2014/11/27 by Rafael Oliveira, Oliveira, Rafael, Amir Shpilka +3
Computer Science · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #cs.CC
paper · pdf · doi:10.48550/arxiv.1411.7492
34 pages
arxiv created 2014/11/27 · arxiv updated 2014/12/01
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(nδ), we give a hitting set of size exp(O(n2/3 + 2δ/3)). This implies a lower bound of exp(Ω(n1/2)) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(nδ), we give a hitting set of size exp(O(n2/3 + 4δ/3)). This implies a lower bound of exp(Ω(n1/4)) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,× gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n1- δ ), for regular depth-d multilinear formulas of size exp(nδ), where δ= O((1)/(√(5)d)). This result implies a lower bound of roughly exp(Ω(n(1)/(√(5)d))) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).