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Technical Perspective: Low-depth Arithmetic Circuits


The computations of polynomials (over a field, which we shall throughout assume is of zero or large enough characteristic) using arithmetic operations of addition and multiplication (and possibly division) are of course as natural as the computation of Boolean functions via logical gates, and capture many natural important tasks including Fourier transforms, linear algebra, matrix computations and more generally symbolic algebraic computations arising in many settings. Arithmetic circuits are the natural computational model for understanding the computational complexity of such tasks just like Boolean circuits are for Boolean functions. The presence of algebraic structure and mathematical tools supplied by centuries of work in algebra were a source of hope that understanding arithmetic circuits will be much faster and easier than their Boolean siblings. And while we generally know more about arithmetic circuits, their power is far from understood, and in particular, the arithmetic analog VP vs. VNP of the Boolean P vs. NP problem as formulated by Valiant8 is wide open.

The past few years have seen a revolution in our understanding of arithmetic circuits. Surprising new upper bounds, combined with new powerful techniques of proving lower bounds, have brought us to recognize the mysterious importance of very shallow circuits to capture the long-term goals of the field, and to pinpointing with uncommon precision the complexity of natural problems in this model. These developments have rejuvenated the hope, and give a concrete program, to the possible separation of Valiant's classes VP and VNP. The following paper of Gupta et. al. on the "chasm at depth 3" is one of the culminations of this new understanding. I will now briefly explain the "chasm" phenomenon, and some of these developments, on which the authors of the paper elaborate.


 

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