Theory of Computing ------------------- Title : A Non-linear Time Lower Bound for Boolean Branching Programs Authors : Miklos Ajtai Volume : 1 Number : 8 Pages : 149-176 URL : https://theoryofcomputing.org/articles/v001a008 Abstract -------- We give an exponential lower bound for the size of any linear-time Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small epsilon > 0, if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2^{\epsilon n} which, for all inputs X\subseteq {0,1,...,n-1}, computes in time kn the parity of the number of elements of the set of all pairs with the property x\in X, y\in X, x < y, x+y\in X. For the proof of this fact we show that if A=(a_{i,j})_{i=0,j=0}^{n} is a random n by n matrix over the field with 2 elements with the condition that ``A is constant on each minor diagonal,'' then with high probability the rank of each (delta n) by (delta n) submatrix of A is at least c\delta |log delta|^{-2}n, where c > 0 is an absolute constant and n is sufficiently large with respect to delta. (A preliminary version of this paper has appeared in the Proceedings of the 40th IEEE Symposium on Foundations of Computer Science.)