Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.

翻译：珀尔的因果层级揭示，观察性、干预性和反事实性查询具有本质区别。我们提出该问题的量化版本：在已知低层级回答的情况下，指定高层级因果答案需要多少额外比特？我们通过查询类描述长度对此进行形式化——即针对某类查询的结构因果模型（SCM）所诱导的答案奥拉克尔的柯尔莫哥洛夫复杂度。主要构造给出了二元无环SCM，其观察性分布具有恒定描述长度，而单变量干预答案奥拉克尔的描述长度为$Θ(n^2)$。一个度敏感的界表明，入度为$d$的有限门模式SCM的观察-干预间隙最多为$O(nd \log(en/d) + n \log n)$，使得二次构造在稠密情形下达到阶最优，而有根树构造在有界入度情形下达到阶最优。对于每个固定$\varepsilon < 1/4$的$\varepsilon$-精确全变差描述，该二次分离依然成立。在更高层级上，完全硬干预奥拉克尔仍可留下$Θ(n)$的反事实描述间隙。一个通用的模糊至比特定理及其香农类比表明，这些间隙等于剩余高层级模糊度的对数（忽略低阶项）。