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Boolean Function Sensitivity


The Boolean function sensitivity of a Boolean function f:{0,1}^n->{0,1} at an input x is the number of individual coordinates whose reversal changes f(x). If x^({i}) reverses coordinate i, then

 s(f,x)=|{i:f(x^({i}))!=f(x)}|.

The sensitivity of f is the maximum s(f)=max_(x)s(f,x) (Virza 2011). A coordinate function f(x)=x_i has sensitivity 1, while f(x)=(x_1+...+x_n)mod2 has sensitivity n because every coordinate reversal changes its value.

Averaging s(f,x) over the discrete uniform distribution on the inputs gives average sensitivity. Allowing disjoint sets of coordinates to reverse together gives block sensitivity.


See also

Average Sensitivity, Block Sensitivity, Boolean Function, Sensitivity Conjecture, Spectral Sensitivity

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References

Virza, M. "Sensitivity Versus Block Sensitivity of Boolean Functions." Inform. Process. Lett. 111, 433-435, 2011. https://doi.org/10.1016/j.ipl.2011.02.001.

Cite this as:

Weisstein, Eric W. "Boolean Function Sensitivity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BooleanFunctionSensitivity.html

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