*lower**bound*holds*for*the permanent polynomial. Finally we obtain an exponential*lower**bound**for*the sum of ROPs computing*a*polynomial in VP defined by Raz and Yehudayoff. ... Further, we thank one of the anonymous reviewers*for*pointing out an outline of argument*for*Lemma 7. ...
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