Front-door adjustment gives a simple closed-form identification formula under the classical front-door criterion, but its applicability is often viewed as narrow. By contrast, the general ID algorithm can identify many more causal effects in arbitrary graphs, yet typically outputs algebraically complex expressions that are hard to estimate and interpret. We show that many such graphs can in fact be reduced to a standard front-door setting via front-door reducibility (FDR), a graphical condition on acyclic directed mixed graphs that aggregates variables into super-nodes $(\boldsymbol{X}^{*},\boldsymbol{Y}^{*},\boldsymbol{M}^{*})$. We characterize the FDR criterion, prove it is equivalent (at the graph level) to the existence of an FDR adjustment, and present FDR-TID, an exact algorithm that finds an admissible FDR triple with correctness, completeness, and finite-termination guarantees. Empirical examples show that many graphs far outside the textbook front-door setting are FDR, yielding simple, estimable adjustments where general ID expressions would be cumbersome. FDR therefore complements existing identification methods 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有限。相比之下，通用的ID算法能在任意图中识别更多因果效应，但通常输出代数形式复杂的表达式，难以估计和解释。我们证明，通过前门可约性这一有向无环混合图上的图条件，许多此类图实际上可约简至标准前门设定。该条件将变量聚合为超节点$(\boldsymbol{X}^{*},\boldsymbol{Y}^{*},\boldsymbol{M}^{*})$。我们刻画了FDR准则，证明其在图层面等价于FDR调整的存在性，并提出FDR-TID算法——一种能正确、完备且有限终止地找到可容许FDR三元组的精确算法。实证案例表明，许多远超出教科书前门设定的图具有FDR性质，可产生简洁、可估计的调整公式，而通用ID表达式则可能极为繁琐。因此，FDR通过优先考虑可解释性与计算简洁性，同时不牺牲混合图的普适性，对现有识别方法形成了有效补充。