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

Neeraj Kayal, Chandan Saha
2016 Computational Complexity  
Shpilka and Wigderson [22] had posed the problem of proving exponential 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.  ... 
doi:10.1007/s00037-016-0132-0 fatcat:s6mnfmxlizhzve5aq4xwit2aum

Approaching the Chasm at Depth Four

Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi
2014 Journal of the ACM  
having fanin 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.  ... 
doi:10.1145/2629541 fatcat:de53fxiayzgvvggpwukfk7z5fa

Approaching the Chasm at Depth Four

Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi
2013 2013 IEEE Conference on Computational Complexity  
having fanin 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.  ... 
doi:10.1109/ccc.2013.16 dblp:conf/coco/0001KKS13 fatcat:udy6adogijecxciazfmr47ojce

Elusive functions and lower bounds for arithmetic circuits

Ran Raz
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 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  ... 
doi:10.1145/1374376.1374479 dblp:conf/stoc/Raz08 fatcat:ccqjajxfjrfivem6k7sor45w7e

Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

Zeev Dvir, Amir Shpilka, Amir Yehudayoff
2010 SIAM journal on computing (Print)  
In this paper we show that 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.  ... 
doi:10.1137/080735850 fatcat:gonpk6xthzgdzldbrgsdqowlha

Hardness-randomness tradeoffs for bounded depth arithmetic circuits

Zeev Dvir, Amir Shpilka, Amir Yehudayoff
2008 Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08  
In this paper we show that 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.  ... 
doi:10.1145/1374376.1374482 dblp:conf/stoc/DvirSY08 fatcat:2meotxjq4zff7bbpqbqkqvekuq

Arithmetic Circuits: A survey of recent results and open questions

Amir Shpilka, Amir Yehudayoff
2009 Foundations and Trends® in Theoretical Computer Science  
As examples we mention the connection between polynomial identity testing and 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 .  ... 
doi:10.1561/0400000039 fatcat:vejtujygx5ddjkm2crbxh2udcq

Functional lower bounds for restricted arithmetic circuits of depth four [article]

Suryajith Chillara
2021 arXiv   pre-print
Thus they argued that 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] ).  ... 
arXiv:2107.09703v1 fatcat:omis7sfwbje2jfpxnj6eamzzzy

Lower Bounds for Multilinear Order-Restricted ABPs

C. Ramya, B. V. Raghavendra Rao, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
As 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.  ... 
doi:10.4230/lipics.mfcs.2019.52 dblp:conf/mfcs/RamyaR19 fatcat:tyxudljc5ba3dmeleu2nsrdq54

Lower bounds for multilinear bounded order ABPs [article]

C.Ramya, B.V.Raghavendra Rao
2019 arXiv   pre-print
Proving 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.  ... 
arXiv:1901.04377v1 fatcat:radqvl3pczeudo3qqfqwyxcsjy

An exponential lower bound for homogeneous depth-5 circuits over finite fields [article]

Mrinal Kumar, Ramprasad Saptharishi
2015 arXiv   pre-print
In this paper, we show exponential 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] .  ... 
arXiv:1507.00177v1 fatcat:s2fof3vribgr7grxz56hx34ece

A Selection of Lower Bounds for Arithmetic Circuits [chapter]

Neeraj Kayal, Ramprasad Saptharishi
2014 Perspectives in Computational Complexity  
We conclude with some recent progress on 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  ... 
doi:10.1007/978-3-319-05446-9_5 fatcat:6crjza2rtvhhbn3ez2ghj2l7uy

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields [article]

Suryajith Chillara, Partha Mukhopadhyay
2013 arXiv   pre-print
The result [GK1998] can only rule out such 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  ... 
arXiv:1401.0189v1 fatcat:qeb2ot7mrraqnnoi2tcteslz6e

Lower Bounds for Tropical Circuits and Dynamic Programs

Stasys Jukna
2014 Theory of Computing Systems  
In this paper we present some 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.  ... 
doi:10.1007/s00224-014-9574-4 fatcat:jswaye4o4jg7tdhz6a34a2j5ay

Limitations of sum of products of Read-Once Polynomials [article]

C. Ramya, B.V. Raghavendra Rao
2015 arXiv   pre-print
We prove an exponential 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.  ... 
arXiv:1512.03607v1 fatcat:s2ijgisiyvgl5dgsi7tilutsdu
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