Amir Yehudayoff

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 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.

Lower Bounds and Separations for Constant Depth Multilinear Circuits

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth d and product-depth d+1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that (1) There exists a multilinear circuit of product-depth d+1 and of polynomial size computing f and (2) Every multilinear circuit of product-depth d computing f has super-polynomial size.

Hardness-Randomness 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

Affine extractors over prime fields

f

Direct Products in Communication Complexity

that cannot be computed by a depth

Balancing Syntactically Multilinear Arithmetic Circuits

d

Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes

arithmetic circuit of small size, then there exists an efficient deterministic black-box algorithm to test whether a given depth

Pseudorandom Generators for Regular Branching Programs

d-5

Pseudorandomness for Width 2 Branching Programs

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

Direct product via round-preserving compression

P(x_1,\dots,x_n,y)\equiv0