Reviewing bounds on the circuit size of the hardest functions
In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be 2n/n(1+log n/n-O(1/n))<= L(n) <= 2n/n(1+3 log n/n+O(1/n)). However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {NOT,boolean AND,boolean OR}. (c) 2005 Elsevier B.V. All rights reserved.
computational complexity