A well-mixed function with circuit complexity 5n ± o(n): tightness of the Lachish-Raz-type bounds

Authors:
Kazuyuki Amano;Jun Tarui
Affiliations:
Dept of Comput Sci, Gunma Univ., Kiryu, Gunma, Japan;Dept of Info & Comm Eng, Univ of Electro-Comm, Chofu, Tokyo, Japan
Venue:
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Year:
2008

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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 U2 is 5n+o(n). This shows that a lower bound method on the size of a U2-circuit that uses the property of k-mixed, which gives the current best lower bound of 5n - o(n) on a U2-circuit size (Iwama, Lachish, Morizumi and Raz [STOC '01, MFCS '02]), has reached the limit.