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=== Do演算的完备性(Completeness) ===

=== Do演算的完备性(Completeness) ===

'''定理''': <math> Q = P(y \mid do(x),z) </math> 是可识别的，当且仅当它可以被Do演算的三条规则转化为一个不包含Do算子的表达式。

==== 定理 ====

<math> Q = P(y \mid do(x),z) </math> 是可识别的，当且仅当它可以被Do演算的三条规则转化为一个不包含Do算子的表达式。

要证明上述定理的正确性，需要分别证明Do演算的可靠性(Soundness)和充分性(Sufficiency)。其中可靠性的证明由Judea Pearl于1995年给出<ref name="pearl:95">{{citation | last = Pearl | first = Judea | doi = 10.1093/biomet/82.4.669 | issue = 4 | journal = Biometrika | pages = 669–710 | title = Causal diagrams for empirical research | volume = 82 | year = 1995| url = https://escholarship.org/uc/item/6gv9n38c }}.</ref>，充分性的证明由Yimin Huang和Marco Valtorta于2006年给出<ref name="huang:06">{{cite journal |last1=Huang |first1=Yimin |last2=Valtorta |first2=Marco |date=2006 |title=Pearl's Calculus of Intervention is Complete |url= |journal=Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence |pages=217–224}}</ref>。

要证明上述定理的正确性，需要分别证明Do演算的可靠性(Soundness)和充分性(Sufficiency)。其中可靠性的证明由Judea Pearl于1995年给出<ref name="pearl:95">{{citation | last = Pearl | first = Judea | doi = 10.1093/biomet/82.4.669 | issue = 4 | journal = Biometrika | pages = 669–710 | title = Causal diagrams for empirical research | volume = 82 | year = 1995| url = https://escholarship.org/uc/item/6gv9n38c }}.</ref>，充分性的证明由Yimin Huang和Marco Valtorta于2006年给出<ref name="huang:06">{{cite journal |last1=Huang |first1=Yimin |last2=Valtorta |first2=Marco |date=2006 |title=Pearl's Calculus of Intervention is Complete |url= |journal=Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence |pages=217–224}}</ref>。