Front-door adjustment provides a simple closed-form identification formula under the classical front-door criterion, but its applicability is often viewed as narrow and strict. Although ID algorithm is very useful and is proved effective for causal relation identification in general causal graphs (if it is identifiable), performing ID algorithm does not guarantee to obtain a practical, easy-to-estimate interventional distribution expression. We argue that the applicability of the front-door criterion is not as limited as it seems: many more complicated causal graphs can be reduced to the front-door criterion. In this paper, We introduce front-door reducibility (FDR), a graphical condition on acyclic directed mixed graphs (ADMGs) that extends the applicability of the classic front-door criterion to reduce a large family of complicated causal graphs to a front-door setting by aggregating variables into super-nodes (FDR triple) $\left(\boldsymbol{X}^{*},\boldsymbol{Y}^{*},\boldsymbol{M}^{*}\right)$. After characterizing FDR criterion, we prove a graph-level equivalence between the satisfication of FDR criterion and the applicability of FDR adjustment. Meanwhile, we then present FDR-TID, an exact algorithm that detects an admissible FDR triple, together with established the algorithm's correctness, completeness, and finite termination. Empirically-motivated examples illustrate that many graphs outside the textbook front-door setting are FDR, yielding simple, estimable adjustments where general ID expressions would be cumbersome. FDR thus complements existing identification method by prioritizing interpretability and computational simplicity without sacrificing generality across mixed graphs.

翻译：前门调整在经典前门准则下提供了简洁的闭式识别公式，但其适用性常被视为狭窄且严格。尽管ID算法非常有用，且被证明在一般因果图中对因果关系识别有效（若可识别），但执行ID算法并不能保证获得实用、易于估计的干预分布表达式。我们认为前门准则的适用性并不像表面那样有限：许多更复杂的因果图可被约简至前门准则。本文引入前门可约性（FDR），这是针对有向无环混合图（ADMG）的图条件，通过将变量聚合为超节点（FDR三元组）$\\left(\\boldsymbol{X}^{*},\\boldsymbol{Y}^{*},\\boldsymbol{M}^{*}\\right)$，将经典前门准则的适用性扩展至一大类复杂因果图，使其约简为前门设定。在刻画FDR准则后，我们证明了FDR准则的满足与FDR调整的适用性在图层面具有等价性。同时，我们提出FDR-TID算法——一种精确检测可容许FDR三元组的算法，并确立了算法的正确性、完备性和有限终止性。基于实证动机的示例表明，许多超出教科书前门设定的图具有FDR性质，可产生简单、可估计的调整公式，而通用ID表达式则会显得繁琐。因此，FDR通过在不牺牲混合图普适性的前提下优先考虑可解释性与计算简洁性，对现有识别方法形成了补充。