Amir Shpilka

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.

Deterministic polynomial identity testing in non commutative models

We give a deterministic polynomial time algorithm for polynomial identity testing in the following two cases: 1. Non commutative arithmetic formulas: the algorithm gets as an input an arithmetic formula in the non-commuting variables x/sub i/,...,x/sub n/ and determines whether or not the output of the formula is identically 0 (as a formal expression). 2. Pure arithmetic circuits: the algorithm gets as an input a pure arithmetic circuit (as defined by N. Nisan and A. Wigderson (1996)) in the variables x/sub i/,...,x/sub n/ and determines whether or not the output of the circuit is identically 0 (as a formal expression). We also give a deterministic polynomial time identity testing algorithm for non commutative algebraic branching programs as defined by N. Nisan (1991). One application is a deterministic polynomial time identity testing for multilinear arithmetic circuits of depth 3. Finally, we observe an exponential lower bound for the size of pure arithmetic circuits for the permanent and for the determinant. (Only lower bounds for the depth of pure circuits were previously known by N. Nisan and A. Wigderson (1996).

Depth-3 arithmetic formulae over fields of characteristic zero

Abstract. In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions.¶The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Neciporuk lower bound on Boolean formulae.

Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits

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

Deterministic polynomial identity testing in non-commutative models

E: \mathbb{F}^n \mapsto \mathbb{F}^m

Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits

is a linear LDC with two queries, then

Testing Fourier Dimensionality and Sparsity

m = \exp(\Omega(n))

Improved Polynomial Identity Testing for Read-Once Formulas

. Previously this was known only for fields of size

Black Box Polynomial Identity Testing of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-In

\ll 2^n

Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

[O. Goldreich e, Comput. Complexity, 15 (2006), pp. 263-296]. (2) We show that from every depth 3 arithmetic circuit (