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

TL;DR

This work investigates the limits of hardness condensation for Boolean function measures by constructing explicit Rubinstein-like functions that resist condensation. It proves incondensability for block sensitivity, certificate complexity, and AND/OR decision-tree complexity: restricting to $O(k^3)$ variables cannot preserve the original scale, with $bs(f_ ho), C(f_ ho)=O(k^2)$ despite $bs(f)=C(f)=k^3$. It contrasts these negative results with a weak, non-lossless condensation for Fourier sparsity, showing a more favorable behavior under restrictions to $O(k^2)$ variables while preserving sparsity $k$. The results illuminate the boundaries of condensation as a tool for proving lower bounds, relate condensability to known measure separations (e.g., Tribes, Rubinstein), and pose open questions about the extent of possible condensation, especially for sparsity-based and monotone measures.

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 $Ω(k)$? 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. In contrast, we prove that Fourier sparsity admits a weak form of condensation.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2: AND-Decision Tree
• Definition 2.3: OR-Decision Tree
• Definition 3.1: Modified Rubinstein's Function
• Lemma 3.2
• proof
• Theorem 3.3