We study the computational complexity of counterfactual reasoning in relation to the complexity of associational and interventional reasoning on structural causal models (SCMs). We show that counterfactual reasoning is no harder than associational or interventional reasoning on fully specified SCMs in the context of two computational frameworks. The first framework is based on the notion of treewidth and includes the classical variable elimination and jointree algorithms. The second framework is based on the more recent and refined notion of causal treewidth which is directed towards models with functional dependencies such as SCMs. Our results are constructive and based on bounding the (causal) treewidth of twin networks -- used in standard counterfactual reasoning that contemplates two worlds, real and imaginary -- to the (causal) treewidth of the underlying SCM structure. In particular, we show that the latter (causal) treewidth is no more than twice the former plus one. Hence, if associational or interventional reasoning is tractable on a fully specified SCM then counterfactual reasoning is tractable too. We extend our results to general counterfactual reasoning that requires contemplating more than two worlds and discuss applications of our results to counterfactual reasoning with a partially specified SCM that is coupled with data. We finally present empirical results that measure the gap between the complexities of counterfactual reasoning and associational/interventional reasoning on random SCMs.

翻译：我们研究了反事实推理的计算复杂度，并将其与结构因果模型（SCMs）中的关联推理和干预推理的复杂度进行了关联分析。研究表明，在两个计算框架下，针对完全指定的SCMs，反事实推理的难度并不高于关联推理或干预推理。第一个框架基于树宽概念，包含经典变量消除算法和联合树算法；第二个框架基于更新的精细化因果树宽概念，该概念针对具有函数依赖关系的模型（如SCMs）而设计。我们的结论是建设性的，通过将双网络（标准反事实推理中用于同时考量真实世界与想象世界的结构）的（因果）树宽约束为基础SCM结构的（因果）树宽来证明。具体而言，我们证明后者（因果）树宽最多是前者的两倍加一。因此，若关联推理或干预推理在完全指定SCM上可解，则反事实推理同样可解。我们将结论推广至需考量多于两个世界的一般反事实推理，并讨论了结合部分指定SCM与数据的反事实推理应用场景。最后，我们通过随机SCM上的实证研究，量化了反事实推理与关联/干预推理复杂度之间的差距。