Is the language $L$ of coded CFG's Turing decidable?

Consider the following language

$$L$$ = {$$$$ | $$G$$ is a CFG and $$w\in L(G)$$}

Now, I wish to prove that $$L$$ is Turing decidable. My gut tells me to construct a Turing machine that simulates a NPDA that decide whether $$w$$ is indeed in $$L(G)$$. The TM will hold two tapes: one for the input and another to simulate the stack.

Am I in the right direction here?

• There are many algorithms that decide whether $w \in L(G)$ for context-free $G$. See for example Wikipedia. – Yuval Filmus Jan 13 at 17:01