Sensitivity, block sensitivity, and ℓ-block sensitivity of boolean functions

AbstractSensitivity is one of the simplest, and block sensitivity one of the most useful, invariants of a boolean function. Nisan [SIAM J. Comput. 20(6) (1991) 999] and Nisan and Szegedy [Comput. Complexity 4(4) (1994) 301] have shown that block sensitivity is polynomially related to a number of measures of boolean function complexity. The main open question is whether or not a polynomial relationship exists between sensitivity and block sensitivity. We define the intermediate notion of ℓ-block sensitivity, and show that, for any fixed ℓ, this new quantity is polynomially related to sensitivity. We then achieve an improved (though still exponential) upper bound on block sensitivity in terms of sensitivity. As a corollary, we also prove that sensitivity and block sensitivity are polynomially related when the block sensitivity is Ω(n)