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Arithmetic Circuits: a survey of recent results and open questions
"... 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 fi ..."
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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
Arithmetic circuits: the chasm at depth four gets wider
"... In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower ..."
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In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or blackbox derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n×n matrices has circuits of size polynomial inn, then it also has depth 4 circuits of sizen O( √ nlogn) If the original circuit uses only integer constants of polynomial size, then the same is true of the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: The existence of nontrivial boolean circuits of constant depth for languages in LOGCFL. Reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree.
Readonce Polynomial Identity Testing
"... An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readon ..."
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Cited by 23 (6 self)
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An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readonce formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non blackbox) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If F  is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a subexponential time deterministic algorithm for
TensorRank and Lower Bounds for Arithmetic Formulas
"... We show that any explicit example for a tensor A: [n] r → F with tensorrank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit superpolynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmet ..."
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Cited by 22 (1 self)
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We show that any explicit example for a tensor A: [n] r → F with tensorrank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit superpolynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply superpolynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any nvariate homogenous polynomial f of degree r, if there exists a (fanin2) ( formula of size s and depth d for f then there exists a homogenous (d+r+1)) formula of size O r · s for f. In particular, for any r ≤ log n, if there exists a polynomial size formula for f then there exists a polynomial size homogenous formula for f. This refutes a conjecture of Nisan and Wigderson [NW95] and shows that superpolynomial lower bounds for homogenous formulas for polynomials of small degree imply superpolynomial lower bounds for general formulas. We show that for any nvariate setmultilinear polynomial f of degree r, if there exists a (fanin2) formula of size s and depth d for f then there exists a setmultilinear formula of size O ((d + 2) r · s) for f. In particular, for any r ≤ log n / log log n, if there exists a polynomial size formula for f then there exists a polynomial size setmultilinear formula for f. This shows that superpolynomial lower bounds for setmultilinear formulas for polynomials of small degree imply superpolynomial lower bounds for general formulas.
Deterministic identity testing of depth 4 multilinear circuits with bounded top fanin
, 2009
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Quasipolynomial hittingset for setdepth formulas
 In STOC
, 2013
"... Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1 ..."
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Abstract. We call a depth4 formula C setdepth4 if there exists a (unknown) partition X1 unionsq · · · unionsq Xd of the variable indices [n] that the top product layer respects, i.e. C(x) = ∑k i=1
Blackbox identity testing of depth4 multilinear circuits
 In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC
, 2011
"... We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth4 circuits with fanin k at the top + gate. We give the first polynomialtime deterministic identity testing algorithm for such circuits. Our results also hold in the blackbox setting. The running time ..."
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Cited by 14 (3 self)
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We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth4 circuits with fanin k at the top + gate. We give the first polynomialtime deterministic identity testing algorithm for such circuits. Our results also hold in the blackbox setting. The running time of our algorithm is (ns)O(k 3), where n is the number of variables, s is the size of the circuit and k is the fanin of the top gate. The importance of this model arises from [AV08], where it was shown that derandomizing blackbox polynomial identity testing for general depth4 circuits implies a derandomization of polynomial identity testing (PIT) for general arithmetic circuits. Prior to our work, the best PIT algorithm for multilinear ΣΠΣΠ(k) circuits [KMSV10] ran in quasipolynomialtime, with the running time being nO(k 6 log(k) log2 s). We obtain our results by showing a strong structural result for multilinear ΣΠΣΠ(k) circuits that compute the zero polynomial. We show that under some mild technical conditions, any gate of such a circuit must compute a sparse polynomial. We then show how to combine the structure theorem with a result by Klivans and Spielman [KS01], on the identity testing for sparse polynomials, to yield the full result.
Approaching the chasm at depth four
 In Proceedings of the 28th Annual IEEE Conference on Computational Complexity (CCC 2013), pages 65– 73, 2013. Full version in the Electronic Colloquium on Computational Complexity (ECCC
"... AgrawalVinay [AV08], Koiran [Koi12] and Tavenas [Tav13] have recently shown that an exp(ω( n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to superpolynomial lower bound for general arithmetic circ ..."
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AgrawalVinay [AV08], Koiran [Koi12] and Tavenas [Tav13] have recently shown that an exp(ω( n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to superpolynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by n computing the permanent (or the determinant) must be of size exp(Ω( n)).
NonCommutative Circuits and the SumofSquares Problem (Extended Abstract)
, 2010
"... We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares p ..."
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We initiate a direction for proving lower bounds on the size of noncommutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of noncommutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sumofsquares problem: find the smallest n such that there exists an identity (x 2 1+x 2 2+ · · ·+x 2 k)·(y 2 1+y 2 2+ · · ·+y 2 k) = f 2 1 +f 2 2 + · · ·+f 2 n, (1) where each fi = fi(X, Y) is bilinear in X = {x1,..., xk} and Y = {y1,..., yk}. Over the complex numbers, we show that a sufficiently strong superlinear lower bound on n in (1), namely, n ≥ k 1+ɛ with ε> 0, implies an exponential lower bound on the size of arithmetic circuits computing the noncommutative permanent. More generally, we consider such sumofsquares identities for any biquadratic polynomial h(X, Y), namely h(X, Y) = f 2 1 + f 2 2 + · · · + f 2 n. (2) Again, proving n ≥ k 1+ɛ in (2) for any explicit h over the complex numbers gives an exponential lower bound for the noncommutative permanent. Our proofs relies on several new structure theorems for noncommutative circuits, as well as a noncommutative analog of Valiant’s completeness of the permanent. We proceed to prove such superlinear bounds in some restricted cases. We prove that n ≥ Ω(k 6/5) in (1), if f1,..., fn are required to have integer coefficients. Over the real numbers, we construct an explicit biquadratic polynomial h such that n in (2) must be at least Ω(k 2). Unfortunately, these results do not imply circuit lower bounds.
A Superpolynomial Lower Bound on the Size of Uniform Nonconstantdepth Threshold Circuits for the Permanent
, 2009
"... We show that the permanent cannot be computed by DLOGTIMEuniform threshold or arithmetic circuits of depth o(log log n) and polynomial size. ..."
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We show that the permanent cannot be computed by DLOGTIMEuniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.