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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
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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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.
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 21 (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 18 (5 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
Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes
 Proc. of the 43rd annual STOC, ACM Press
, 2011
"... A (q, k, t)design matrix is an m × n matrix whose pattern of zeros/nonzeros satisfies the following designlike condition: each row has at most q nonzeros, each column has at least k nonzeros and the supports of every two columns intersect in at most t rows. We prove that for m ≥ n, the rank of ..."
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Cited by 14 (8 self)
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A (q, k, t)design matrix is an m × n matrix whose pattern of zeros/nonzeros satisfies the following designlike condition: each row has at most q nonzeros, each column has at least k nonzeros and the supports of every two columns intersect in at most t rows. We prove that for m ≥ n, the rank of any (q, k, t)design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n − ( ) 2 qtn 2k Using this result we derive the following applications: Impossibility results for 2query LCCs over large fields. A 2query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are
Derandomizing polynomial identity testing for multilinear constantread formulae
 Electronic Colloquium on Computational Complexity, Tech. Rep
"... Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subex ..."
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Abstract—We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subexponentialtime deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasipolynomial time in general, and polynomial time for constant depth. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of readonce formulae, and for multilinear depthfour circuits. Keywordsarithmetic circuit; boundeddepth circuit; derandomization; polynomial identity testing; I.
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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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.
Jacobian hits circuits: Hittingsets, lower bounds for depthD occurk formulas & depth3 transcendence degreek circuits
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Simple Affine Extractors using Dimension Expansion
, 2009
"... Let Fq be the field of q elements. An (n, k)affine extractor is a mapping D: F n q → {0, 1} such that for any kdimensional affine subspace X ⊆ F n q, D(x) is an almost unbiased bit when x is chosen uniformly from X. Loosely speaking, the problem of explicitly constructing affine extractors gets ha ..."
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Let Fq be the field of q elements. An (n, k)affine extractor is a mapping D: F n q → {0, 1} such that for any kdimensional affine subspace X ⊆ F n q, D(x) is an almost unbiased bit when x is chosen uniformly from X. Loosely speaking, the problem of explicitly constructing affine extractors gets harder as q gets smaller and easier as k gets larger. This is reflected in previous results: When q is ‘large enough’, specifically q = Ω(n 2), Gabizon and Raz [3] construct affine extractors for any k ≥ 1. In the ‘hardest case’, i.e. when q = 2, Bourgain [2] constructs affine extractors for k ≥ δn for any constant (and even slightly subconstant) δ> 0. Our main result is the following: Fix any k ≥ 2 and let d = 5n/k. Then whenever q> 2 · d 2 and p = char(Fq)> d, we give an explicit (n, k)affine extractor. For example, when k = δn for constant δ> 0, we get an extractor for a field of constant size Ω ( () 1 2). δ Thus our result may be viewed as a ‘fieldsize/dimension ’ tradeoff for affine extractors. Although for large k we are not able to improve (or even match) the previous result of [2], our construction and proof have the advantage of being very simple: Assume n is prime and d is odd, and fix any nontrivial linear map T: Fn q ↦ → Fq. Define QR: Fq ↦ → {0, 1} by QR(x) = 1 if and only if x is a quadratic residue. Then, the function D: F n q ↦ → {0, 1} defined by D(x) � QR(T (x d)) is an (n, k)affine extractor. Our proof uses a result of Heur, Leung and Xiang [4] giving a lower bound on the dimension of products of subspaces. 1
Improved rank bounds for design matrices and a new proof of Kelly’s theorem
, 2012
"... We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et al. [Rank bounds for design matrices with applications to combinatorial geometry and locall ..."
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We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519–528] in which they were used to answer questions regarding point configurations. In this work, we derive nearoptimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly’s theorem, which is the complex analog of the Sylvester–Gallai theorem. 2010 Mathematics Subject Classification: primary 52C35; secondary 68Q99, 94B65. 1.