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 discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.
We give a deterministic polynomial time algorithm for polynomial identity testing in the following two cases:
This paper studies the parameters for which binary ReedMuller (RM) codes can be decoded successfully on the BEC and BSC, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF (2) polynomials on random sets of inputs.For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity.The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about E(m, r), the matrix whose rows are truth tables of all monomials of degree ≤ r in m variables. What is the most (resp. least) number of random columns in E(m, r) that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes.Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate we construct a new code C obtained by tensoring C, such that for every subset S of coordinates, if C can recover from erasures in S, then C can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate.Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [27] bounds from constant degree to linear degree polynomials. * See [2] for a full version of this paper.
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 locally decodable codes and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: is a linear LDC with 2 queries then m = exp(Ω(n)). Previously this was only known for fields of size <<2. We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct a locally decodeable code. 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 2 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 fan-in, that compute the zero polynomial. In particular we show that if such a circuit is simple and minimal and of polynomial size then its rank, r, is only polylogarithmic in the number of variables (a priory it could have been linear).4. We give new PIT algorithms for ΣΠΣ circuits with a bounded top fan-in:(a) A deterministic algorithm that runs in quasipolynomial time.(b) A randomized algorithm that runs in polynomial time and uses only polylogarithmic number of random bits.Moreover, when the circuit is multilinear our deterministic algorithm runs in polynomial time. Previously deterministic subexponential time algorithms for PIT in bounded depth circuits were known only for depth 2 circuits (in the black box model) [GKS90, BOT88, KS01]. In particular, for the special case of depth 3 circuits with 3 multiplication gates our result resolves an open question asked by Klivans and Spielman [KS01].
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 that cannot be computed by a depth d arithmetic circuit of small size, then there exists an efficient deterministic black-box algorithm to test whether a given depth d−5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not. 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 hardness-randomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the arithmetic Nisan-Wigderson generator of Kabanets and Impagliazzo 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 (x 1 , . . . , xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f (x 1 , . . . , xn) satisfies P (x 1 , . . . , xn, f) ≡ 0, then f has a circuit of depth d + 3 and size poly(s, m r ), where m is the total degree of f and r is the degree of y in P . This circuit for f can be found probabilistically in time poly(s, m r ). In the other direction we observe that the methods of Kabanets and Impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits. More formally, if we can derandomize polynomial identity testing for bounded depth circuits, then NEXP does not have bounded depth arithmetic circuits. That is, either NEXP ⊆ P/poly or the Permanent is not computable by polynomial size bounded depth arithmetic circuits.
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