Dyck Languages in NC$1$

The ordinary Dyck Language D$1$ is essentially the language we have called $$Paren$$ in the lectures $$ here we$$ll redefine it to have alphabet $\{$a$,$b$\}$ and context$-$free grammar S $->$ aBb, S $->$ SS$,$ and S $->\backslash$epsi $.$ The Dyck$-2$ Language D$2$ is similar, but has two types of parentheses which must be nested properly. Its alphabet is $\{$a$,$ b$,$ c$,$ d$\}$ and its context$-$free grammar is S $->$ aSb, S $->$ cSd$,$ S $->$ SS$,$ and S $->\backslash$epsi $.$

$($a$)$ Prove that D$1$ is in the class NC$1.($b$)$ Prove that D$2$ is in the class NC$1.$

In fact both languages are in the perhaps smaller class TC$0,$ and you should concentrate on methods using the MAJORITY language that we will prove to be in NC$1$ in Week $13.$ You may also find it useful to think about why these languages are in LOGSPACE, but you will have to adapt those methods to use them in NC$1$