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

Sensitivity 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. Themain open question is whether or not a polynomial relationship exists between sensitivity and block sensitivity. We define the intermediate notion of -block sensitivity, and show that, for any fixed , 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 (n). © 2003 Elsevier Inc. All rights reserved.