An exponential lower bound for depth 3 arithmetic circuits

Grigoriev, Dima; Karpiński, Marek

doi:10.1145/276698.276872

Cited by 83 publications

(87 citation statements)

References 7 publications

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“…Recall that depth-3 ΣΠΣ circuits compute a sum of products of linear forms (there is also the less interesting case of ΠΣΠ circuits, that are "closer" to depth-2 circuits, which we do not discuss here). Over finite fields, Grigoriev and Karpinski [GK98] and Grigoriev and Razborov [GR00] proved exponential lower bound on the size of ΣΠΣ circuits. In fact, their lower bound holds for any circuit computing the function MOD q (where q is a prime number different than the characteristic of the field), which is stronger than a lower bound for computing some specific polynomial-representation of the function (recall that the MOD q function is defined as MOD q (x 1 , .…”

Section: Constant Depth Circuitsmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. 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. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. 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 lower bounds: Geometric Complexity Theory and Elusive Functions.The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

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Section: Constant Depth Circuitsmentioning

confidence: 99%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

293

275

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“…We know that such a depth reduction is not possible over small finite fields. Lower bounds of the form 2 Ω(n) were shown for depth 3 (non-homogeneous) circuits over small finite fields (even for the determinant) by Grigoriev and Karpinksi [GK98] and Grigoriev and Razborov [GR98] 3 . Thus at least for depth 3 circuits, we know that there is a vast difference between the computational power of circuits for different fields.…”

Section: Introductionmentioning

confidence: 99%

On the Power of Homogeneous Depth 4 Arithmetic Circuits

Kumar

Saraf²

2014

2014 IEEE 55th Annual Symposium on Foundations of Computer Science

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We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in VP. Our results hold for the Iterated Matrix Multiplication polynomial -in particular we show that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension n O(1) must have size n Ω( √ n) . Our results strengthen previous works in two significant ways.1. Our lower bounds hold for a polynomial in VP. Prior to our work, Kayal et al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in VNP. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in VP was the bound of n Ω(log n) by [LSS, KLSS14].Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they imply that the depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even for reductions to general homogeneous depth 4 circuits (without the restriction of bounded bottom fanin).2. Our lower bound holds over all fields. The lower bound of [KLSS14] worked only over fields of characteristic zero. Prior to our work, the best lower bound for homogeneous depth 4 circuits over fields of positive characteristic was n Ω(log n) [LSS,KLSS14].

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“…Exponential lower bounds for depth-3 arithmetic circuits over finite fields were shown in [47,48]. On the other hand, for depth-3 arithmetic circuits over fields of characteristic zero, only quadratic lower bounds are known [49].…”

Section: Consequences Of Pathetic Arithmetic Circuit Lower Boundsmentioning

confidence: 99%

Uniform derandomization from pathetic lower bounds

Allender

Arvind

Santhanam

et al. 2012

Phil. Trans. R. Soc. A.

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The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where 'pathetic' lower bounds of the form n 1+e would suffice to derandomize interesting classes of probabilistic algorithms. We show the following:-If the word problem over S 5 requires constant-depth threshold circuits of size n 1+e for some e > 0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size). -If there are no constant-depth arithmetic circuits of size n 1+e for the problem of multiplying a sequence of n 3 × 3 matrices, then, for every constant d, black-box identity testing for depth-d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

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An exponential lower bound for depth 3 arithmetic circuits

Cited by 83 publications

References 7 publications

Arithmetic Circuits: A survey of recent results and open questions

Arithmetic Circuits: A survey of recent results and open questions

On the Power of Homogeneous Depth 4 Arithmetic Circuits

Uniform derandomization from pathetic lower bounds

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