$\mathbf{NC_2}$ is closed under log-space reduction I actually have to prove the following :

$\mathbf{NL} \subseteq \mathbf{NC_2}$ 

I have the following approach :


I will prove that $\mathbf{PATH} = \{〈D, s, t〉 | \text{D is a directed graph with a path from vertex s to t}\} \in \mathbf{NC_2}$.
I will show that $\mathbf{NC_2}$ is closed under log-space reductions i.e:


$$(1): B \in \mathbf{NC_2} \hbox{ and } A \le_l B \Longrightarrow A \in \mathbf{NC_2}$$ 

where $\le_l$ is the logspace reduction defined as

$$A \le_l B :\Longleftrightarrow (\exists M \hbox{ TM}, \forall x)[x \in A \Longleftrightarrow M(x) \in B]$$

($M$ is a TM which runs in logarithmic space).


Since $\mathbf{PATH}$ is an $\mathbf{NL}$-complete problem the proof will be done.


Proving The 1st part was easy, i am stuck at the second part and have no idea how to proceed.

Any help?
We want to show that if $A\le_lB$ and $B\in NC_2$ then $A\in NC_2$.

Since $A\le_l B$ we have a Turing machine $M_{f,i}(x)$ calculating the i'th bit of the reduction output $f(x)_i$, using logarithmic space. 

$M_{f,i}(x)=1 \iff \left( G_{M_{f,i}(x)},c_{start},c_{acc} \right)\in PATH$

Where $G_{M_{f,i}(x)}$ is the configuration graph of the computation of the logspace machine $M_{f,i}$ on $x$,and $c_{start},c_{acc}$ are the initial and accepting configurations, correspondingly. 

Now your $NC_2$ circuit for $A$ will, "in parallel", compute all the bits of the reduction $f(x)$ (this can be done since you proved $PATH\in NC_2$) and send the result as input to the $NC_2$ circuit for $B$. This results in an $NC_2$ circuit for $A$.