On Condensation of Block Sensitivity, Certificate Complexity and the $\mathsf{AND}$ (and $\mathsf{OR}$) Decision Tree Complexity

Abstract

Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f)=k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity measure is $\Omega (k)$? In this work, we study the condensability of block sensitivity, certificate complexity, AND (and OR) query complexity and Fourier sparsity. We show that block sensitivity does not condense under restrictions, unlike sensitivity: there exists a Boolean function $f$ with query complexity $k$ such that any restriction of $f$ to $O(k)$ variables has block sensitivity $O(k^{\frac{2}{3}})$. This answers an open question in Göös, Newman, Riazanov, and Sokolov (2024) in the negative. The same function yields an analogous incondensable result for certificate complexity. We further show that $\mathsf{AND}$(and $\mathsf{OR}$) decision trees are also incondensable.

v2: Removed a proof that was a minor modification of a result from Hart, Hosseini, and Iosevich (Condensing Hardness in Boolean Functions)