0
$\begingroup$

Problem of determining whether a context sensitive language is context free is undecidable.

How to prove it

$\endgroup$

1 Answer 1

0
$\begingroup$

This can be done in a similar way to the proof that it is undecidable whether the language of a context-free grammar is regular. A general statement of such undecidability properties is Greibach's theorem observed by Sheila Greibach in 1968.

Let $C$ be a family of languages effectively closed under union and concatenation with regular sets, and for which equality to $\Sigma^*$ is undecidable (for sufficiently large $\Sigma$). Let $P$ be a proper subset of $C$, which contains all regular languages, and is closed under single letter quotient. Then membership of $P$ is undecidable for $C$.

So, we have to check whether the context-sensitive and context-free languages meet the criteria of Greibach for $C$ and $P$.

$\endgroup$
2
  • $\begingroup$ Thank you. I am new to this. I was wondering can this be proven by contradiction, NP, pumping lemma. $\endgroup$ Dec 8, 2018 at 6:50
  • $\begingroup$ Basically this uses contradiction. If your question would be decidable, then one could decide that $L(G) = \Sigma^*$ for a context-free grammar $G$, which contradicts the fact that we can't. It is a complete puzzle to me how NP or pumping lemma are related to this problem? $\endgroup$ Dec 9, 2018 at 15:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.