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

TL;DR

This paper investigates whether certain Boolean function complexity measures can be preserved under variable restrictions, finding that block sensitivity, certificate complexity, and AND/OR decision tree complexities generally cannot be condensed, unlike sensitivity.

Contribution

The paper demonstrates that block sensitivity, certificate complexity, and AND/OR decision tree complexities are incondensable under restrictions, answering open questions and contrasting with sensitivity.

Findings

Block sensitivity does not condense under restrictions.

Certificate complexity is also incondensable.

AND/OR decision tree complexities cannot be condensed.

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 G\"o\"os, 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.