Recently I am interested in DAG and the causal inference as to Judea Pearl's work. Here is a record of my understanding.

Concepts

Front door criterion and front door path

The first thing to clarify is so-called the "Front door criterion and front door path", following is my version of understanding.
Basically, the front door criterion requires that:

1. $X_i -> X_6 -> Y$ is a front door path, and in this path $X_6$ blocks $X_i$ to $Y$.
2. All "back door paths" from $X_6$ to $Y$ are blocked. Like $X_6$ <- A_set_of_vars -> $Y$, basically a fork like path. Here, this is indeed the case.
   And basically, if we have observed a set of variables that satiesfying front door criterion, then the causal relationship between x and y is identifiable.

Back door criterion and back door path

The other concept is back door criterion and corresponding back door path. Here, we say that $(X_3,X_4)$ or $(X_5, X_6)$ are the "Z" set that meets the back door criterion relative to the causal inference from $X_i$ to $X_j$. And we then list the conditions.

1. $(X_3, X_4)$ is not the decendents of $X_i$, my understanding is that there is no path like $X_i -> X_3$. But the reverse could be true.
2. $(X_3, X_4)$ blocks all paths (regardless of the directions, as long as it connected) pointing to $X_i$ and also pointing or pointed at $X_j$.

And backdoor criterion also provides an indentification tool.