How can you express the average treatment effect on the treated (ATT) in Pearl's do notation? How can you express the average treatment effect on the treated (ATT) in Pearl's do notation?
Would it be $E(Y|X=1,do(X)=1)-E(Y|X=1,do(X)=0)$?

As Julian and Noah have pointed out, the $\operatorname{do}$ notation is not appropriate for expressing the $\operatorname{ATT}$ (or $\operatorname{ETT}$ - the Effect of Treatment on the Treated - as Pearl calls it). Pearl expresses the $\operatorname{ETT}$ as this:
$$\operatorname{ETT}:=E[Y_1-Y_0|X=1].$$
The notation $Y_1$ and $Y_0$ means, respectively, the (possibly counterfactual) values of $Y$ had $X=1$ or $0.$
So, while the $\operatorname{do}$ framework is too crude to express the $\operatorname{ETT},$ counterfactuals suffice.
For reference, see Causal Inference in Statistics: A Primer, p. 106.
