I am working on my midterm sample problems and got stuck on this one. Is this language undecidable?

$\qquad L = \{\langle G_1, G_2 \rangle: G_1, G_2 \text{ are CFG and have strings that can be parsed in both grammars }\}$

Is this one the same as $L' = \{\langle G_1, G_2 \rangle: L(G_1) \cap L(G_2) \ne \emptyset\}$? My professor said the above problem can be reduced to Post Correspondence Problem but I found it closer to the decidable of $L'$.

  • $\begingroup$ I am asking if the two are the same or not, because I am confused; if it is the same, $L'$ can be prove without using PCP. But my professor said it can be reduced to PCP. $\endgroup$ – phhoang Nov 16 '15 at 22:20
  • $\begingroup$ parsed by the grammar = accepted in the language, at least in my terminology. And yes, PCP can be reduced to this problem. $\endgroup$ – Hendrik Jan Nov 16 '15 at 22:24
  • $\begingroup$ Recall that an intersection of CFL's is not necessarily a CFL. $\endgroup$ – Ran G. Nov 16 '15 at 22:26
  • $\begingroup$ My proof is set $G_2$ to $\Sigma^{*}$ and $L(G_1) \cap L(G_2) \ne \emptyset \Leftrightarrow L(G_1) \ne \emptyset$, which is undecidable ($E_{TM}$ in Sipser's book). But I'm stuck with a proof with PCP. $\endgroup$ – phhoang Nov 16 '15 at 22:26
  • $\begingroup$ Actually emptiness of context-free languages is decidable (a single one, not the intersection of two). Emptiness of TM languages, that is indecidable. Back to the drawing board. $\endgroup$ – Hendrik Jan Nov 17 '15 at 2:30

Yes, they are the same languages: $L=L'$ (at least the way that I understand the definition of $L$). We say that the grammar $G_1$ can parse a string $x$ iff $x \in L(G_1)$. It follows that $L=L'$. $L$ is defined in a somewhat informal way; good for you for finding a way to provide a more precise formal statement of it.

Indeed, the problem of deciding whether two CFLs have a non-empty intersection is undecidable, so neither $L$ nor $L'$ is decidable.

  • $\begingroup$ Thanks, I got it. Could you reduce to PCP? $\endgroup$ – phhoang Nov 17 '15 at 0:44

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