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12
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
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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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
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
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.
Derandomizing ArthurMerlin games and approximate counting implies exponentialsize lower bounds
 In Proceedings of the IEEE Conference on Computational Complexity
, 2010
"... Abstract. We show that if ArthurMerlin protocols can be derandomized, then there is a language computable in deterministic exponentialtime with access to an NP oracle, that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that hav ..."
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Abstract. We show that if ArthurMerlin protocols can be derandomized, then there is a language computable in deterministic exponentialtime with access to an NP oracle, that requires circuits of exponential size. More formally, if every promise problem in prAM, the class of promise problems that have ArthurMerlin protocols, can be computed by a deterministic polynomialtime algorithm with access to an NP oracle then there is a language in ENP that requires circuits of size Ω(2n /n). The lower bound in the conclusion of our theorem suffices to construct pseudorandom generators with exponential stretch. We also show that the same conclusion holds if the following two related problems can be computed in polynomial time with access to an NPoracle: (i) approximately counting the number of accepted inputs of a circuit, up to multiplicative factors; and (ii) recognizing an approximate lower bound on the number of accepted inputs of a circuit, up to multiplicative factors.
Uniform Derandomization from Pathetic Lower Bounds
, 2009
"... A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed ..."
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A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic (linearsize lower bounds for general circuits [IM02], nearly cubic lower bounds for formula size [H˚as98], nearly n log log n size lower bounds for branching programs [BSSV03], n 1+cd for depth d threshold circuits [IPS97]). Here, we present two instances where “pathetic ” lower bounds of the form n 1+ɛ would suffice to derandomize interesting classes of probabilistic algorithms. We show: • 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 is 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 by a uniform family of deterministic constantdepth AC0 circuits of subexponential size.
WEAKENING ASSUMPTIONS FOR DETERMINISTIC SUBEXPONENTIAL TIME NONSINGULAR MATRIX COMPLETION
, 2010
"... Abstract. Kabanets and Impagliazzo [9] show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family {fm}m≥1 for arithmetic circuits. In this paper, a special case of CPIT is considered ..."
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Abstract. Kabanets and Impagliazzo [9] show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family {fm}m≥1 for arithmetic circuits. In this paper, a special case of CPIT is considered, namely nonsingular matrix completion (NSMC) under a lowindividualdegree promise. For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of the determinantal complexity dc(fm) of fm. Building on work by Agrawal [17], hardnessrandomness tradeoffs will also be shown in the converse direction, in an effort to make progress on Valiant’s VP versus VNP problem. To separate VP and VNP, it is known to be sufficient to prove that the determinantal complexity of the m × m permanent is m ω(log m). In this paper it is shown, for an appropriate notion of explicitness, that the existence of an explicit multilinear polynomial family {fm}m≥1 with dc(fm) = m ω(log m) is equivalent to the existence of an efficiently computable generator {Gn}n≥1 for multilinear NSMC with seed length O(n 1/ √ log n). The latter is a combinatorial object that provides an efficient deterministic blackbox algorithm for NSMC. “Multilinear NSMC ” indicates that Gn only has to work for matrices M(x) of poly(n) size in n variables, for which det(M(x)) is a multilinear polynomial. 1.
Academy of Sciences of the Czech Rep. Prague, Czech Rep.
"... Devising an efficient deterministic – or even a nondeterministic subexponential time – algorithm for testing polynomial identities is a fundamental problem in algebraic complexity and complexity at large. Motivated by this problem, as well as by results from proof complexity, we investigate the com ..."
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Devising an efficient deterministic – or even a nondeterministic subexponential time – algorithm for testing polynomial identities is a fundamental problem in algebraic complexity and complexity at large. Motivated by this problem, as well as by results from proof complexity, we investigate the complexity of proving polynomial identities. To this end, we study a class of equational proof systems, of varying strength, operating with polynomial identities written as arithmetic formulas over a given ring. A proof in these systems establishes that two arithmetic formulas compute the same polynomial, and consists of a sequence of equations between polynomials, written as arithmetic formulas, where each equation in the sequence is derived from previous equations by means of the polynomialring axioms. We establish the first nontrivial upper and lower bounds on the size of equational proofs of polynomial identities, as follows: 1) Polynomialsize upper bounds on equational proofs of identities involving symmetric polynomials and interpolationbased identities. In particular, we show that basic properties of the elementary symmetric polynomials are efficiently provable already in equational proofs operating with depth4 formulas, over infinite fields. This also yields polynomialsize depth4 proofs of the Newton identities, providing a positive answer to a question posed by Grigoriev and Hirsch [6]. 2) Exponentialsize lower bounds on (full, unrestricted) equational proofs of identities over certain specific rings. 3) Exponentialsize lower bounds on analytic proofs operating with depth3 formulas, under a certain regularity condition. The “analytic ” requirement is, roughly, a condition that forbids introducing arbitrary formulas in a proof and the regularity condition is an additional structural restriction. 4) Exponentialsize lower bounds on oneway proofs (of unrestricted depth) over infinite fields. Here, oneway
On Lower Bounds for Constant Width Arithmetic Circuits
, 907
"... Abstract. The motivation for this paper is to study the complexity of constantwidth arithmetic circuits. Our main results are the following. 1. For every k> 1, we provide an explicit polynomial that can be computed by a linearsized monotone circuit of width 2k but has no subexponentialsized monot ..."
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Abstract. The motivation for this paper is to study the complexity of constantwidth arithmetic circuits. Our main results are the following. 1. For every k> 1, we provide an explicit polynomial that can be computed by a linearsized monotone circuit of width 2k but has no subexponentialsized monotone circuit of width k. It follows, from the definition of the polynomial, that the constantwidth and the constantdepth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardnessrandomness tradeoffs for identity testing constantwidth commutative circuits analogous to [KI03,DSY08]. 1
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).