Smallest Formulas for Parity of 2k Variables Are Essentially Unique

Authors:
Jun Tarui
Affiliations:
University of Electro-Comm,Chofu, Tokyo, Japan 182-8585
Venue:
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Year:
2008

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Abstract

For n= 2k, we know that the size of a smallest AND/OR/ NOT formula computing the Boolean function is exactly n2: For any n, it is at least n2by classical Khrapchenko's bound, and for n= 2kwe easily obtain a formula of size n2by writing and recursively expanding We show that for n= 2kthe formula obtained above is an essentially unique one that computes with size n2. In the equivalent framework of the Karchmer-Wigderson communication game, our result means that an optimal protocol for Parity of 2kvariables is essentially unique.