Decidability of equivalence of two context free grammars I got a question regarding the decidability of equivalence of two context free grammars:


  Construct a Turing machine that decides whether $L(G) = L(H)$, where $G$ and $H$ are two context free grammars.


This question is taken from Sipser's book on theory of computation. 

My current idea is given that $G$ and $H$ are CFG's, we know that there exists push down automata that accept the languages described by $G$ and $H$. We can simulate a PDA on a Turing machine and hence, we can convert the problem to the equality of Turing machines, such that $M_1$ and $M_2$ are TMs and $L(M_1) = L(M_2)$ which is known to be undecidable.

My question is, are these steps fine to do (with a bit more formalism)?
Let me prove that the problem of determining whether $x = 0$ is undecidable.

Let $A$ be a Turing machine that accepts only $x$, and let $B$ be a Turing machine that accepts only $0$. Then $x = 0$ iff $L(A) = L(B)$, so according to your argument, determining whether $x = 0$ should be undecidable.

I hope that this example helps you understand why your reasoning doesn't work. Instead, try to to use the fact that CFG universality (given a grammar $G$ over $\Sigma$, does $L(G)$ equal $\Sigma^*$?) is undecidable.