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

How to prove it

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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 $P$.

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  • $\begingroup$ Thank you. I am new to this. I was wondering can this be proven by contradiction, NP, pumping lemma. $\endgroup$ – Adarsh Bahadur Dec 8 '18 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? $\endgroup$ – Hendrik Jan Dec 9 '18 at 15:24

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