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Block Sensitivity


The block sensitivity of a Boolean function f:{0,1}^n->{0,1} at an input x is the largest number of pairwise disjoint sets of coordinates B_1,...,B_m, each nonempty, such that f(x^(B_j))!=f(x) for every j, where x^(B_j) reverses the coordinates in B_j. The block sensitivity of f is bs(f)=max_(x)bs(f,x) (Virza 2010).

Blocks consisting of singleton sets show that s(f)<=bs(f)<=n, where s(f) is Boolean function sensitivity. For example, the disjunction of n inputs has both quantities equal to n at the all-zero input. The sensitivity conjecture concerns a bound on block sensitivity that is polynomial in s(f).

The quadratic bound in terms of spectral sensitivity has a reported counterexample (Meiburg 2026).


See also

Boolean Function, Boolean Function Sensitivity, Sensitivity Conjecture, Spectral Sensitivity

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References

Meiburg, A. "Block Sensitivity Can Exceed Spectral Sensitivity Squared." 1 Aug 2026. https://arxiv.org/abs/2608.00851.VibeMathed. "Is Block Sensitivity at Most Quadratic in Spectral Sensitivity? (Aaronson, Ben-David, Kothari, Rao and Tal)." Oct. 7, 2026. https://vibemathed.com/problem/block-sensitivity-spectral-sensitivity-squared.Virza, M. "Sensitivity versus Block Sensitivity of Boolean Functions." 8 Dec 2010. https://arxiv.org/abs/1008.0521v2.

Cite this as:

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

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