The complexity of Boolean functions
The complexity of Boolean functions
An extension of Khrapchenko's theorem
Information Processing Letters
The complexity of finite functions
Handbook of theoretical computer science (vol. A)
Improvements on Khrapchenko's theorem
Theoretical Computer Science
Communication complexity
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The Shrinkage Exponent of de Morgan Formulas is 2
SIAM Journal on Computing
THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS
Computational Complexity
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
| Hi-index | 5.24 |
For n=2^k, we know that the size of a smallest AND/OR/NOT formula computing the Boolean function Parity(x"1,...,x"n)=Odd(x"1,...,x"n) is exactly n^2: For any n, it is at least n^2 by the classical Khrapchenko bound, and for n=2^k we easily obtain a formula of size n^2 by writing and recursively expanding Odd(x"1,...,x"n)=[Odd(x"1,...,x"n"/"2)@?Even(x"n"/"2"+"1,...,x"n)]@?[Even(x"1,...,x"n"/"2)@?Odd(x"n"/"2"+"1,...,x"n)]. We show that for n=2^k the formula obtained above is an essentially unique one that computes Parity(x"1,...,x"n) with size n^2. In the equivalent framework of the Karchmer-Wigderson communication game, our result means that an optimal protocol for the parity of 2^k variables is essentially unique.