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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
Lower Bounds and Separations for Constant Depth Multilinear Circuits
"... We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between ..."
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Cited by 33 (5 self)
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We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a superpolynomial separation between the size of productdepth 1 d and productdepth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that • There exists a multilinear circuit of productdepth d + 1 and of polynomial size computing f. • Every multilinear circuit of productdepth d computing f has superpolynomial size. 1
HardnessRandomness Tradeoffs for Bounded Depth Arithmetic Circuits
"... In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of sma ..."
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Cited by 22 (5 self)
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In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d − 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardnessrandomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the the arithmetic NisanWigderson generator of [KI04] together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f(x1,..., xn)) ≡ 0 then f has a circuit of depth d + 3 and size O(s · r + m r), where m is the total degree of f and r is the degree of y in P.
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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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.
Uniform Derandomization from Pathetic Lower Bounds
, 2009
"... 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 nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed ..."
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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 nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic (linearsize lower bounds for general circuits [IM02], nearly cubic lower bounds for formula size [H˚as98], nearly n log log n size lower bounds for branching programs [BSSV03], n 1+cd for depth d threshold circuits [IPS97]). Here, we present two instances where “pathetic ” lower bounds of the form n 1+ɛ would suffice to derandomize interesting classes of probabilistic algorithms. We show: • If the word problem over S5 requires constantdepth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomialsize probabilistic threshold circuits is accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size. • If there are no constantdepth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3by3 matrices, then for every constant d, blackbox identity testing for depthd arithmetic circuits with bounded individual degree can be performed by a uniform family of deterministic constantdepth AC0 circuits of subexponential size.
Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth
 In ICALP
, 2011
"... Abstract. We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constantfree 1 succinct arithmetic circuit family {Φn}, where Φn has size at most p(n) and depth O(1), such that Φn computes the n × n permanent. A circuit family {Φn} is succinct if there exists ..."
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Abstract. We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constantfree 1 succinct arithmetic circuit family {Φn}, where Φn has size at most p(n) and depth O(1), such that Φn computes the n × n permanent. A circuit family {Φn} is succinct if there exists a nonuniform Boolean circuit family {Cn} with O(log n) many inputs and size n o(1) such that that Cn can correctly answer direct connection language queries about Φn succinctness is a relaxation of uniformity. To obtain this result we develop a novel technique that further strengthens the connection between blackbox derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy. 1
Reconstruction of depth4 multilinear circuits with top fanin 2
 In Proceedings of the 44th Annual STOC
, 2012
"... We present a randomized algorithm for reconstructing multilinear ΣΠΣΠ(2) circuits, i.e. multilinear depth4 circuits with fanin 2 at the top + gate. The algorithm is given blackbox access to a polynomial f ∈ F[x1,..., xn] computable by a multilinear ΣΠΣΠ(2) circuit of size s and outputs an equivale ..."
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We present a randomized algorithm for reconstructing multilinear ΣΠΣΠ(2) circuits, i.e. multilinear depth4 circuits with fanin 2 at the top + gate. The algorithm is given blackbox access to a polynomial f ∈ F[x1,..., xn] computable by a multilinear ΣΠΣΠ(2) circuit of size s and outputs an equivalent multilinear ΣΠΣΠ(2) circuit, runs in time poly(n, s), and works over any field F. This is the first reconstruction result for any model of depth4 arithmetic circuits. Prior to our work, reconstruction results for bounded depth circuits were known only for depth2 arithmetic circuits (Klivans & Spielman, STOC 2001), ΣΠΣ(2) circuits (depth3 arithmetic circuits with top fanin 2) (Shpilka, STOC 2007), and ΣΠΣ(k) with k = O(1) (Karnin & Shpilka, CCC 2009). Moreover, the running times of these algorithms have a polynomial dependence on F  and hence do not work for infinite fields such as Q. Our techniques are quite different from the previous ones for depth3 reconstruction and rely on a polynomial operator introduced by Karnin et al. (STOC 2010) and Saraf & Volkovich (STOC 2011) for devising blackbox identity tests for multilinear ΣΠΣΠ(k) circuits. Some other ingredients of our algorithm include the classical multivariate blackbox factoring algorithm by Kaltofen & Trager (FOCS 1988) and an algorithm for reconstructing setmultilinear ΣΠΣ(2) circuits by Kayal.
Noncommutative circuits and the sumofsquares problem
 J. Amer. Math. Soc
"... 1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of ..."
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1.1. Noncommutative computation. Arithmetic complexity theory studies the computation of formal polynomials over some field or ring. Most of this theory is concerned with the computation of commutative polynomials. The basic model of computation is that of an arithmetic circuit. Despite decades of work, the best
A NonLinear Lower Bound for Constant Depth Arithmetical Circuits via the Discrete Uncertainty Principle
, 2006
"... We prove a nonlinear lower bound on the size of a bounded depth bilinear arithmetical circuit computing the circular convolution mapping in case the input vectors are of prime length. For this proof we utilize a strengthing of the DonohoStark uncertainty principle [DS89], as given by Tao [Tao05], ..."
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We prove a nonlinear lower bound on the size of a bounded depth bilinear arithmetical circuit computing the circular convolution mapping in case the input vectors are of prime length. For this proof we utilize a strengthing of the DonohoStark uncertainty principle [DS89], as given by Tao [Tao05], and a combinatorial lemma by Raz and Shpilka [RS03].A new proof is given of the DonohoStark uncertainty principle.