Is the decidablity of the following question known?

Given a CFG G, is L(G) regular?

I've seen a bunch of decidability results in the world of CFLs, but I don't think I've ever seen this one, and can't find anything about it.

  • 3
    $\begingroup$ See Hopcroft and Ullman, 1979, page 281. $\endgroup$
    – Kaveh
    Feb 2, 2012 at 23:33

1 Answer 1


By Wikipedia it is undecidable.

  • $\begingroup$ Fair enough; I guess I only looked at the page for CFLs and not CFGs. Still, the page you cite doesn't give any references for that claim, and even a more directed search in Sipser (using the sections in the "further reading" citations and a better index term) doesn't yield anything. Closest it has is "does this TM define a regular language", which falls right out of Rice's thm. $\endgroup$
    – EvanED
    Feb 2, 2012 at 19:14
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    $\begingroup$ It's a consequence of Greibach's Theorem, see last pages here: cis.upenn.edu/~jean/gbooks/PCPh04.pdf $\endgroup$
    – Shir
    Feb 2, 2012 at 19:37
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    $\begingroup$ I found a reference to include in the Wikipedia article. (The source Shir posted looks better than the one I found, in the sense that it justifies the claim of undecidability rather than just stating it, but unfortunately it doesn't seem to be published yet.) $\endgroup$ Feb 2, 2012 at 21:34
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    $\begingroup$ If the alphabet is unary it is decidable. ;) $\endgroup$ Feb 2, 2012 at 22:59
  • $\begingroup$ @David, see my comment below the question for a published reference. $\endgroup$
    – Kaveh
    Feb 2, 2012 at 23:35

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