Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)
Smallest Formulas for Parity of 2k Variables Are Essentially Unique
For n = 2 k , we know that the size of a smallest AND/OR/ NOT formula computing the Boolean function Open image in new window is exactly n 2: For any n, it is at least n 2 by classical Khrapchenko’s bound, and for n = 2 k we easily obtain a formula of size n 2 by writing and recursively expanding Open image in new window with size n 2. In the equivalent framework of the Karchmer-Wigderson communication game, our result means that an optimal protocol for Parity of 2 k variables is essentially unique.
KeywordsBoolean Function Formula Computing Formula Size Exact Complexity Dual Case
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