A 3n-Lower Bound on the Network Complexity of Boolean Functions

Abstract Let f n :{0, 1} 2⌜lgn⌝+1+n →{0, 1} be the Boolean function f n (a,b,q,z 1 …,z n )=q ⋁ j=1 n z j ( a =j∨ b =j)∨ ⊕ j=1 n z j ( a =j∨ b =j) where a → a is any surjective map B ⌜ lgn ⌝ → {1, 2, …, n }. We prove C ( f n )⩾3 n −2 where C ( f n ) is the minimal size of a Boolean network which computes f n over the base of all 16 binary Boolean operations. This lower bound corresponds to an upper bound of 3 n provided that we count only those gates that depend on some variable z j .