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：对于任意布尔函数 $f:\{0,1\}^n \to \{0,1\}$，若其复杂度度量值为 $k \ll n$，是否可能将 $f$ 限制到 $\Theta(k)$ 个变量上，同时保持复杂度为 $\Theta(k)$？在查询复杂度的背景下，此问题近期由 G{ö}{ö}s、Newman、Riazanov 和 Sokolov（STOC 2024）研究。他们指出，除其他结果外，查询复杂度无法无损凝缩。他们进一步询问，诸如块灵敏度或无歧认证书复杂度等复杂度度量是否能够无损凝缩？本文证明了决策树度量（如块灵敏度和证书复杂度）无法无损凝缩。即存在一个布尔函数 $f$，使得 $f$ 到 $O(\mathcal{M}(f))$ 个变量的任意限制，其 $\mathcal{M}(\cdot)$-复杂度至多为 $\tilde{O}(\mathcal{M}(f)^{2/3})$，其中 $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$。该结果还改进了 G{ö}{ö}s、Newman、Riazanov 和 Sokolov（STOC 2024）的一项结论。此外，我们通过有损凝缩的正向结果对无损凝缩的负面结论进行了补充。具体而言，对于任意布尔函数 $f$，存在 $f$ 到 $O(\mathcal{M}(f))$ 个变量的一个限制，使得其 $\mathcal{M}(\cdot)$-复杂度至少为 $\Omega(\mathcal{M}(f)^{1/2})$，其中 $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$。我们还针对随机化和量子查询复杂度给出了稍弱的正向结果。