Alternation, Sparsity and Sensitivity: Combinatorial Bounds and Exponential Gaps

Sensitivity, Affine Transforms and Quantum Communication Complexity

“…It is conceivable that salt(f ) is much smaller compared to alt(f ) for a Boolean function f and hence that salt(f ) can potentially be upper bounded by poly(s(f )) thereby settling the Sensitivity Conjecture. However, we rule this out by showing the following stronger gap, about the same family of functions demonstrated in [DS18] (see also [GSW16]).…”

“…The authors [DS18] have shown that for any f ∈ F, there exists of a chain of large alternation in f . However, this is not sufficient to argue existence of a chain of large alternation under every linear shift.…”

“…It is conceivable that salt(f ) is much smaller compared to alt(f ) for a Boolean function f and hence that salt(f ) can potentially be upper bounded by poly(s(f )) thereby settling the Sensitivity Conjecture. However, we rule this out by showing the following stronger gap, about the same family of functions demonstrated in [DS18] (see also [GSW16] Block Sensitivity under Affine Transformations : We now generalize our theme of study to the affine transforms over F n 2 . In particular, we explore how to design affine transformations in such a way that block sensitivity of the original function (f ) is upper bounded by the sensitivity of the new function (g).…”

“…This bound for bs(f ), implies that to settle the Sensitivity Conjecture, it suffices to show that alt(f ) is upper bounded by poly(s(f )) for all Boolean functions f . However, the authors [DS18] ruled this out, by exhibiting a family of functions where alt(f ) is at least 2 Ω(s(f )) .…”

“…Proof. We remark that for the family of functions f k ∈ F (Definition 3.2), alt(f k ) ≥ 2 (log sparsity(f k ))/2 − 1 [DS18]. We now use this family F to describe the family of functions g k .…”