A well-mixed function with circuit complexity 5n ±o(n): Tightness of the lachish-raz-type bounds

Abstract

A Boolean function on n variables is called k-mixed if for any two different restrictions fixing the same set of k variables must induce different functions on the remaining n-k variables. In this paper, we give an explicit construction of an n-o(n)-mixed Boolean function whose circuit complexity over the basis U 2 is 5n+o(n). This shows that a lower bound method on the size of a U 2-circuit that uses the property of k-mixed, which gives the current best lower bound of 5n-o(n) on a U 2-circuit size (Iwama, Lachish, Morizumi and Raz [STOC '01, MFCS '02]), has reached the limit. © 2008 Springer-Verlag Berlin Heidelberg.