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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
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
Quasipolynomialtime Identity Testing of NonCommutative and ReadOnce Oblivious Algebraic Branching Programs
, 2012
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for readonce oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), ..."
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Cited by 14 (4 self)
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for readonce oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work had no known such blackbox algorithm. Here we obtain the first quasipolynomial sized hitting sets for this class, when the order of the variables is known. This work can be seen as an algebraic analogue of the results of Nisan [Nis92] and ImpagliazzoNisanWigderson [INW94] for spacebounded pseudorandom generators. We also show that several other circuit classes can be blackbox reduced to readonce oblivious ABPs, including setmultilinear ABPs (a generalization of depth 3 setmultilinear formulas), noncommutative ABPs (generalizing noncommutative formulas), and (semi)diagonal depth4 circuits (as introduced by Saxena [Sax08], and recently shown by Mulmuley [Mul12] to have implications for derandomizing Noether’s Normalization Lemma). For setmultilinear ABPs and noncommutative ABPs, we give quasipolynomialtime blackbox PIT algorithms, where the latter case involves evaluations over the algebra of (D + 1) × (D + 1) matrices, where D is the depth of the ABP. For (semi)diagonal depth4 circuits, we obtain a blackbox PIT algorithm (over any characteristic) whose runtime is quasipolynomial in the runtime of Saxena’s whitebox algorithm, matching the concurrent work of Agrawal, Saha, and Saxena [ASS12]. Finally, by combining our results with the reconstruction algorithm of Klivans and Shpilka [KS06], we obtain deterministic reconstruction algorithms for the above circuit classes.
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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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.
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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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
Identity Testing, multilinearity testing, and monomials in ReadOnce/Twice Formulas and Branching Programs ⋆
"... Abstract. We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero (ACIT). We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice formulas. This algorithm also computes the MLIN predicate, testing if the input ..."
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Abstract. We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero (ACIT). We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice formulas. This algorithm also computes the MLIN predicate, testing if the input circuit computes a multilinear polynomial. We further study two related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, and 2) MonCount: compute the number of monomials in C. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH). We address the above problems on readrestricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits. 1
Monomials, Multilinearity and Identity Testing in Simple ReadRestricted Circuits∗
, 2013
"... We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice or readthrice formulas. In the process, these algorithms also test if the input circuit is compu ..."
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We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice or readthrice formulas. In the process, these algorithms also test if the input circuit is computing a multilinear polynomial. We further study three related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, 2) MonCount: compute the number of monomials in C, and 3) MLIN: test if C computes a multilinear polynomial or not. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH). We address the above problems on readrestricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits.