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

2026-02-01

AI Summary

This work investigates whether key complexity measures of Boolean functions—such as block sensitivity, certificate complexity, and AND/OR decision tree complexity—exhibit “shrinkage” under variable restrictions, i.e., whether their complexity can be preserved on a small subset of variables. By constructing explicit counterexamples and employing Fourier-analytic techniques, the study provides the first proof that none of these three measures are shrinkable, thereby resolving an open question posed by Göös et al. in the negative. Additionally, it uncovers a weak form of shrinkage for Fourier sparsity. A central contribution is the construction of a Boolean function with query complexity $k$ whose block sensitivity drops to $O(k^{2/3})$ under any restriction to $O(k)$ variables, accompanied by matching lower bounds. These results significantly advance the understanding of the interplay between local and global complexity in Boolean functions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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. In contrast, we prove that Fourier sparsity admits a weak form of condensation.

Problem

• block sensitivity
• certificate complexity
• decision tree complexity
• Fourier sparsity
• hardness condensation

Innovation

• block sensitivity
• certificate complexity
• decision tree complexity
• Fourier sparsity
• hardness condensation