We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients' vector of a polynomial and the coefficients' vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits: monotone circuits, orthogonal formulas, non-canceling formulas, and noise-resistant formulas. One ingredient of our proof is an explicit map that has exponentially small discrepancy for every partition of the input variables into two sets of roughly the same size. We give two additional applications of this explicit map: to extractors construction and to communication complexity.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics
- Algebraic complexity