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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 62 (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
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 23 (4 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 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
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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Cited by 12 (3 self)
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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.
Algebraic Independence and Blackbox Identity Testing
 ICALP
, 2011
"... Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1,..., fm} ⊂ F[x1,..., xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper ..."
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Cited by 7 (4 self)
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Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1,..., fm} ⊂ F[x1,..., xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(fi)}i = r, assuming fi’s sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing: 1. Given a circuit C and sparse subcircuits f1,..., fm of trdeg r such that D: = C(f1,..., fm) has polynomial degree, we can test blackbox D for zeroness in poly(size(D))r time. 2. Define a ΣΠΣΠδ(k, s, n) circuit C to be of the form ∑k i=1
Pseudorandomness for multilinear readonce algebraic branching programs
 in any order. Electronic Colloquium on Computational Complexity (ECCC
"... We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. F ..."
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Cited by 3 (2 self)
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We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. Furthermore, our result has no known analogue in the model of readonce oblivious boolean branching programs with unknown order, as despite recent work (eg. [BPW11, IMZ12, RSV13]) there is no known pseudorandom generator for this model with subpolynomial seedlength (for unboundedwidth branching programs). This result extends and generalizes the result of Forbes and Shpilka [FS12b] that obtained a nO(lgn)time algorithm when given the order. We also extend and strengthen the work of Agrawal, Saha and Saxena [ASS12] that gave a blackbox algorithm running in time exp((lg n)Ω(d)) for setmultilinear formulas of depth d. We note that the model of multilinear ROABPs contains the model of setmultilinear algebraic branching programs, which itself contains the model of setmultilinear formulas of arbitrary depth. We obtain our results by recasting,
Technion
, 2010
"... 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. Our algorithm runs in time sO(1) ..."
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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. Our algorithm runs in time sO(1) · nk O(k), where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. 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 time nk O(k)+O(k logn) in general, and time nk O(k2)+O(kd) for depth d. 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 formulae. ∗Partially supported by NSF grants 0728809 and 1017597. †Partially supported by NSF grants 0728809 and 1017597.