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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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Cited by 65 (5 self)
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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
Polynomial identity testing for depth 3 circuits
 in Proceedings of the twentyfirst Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main ..."
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Cited by 51 (11 self)
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Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai
An Almost Optimal Rank Bound for Depth3 Identities
"... Abstract—We show that the rank of a depth3 circuit (over any field) that is simple, minimal and zero is at most O(k 3 log d). The previous best rank bound known was 2 O(k2) (log d) k−2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we als ..."
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Cited by 27 (11 self)
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Abstract—We show that the rank of a depth3 circuit (over any field) that is simple, minimal and zero is at most O(k 3 log d). The previous best rank bound known was 2 O(k2) (log d) k−2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Ω(k log d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic blackbox identity tests for depth3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth3 circuit (over any field) is at most O(k 3 log d). The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth3 circuits. We prove interesting structural results about depth3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits. I.
FROM SYLVESTERGALLAI CONFIGURATIONS TO RANK BOUNDS: IMPROVED BLACKBOX IDENTITY TEST FOR DEPTH3 CIRCUITS
"... Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over ratio ..."
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Cited by 27 (6 self)
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Abstract. We study the problem of identity testing for depth3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d kO(k) time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(k2)time. Our structure theorem essentially says that the number of independent variables in a real depth3 identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for blackbox identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional SylvesterGallai theorems and the rank of depth3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of SylvesterGallai rank bounds for any field, and show the intimate connection between this and depth3 identity rank bounds. We also prove the first ever theorem about high dimensional SylvesterGallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional SylvesterGallai configuration. 1.
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
Blackbox identity testing for bounded top fanin depth3 circuits: the field doesn’t matter
 In Proceedings of the 43rd annual ACM Symposium on Theory of Computing (STOC
, 2011
"... Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed ..."
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Cited by 20 (7 self)
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Abstract. Let C be a depth3 circuit with n variables, degree d and top fanin k (called ΣΠΣ(k, d, n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(n)dk, regardless of the base field. The only field for which polynomial time algorithms were previously known is F = Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a ΣΠΣ(k, d, n) circuit to k variables, but preserves the identity structure. Key words. depth3 circuits; polynomial identity testing; derandomization; blackbox; Chinese remaindering; algebra homomorphism
On Identity Testing of Tensors, Lowrank Recovery and Compressed Sensing
, 2011
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms for depth3 setmultilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [ ..."
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Cited by 18 (5 self)
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms for depth3 setmultilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but has no known such blackbox algorithm. We recast this problem as a question of finding a lowdimensional subspace H, spanned by rank 1 tensors, such that any nonzero tensor in the dual space ker(H) has high rank. We obtain explicit constructions of essentially optimalsize hitting sets for tensors of degree 2 (matrices), and obtain quasipolynomial sized hitting sets for arbitrary tensors (but this second hitting set is less explicit). We also show connections to the task of performing lowrank recovery of matrices, which is studied in the field of compressed sensing. Lowrank recovery asks (say, over R) to recover a matrix M from few measurements, under the promise that M is rank ≤ r. In this work, we restrict our attention to recovering matrices that are exactly rank ≤ r using deterministic, nonadaptive, linear measurements, that are free from noise. Over R, we provide a set (of size 4nr) of such measurements, from which M can be recovered in O(rn 2 + r 3 n) field operations,
Deterministically Testing Sparse Polynomial Identities of Unbounded Degree
, 2008
"... We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representa ..."
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Cited by 17 (1 self)
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We present two deterministic algorithms for the arithmetic circuit identity testing problem. The running time of our algorithms is polynomially bounded in s and m, where s is the size of the circuit and m is an upper bound on the number monomials with nonzero coefficients in its standard representation. The running time of our algorithms also has a logarithmic dependence on the degree of the polynomial but, since a circuit of size s can only compute polynomials of degree at most 2 s, the running time does not depend on its degree. Before this work, all such deterministic algorithms had a polynomial dependence on the degree and therefore an exponential dependence on s. Our first algorithm works over the integers and it requires only blackbox access to the given circuit. Though this algorithm is quite simple, the analysis of why it works relies on Linnik’s Theorem, a deep result from number theory about the size of the smallest prime in an arithmetic progression. Our second algorithm, unlike the first, uses elementary arguments and works over any integral domains, but it uses the circuit in a less restricted manner. In both cases the running time has a logarithmic dependence on the largest coefficient of the polynomial.
Black box polynomial identity testing of generalized depth3 arithmetic circuits with bounded top fanin
 in IEEE Conference on Computational Complexity
"... In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in ..."
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Cited by 16 (4 self)
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In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f(¯x) = ∑ k i=1 hi(¯x) · gi(¯x), where each hi is a polynomial that depends on only ρ linear functions, and each gi is a product of linear functions (when hi = 1, for each i, then we get the class of depth3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi(deg(hi · gi)) = d we say that f is computable by a ΣΠΣ(k, d, ρ) circuit. We obtain the following results. 1. A deterministic blackbox identity testing algorithm for ΣΠΣ(k, d, ρ) circuits that runs in quasipolynomial time (for ρ = polylog(n + d)). In particular this gives the first blackbox quasipolynomial time PIT algorithm for depth3 circuits with k multiplication gates. 2. A deterministic blackbox identity testing algorithm for readk ΣΠΣ circuits (depth3 circuits where each variable appears at most k times) that runs in time n 2O(k2). In particular