Determining whether a CFG is $LL(k)$ for any $k$?

Question

In Knuth's original paper on $LR(k)$ grammars, he proved that the decision problem "Given a CFG $G$, is there a $k$ such that $G$ is an $LR(k)$ grammar?" is undecidable.

Is there a similar result showing that it is undecidable whether a given CFG is an $LL(k)$ grammar for some choice of $k$? Or is this problem known to be decidable?

I know that the $LL(1)$ property can be checked. See these slides (48++). Have you tried adapting Knuth's proof to $LL$? — Raphael, Jul 27 '12 at 22:54

score 5 · Accepted Answer · answered Oct 07 '12 at 18:36

This is undecidable. It is theorem 11 in Properties of deterministic top down grammars by Rosenkrantz and Stearns. Theorem 12 in the same paper however states that the problem becomes decidable if the grammar is known to be $LR(k')$ for some (possibly different) $k'$.

Determining whether a CFG is $LL(k)$ for any $k$?

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