Lower Bounds on Synchronous Combinational Complexity

Synchronous combinational complexity, a measure of the size of logic circuits without races, is investigated in this paper. The first author has presented a method for obtaining an $O(n\log n)$ lower bound to synchronous combinational complexity and has shown that this bound applies to “almost all” Boolean functions in n variables. However, he could not constructively exhibit functions to which the lower bound applied (although Wolfgang Paul did produce an example). In this paper we weaken and extend the hypothesis of the lower bound so that a larger class of functions satisfies it and apply it to the determinant and marriage functions of $GF(2)$.