Two lower bounds for branching programs
Miklós Ajtai, László Babai, Peter I. Hajnal, János Komlós, Pavel Pudlák · 1986
The first result concerns branching programs having width (log n) °{*).We give an fl(n log n~ log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean fnnction and in particular the following explicit fnnction: "the sum of the input variables is a quadratic residue mod p" where p is any given prime between n 1/4 and n 1/3.This is a strengthening of previous nonlinear lower bounds obtained by Chandra, Furst, Lipton and by Pudlgk.We mention that by iterating our method the result can be further strengthened to lfl(nlog n).The second result is a C" lower bound for read-onceonly branching programs computing an explicit Boolean function.For n = (~), the function computes the parity of the number of triangles in a graph on v vertices.This improves previous exp(cx/n ) lower bounds for other graph functions by Wegener and Z£k.The result implies a linear lower bound for the space complexity of this Boolean function on "eraser machines", i.e. machines that erase each input bit immediately after having read it.