Bounds for Hardness Condensation in the Query Model

TL;DR

This paper investigates the limits of reducing the number of variables in Boolean functions while preserving certain complexity measures, showing that lossless condensation is impossible for key measures but some lossy forms are achievable.

Contribution

It proves that decision tree complexity measures cannot be losslessly condensed, and provides bounds for lossy condensation, advancing understanding of complexity measure preservation under restrictions.

Findings

Lossless condensation of block sensitivity and certificate complexity is impossible.

Any restriction to O(𝓜(f)) variables reduces complexity to at most 𝑂̃(𝓜(f)^{2/3}).

Existence of restrictions maintaining at least Ω(𝓜(f)^{1/2}) complexity for various measures.

Abstract

For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $\Theta(k)$ variables while keeping the complexity preserved at $\Theta(k)$? This question, in the context of query complexity, was recently studied by G{\"{o}}{\"{o}}s, Newman, Riazanov and Sokolov (STOC 2024). They showed, among other results, that query complexity can not be condensed losslessly. They asked if complexity measures like block sensitivity or unambiguous certificate complexity can be condensed losslessly? In this work, we show that decision tree measures like block sensitivity and certificate complexity, cannot be condensed losslessly. That is, there exists a Boolean function $f$ such that any restriction of $f$ to $O(\mathcal{M}(f))$ variables has $\mathcal{M}(\cdot)$-complexity at most $\tilde{O}(\mathcal{M}(f)^{2/3})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$. This also improves upon a result of G{\"{o}}{\"{o}}s, Newman, Riazanov and Sokolov (STOC 2024). We also complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function $f$ there exists a restriction of $f$ to $O(\mathcal{M}(f))$ variables such that its $\mathcal{M}(\cdot)$-complexity is at least $\Omega(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},\lambda\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.