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27
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
 SIAM J. COMPUT
, 2007
"... In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given ..."
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Cited by 30 (9 self)
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In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fanin that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fanin, that
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 12 (3 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
Diagonal Circuit Identity Testing and Lower Bounds
, 2007
"... In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent onl ..."
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Cited by 12 (3 self)
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In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results: 1. Suppose we are given a depth3 circuit (over any field F) of the form: C(x1,..., xn):= k� i=1 ℓ ei,1 i,1 · · · ℓei,s i,s where, the ℓi,j’s are linear functions living in F[x1,..., xn]. We can test whether C is zero deterministically in poly (nk, max{(1 + ei,1) · · · (1 + ei,s)  1 � i � k}) field operations. This immediately gives a deterministic poly(nk2 d) time identity test for general depth3 circuits of degree d. 2. We prove that if the above circuit C(x1,..., xn) computes the determinant � (or permanent) of an m × m formal matrix with a “small ” s = o then � m log m k = 2 Ω(m). Our lower bounds work for all fields F. (Previous exponential lower bounds for depth3 only work for nonzero characteristic.) 3. We present applications of our ideas to depth4 circuits and an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(n k log(d)) field operations over finite fields).
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 12 (2 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.
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 11 (2 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 11 (4 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
The ideal membership problem and polynomial identity testing
, 2007
"... Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then ..."
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Cited by 10 (1 self)
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Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then there is a randomized polynomialtime membership algorithm to test if f ∈ I. This result holds even for f given by a blackbox, when f is of small degree. (b) When I = 〈m1, m2, · · · , mk 〉 for a constant k and f is computed by a ΣΠΣ circuit with output gate of bounded fanin we can test whether f ∈ I in deterministic polynomial time. This generalizes the KayalSaxena result [KS07] of deterministic polynomialtime identity testing for ΣΠΣ circuits with bounded fanin output gate. (c) When k is not constant the problem is coNPhard. However, the problem is upper bounded by coAM PP over the field of rationals, and by coNP ModpP over finite fields. (d) Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I = 〈f1, · · · , fℓ 〉 where each fi ∈ F[x1, · · · , xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above. 1
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 7 (2 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.
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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Cited by 5 (0 self)
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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
Progress on Polynomial Identity Testing
"... Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this ..."
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Cited by 5 (1 self)
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Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem but a complete solution might take a while. In this article we give a soft survey exhibiting the ideas that have been useful. 1