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Arithmetic Circuits: A Chasm at Depth Four
, 2008
"... We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete blackbox derandomization ..."
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We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete blackbox derandomization of Identity Testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general Identity Testing. 1
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 23 (5 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
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 13 (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.
Towards Dimension Expanders Over Finite Fields
"... In this paper we study the problem of explicitly constructing a dimension expander raised by [BISW04]: Let Fn be the n dimensional linear space over the field F. Find a small (ideally constant) set of linear transformations from Fn to itself {Ai}i∈I such that for every linear subspace V ⊂ Fn of dime ..."
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Cited by 3 (1 self)
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In this paper we study the problem of explicitly constructing a dimension expander raised by [BISW04]: Let Fn be the n dimensional linear space over the field F. Find a small (ideally constant) set of linear transformations from Fn to itself {Ai}i∈I such that for every linear subspace V ⊂ Fn of dimension dim(V) < n/2 we have dim Ai(V) ≥ (1 + α) · dim(V), i∈I where α> 0 is some constant. In other words, the dimension of the subspace spanned by {Ai(V)}i∈I should be at least (1 + α) · dim(V). For fields of characteristic zero Lubotzky and Zelmanov [LZ04] completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. 1. We give a constant number of matrices that expand the dimension of every subspace of dimension d < n/2 by a factor of (1 + 1 / log n). 2. We give a set of O(log n) matrices with expanding factor of (1 + α), for some constant α> 0. Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group Z/Zn and smalldiameter Cayley graphs for the group SL2(p).
Classifying polynomials and identity testing
, 2009
"... email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct repre ..."
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email: One of the fundamental problems of computational algebra is to classify polynomials according to the hardness of computing them. Recently, this problem has been related to another important problem: Polynomial identity testing. Informally, the problem is to decide if a certain succinct representation of a polynomial is zero or not. This problem has been extensively studied owing to its connections with various areas in theoretical computer science. Several efficient randomized algorithms have been proposed for the identity testing problem over the last few decades but an efficient deterministic algorithm is yet to be discovered. It is known that such an algorithm will imply hardness of computing an explicit polynomial. In the last few years, progress has been made in designing deterministic algorithms for restricted circuits, and also in understanding why the problem is hard even for small depth. In this article, we survey important results for the polynomial identity testing problem and its connection with classification of polynomials. 1.
Lecturer: Alistair Sinclair Based on scribe notes by:
"... Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. 2.1 Testing Polynomial Identities Randomized algorithms can be dramatically more efficient than their best kn ..."
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Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. 2.1 Testing Polynomial Identities Randomized algorithms can be dramatically more efficient than their best known deterministic counterparts. An example of a realworld problem for which there exists a simple and efficient randomized algorithm, but for which there is no known polynomial time deterministic algorithm is that of testing polynomial identities. The problem takes as input two polynomials Q and R over n variables, with coefficients in some field, and decides whether Q ≡ R. For example, if we had Q(x1, x2) = (1+x1)(1+x2) and R(x1, x2) = 1+x1+x2+x1x2, the algorithm should output ”Yes. ” This problem arises in many contexts, and is a key primitive in computer algebra packages such as Maple and Mathematica. An obvious way to attack this problem would be to expand both Q and R as sums of monomials and compare coefficients. (Thus, for example, Q(x1, x2) above would be expanded to 1 + x1 + x2 + x1x2, while R(x1, x2) is already in the correct form.) However, this method in general requires time exponential in the size of the representation of the input polynomials: consider, e.g., the polynomial ∏ n i=1 (xi + xi+1), which has length
On Circuit Complexity Classes and Iterated Matrix Multiplication
, 2012
"... In this thesis, we study small, yet important, circuit complexity classes within NC 1, such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We sho ..."
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In this thesis, we study small, yet important, circuit complexity classes within NC 1, such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated Matrix Multiplication and its implications in low levels of algebraic complexity theory. More concretely, • We show that extremely modestsounding lower bounds for certain problems can lead to nontrivial derandomization results. – If the word problem over S5 requires constantdepth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomialsize probabilistic threshold circuits can be solved in subexponential time (and more strongly, can be accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size.) – If there are no constantdepth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3by3 matrices, then for every constant d, blackbox identity testing for depthd arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constantdepth AC circuits of subexponential size).