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 $Θ(k)$ variables while keeping the complexity preserved at $Θ(k)$? This question, in the context of query complexity, was recently studied by G{ö}{ö}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{ö}{ö}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 $Ω(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.

翻译：对于任意复杂度度量值为 $k \ll n$ 的布尔函数 $f:\{0,1\}^n \to \{0,1\}$，是否可能将函数 $f$ 限制在 $Θ(k)$ 个变量上，同时保持其复杂度为 $Θ(k)$？G{ö}{ö}s、Newman、Riazanov 和 Sokolov（STOC 2024）最近在查询复杂度的背景下研究了这个问题。他们的结果表明，查询复杂度无法实现无损凝聚。他们进一步提出疑问：诸如块敏感度或明确证书复杂度等度量能否实现无损凝聚？在本工作中，我们证明了块敏感度和证书复杂度等决策树度量无法实现无损凝聚。即存在一个布尔函数 $f$，使得对于任意将 $f$ 限制在 $O(\mathcal{M}(f))$ 个变量上的限制，其 $\mathcal{M}(\cdot)$-复杂度至多为 $\tilde{O}(\mathcal{M}(f)^{2/3})$，其中 $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$。该结果也改进了 G{ö}{ö}s、Newman、Riazanov 和 Sokolov（STOC 2024）的结论。我们还在无损凝聚的否定性结果基础上，补充了关于有损凝聚的肯定性结果。具体而言，我们证明对于任意布尔函数 $f$，都存在一个将其限制在 $O(\mathcal{M}(f))$ 个变量上的限制，使得其 $\mathcal{M}(\cdot)$-复杂度至少为 $Ω(\mathcal{M}(f)^{1/2})$，其中 $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$。此外，我们还针对随机化查询复杂度和量子查询复杂度给出了稍弱的肯定性结果。