Abstract
For n = 2k, we know that the size of a smallest AND/OR/ NOT formula computing the Boolean function 
is exactly n
2: For any n, it is at least n
2 by classical Khrapchenko’s bound, and for n = 2k we easily obtain a formula of size n
2 by writing and recursively expanding
We show that for n = 2k the formula obtained above is an essentially unique one that computes 
with size n
2. In the equivalent framework of the Karchmer-Wigderson communication game, our result means that an optimal protocol for Parity of 2k variables is essentially unique.
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Tarui, J. (2008). Smallest Formulas for Parity of 2k Variables Are Essentially Unique. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_10
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DOI: https://doi.org/10.1007/978-3-540-69733-6_10
Publisher Name: Springer, Berlin, Heidelberg
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