Arithmetic Circuits: A Chasm at Depth Four

Agrawal, Manindra; Vinay, V.

doi:10.1109/focs.2008.32

Cited by 167 publications

(219 citation statements)

References 24 publications

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“…Namely, an n-variate degree n polynomial can be computed by a sub-exponential arithmetic circuit if and only if it can be computed by a sub-exponential ΣΠΣΠ circuit. This result, due to Agrawal and Vinay [AV08], is discussed in Section 2. 4.…”

Section: Arithmetic Circuit Classesmentioning

confidence: 78%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

290

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: Arithmetic Circuit Classesmentioning

confidence: 78%

Arithmetic Circuits: A survey of recent results and open questions

Shpilka

Yehudayoff

2009

FNT in Theoretical Computer Science

290

275

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“…A series of works (Valiant et al, 1983;Agrawal and Vinay, 2008;Koiran, 2012;Gupta et al, 2013;Tavenas, 2013) on depth three circuits computing a VNP-polynomial, then a superpolynomial lower bound on general circuits immediately follows, proving VP  VNP. An important point here, relevant to the preceding discussion, is that the depth three circuit resulting from the depth reduction potentially has as high a formal degree as…”

Section: Depth Reductionmentioning

confidence: 99%

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Hegde

2017

Proceedings of the Indian National Science Academy

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An arithmetic circuit (respectively, formula) is a rooted graph (respectively, tree) whose nodes are addition or multiplication gates and input variables/nodes. It computes a polynomial in a natural way. The formal degree of an addition (respectively, multiplication) gate with respect to a variable x is defined as the maximum (respectively, sum) of the formal degrees of its children, with respect to x. The formal degree of an input node with respect to x is 1 if the node is labelled with x, and 0 otherwise. In a multi-r-ic formula, the formal degree of every gate with respect to every variable is at most r. Multi-r-ic formulas make an intermediate model between multilinear formulas (the r = 1 case), for which lower bounds are relatively well-understood, and general formulas (the unbounded-r case), which are conjectured to have superpolynomial size lower bound.On depth four multi-r-ic formulas/circuits computing IMM n,d -the product of d symbolic metrices of size n x n each,

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“…It is interesting to note that the PIT problem becomes very difficult already for depth-4 circuits. Indeed, [AV08] proved that a polynomial time black-box PIT algorithm for depth-4 circuits implies an exponential lower bound for general arithmetic circuits (and hence using the ideas of [KI04] a quasi-polynomial time deterministic PIT algorithm for general circuits).…”

Section: Polynomial Identity Testingmentioning

confidence: 99%

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

Shpilka

Volkovich

2010

Automata, Languages and Programming

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We say that a polynomial f (x 1 , . . . , x n ) is indecomposable if it cannot be written as a product of two polynomials that are defined over disjoint sets of variables. The polynomial decomposition problem is defined to be the task of finding the indecomposable factors of a given polynomial. Note that for multilinear polynomials, factorization is the same as decomposition, as any two different factors are variable disjoint.In this paper we show that the problem of derandomizing polynomial identity testing is essentially equivalent to the problem of derandomizing algorithms for polynomial decomposition. More accurately, we show that for any reasonable circuit class there is a deterministic polynomial time (black-box) algorithm for polynomial identity testing of that class if and only if there is a deterministic polynomial time (black-box) algorithm for factoring a polynomial, computed in the class, to its indecomposable components.An immediate corollary is that polynomial identity testing and polynomial factorization are equivalent (up to a polynomial overhead) for multilinear polynomials. In addition, we observe that derandomizing the polynomial decomposition problem is equivalent, in the sense of Kabanets and Impagliazzo [KI04], to proving arithmetic circuit lower bounds to NEXP.Our approach uses ideas from [SV08], that showed that the polynomial identity testing problem for a circuit class C is essentially equivalent to the problem of deciding whether a circuit from C computes a polynomial that has a read-once arithmetic formula.

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Arithmetic Circuits: A Chasm at Depth Four

Cited by 167 publications

References 24 publications

Arithmetic Circuits: A survey of recent results and open questions

Arithmetic Circuits: A survey of recent results and open questions

Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

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