Two lower bounds for branching programs

The first result concerns branching programs having width (log n) °{*). We give an f l (n log n~ log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean fnnction and in part icular 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 i terat ing 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 af-