It is a fundamental problem in the study of computational complexity to understand if access to more resources would mean strictly more computational power. In classical complexity, we have seen such examples in terms of Time Hierarchy and Space Hierarchy theorems. Here, time and space are the resources. It is natural to ask such a question in the setting of algebraic complexity setting. Here, access to more resources translates to allowing the model to perform more number of operations which in turn is allowing the "algebraic formulas" to have larger size.
Arithmetic formulas are directed trees where the leaves are labelled by variables and elements of the field, and internal vertices are labelled by + or x. Every internal vertex computes a polynomial by operating on its inputs. It is easy to see that they are a natural model of computation of polynomials, syntactically (symbolically). The size of the arithmetic formulas refers to the number of + and x operations needed to compute the polynomial.
Rephrasing the question in the algebraic setting we ask the following: for any s, \varepsilon >0, are there polynomial computations that can be efficiently computed by arithmetic formulas of size s but not by any arithmetic formula of size s^{1-\varepsilon}?
In this talk, we will restrict ourselves to arithmetic formulas where computation at every vertex is a multilinear polynomial and here we show explicit separations between the expressive powers of multilinear formulas of small-depth and all polynomial sizes. The multilinear restriction is a reasonable one since most polynomials of interest are indeed multilinear.
Formally, we will show that for any s = poly(n) and \delta > 0, we show that there are explicit polynomial families that have multilinear formulas of size at most s but no 'small'-depth multilinear formulas of size less than s^{0.5 - \delta}. Our proof can be viewed as a derandomization of a lower bound technique of Raz (JACM 2009) using \epsilon-biased spaces.
(Joint work with Nutan Limaye and Srikanth Srinivasan. Appeared in the proceedings of ICALP 2018.) |