$\mathrm{CFL}$ is the class of context-free languages.


Is $\mathrm{CFL}$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $\mathrm{DCFL}$?

  • $\begingroup$ Also see related question: cstheory.stackexchange.com/questions/14873/… $\endgroup$ Jan 14, 2018 at 18:18
  • $\begingroup$ Yes. This is Theorem 4 in the following paper. Philip M. Lewis II, Richard Edwin Stearns, Juris Hartmanis: Memory bounds for recognition of context-free and context-sensitive languages. SWCT (FOCS) 1965: 191-202. Edit: I am sorry. My answer is not helpful. I missed the "small o". Stupid... I am not sure what to do. Delete the answer? $\endgroup$
    – Thomas S
    Jan 16, 2018 at 19:25
  • $\begingroup$ Theorem 4 only shows the $\log^2 n$ bound? $\endgroup$ Jan 16, 2018 at 20:12
  • $\begingroup$ Thank you @Hsien-ChihChang張顯之 ! Yes, I might not have made my question clear enough. I was asking if we can do better than $\log^2(n)$ space. I updated the question changing "less than" to little oh to hopefully clarify. $\endgroup$ Jan 16, 2018 at 20:18
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    $\begingroup$ If $L$ is CF then $L = \varphi( R \cap Dick_k)$ where R is regular, Dick_k is the Dyck language over k symbols and $\varphi$ is an homomorphism. Didn't think about it too much, but $R \cap Dick_k$ should be solvable in $DSPACE(\log(n))$. So a parallel more general problem could be: given $L \in CFL$ and $\varphi$ homomorphism, if $L$ is recognized in $NSPACE(o(\log^2(n)))$, can $\varphi^{-1}(L)$ be recognized in $NSPACE( o(\log^2(n)) )$? $\endgroup$ Sep 3, 2021 at 12:55

1 Answer 1


It is not known whether CFL is contained in space o(log^2(n)). If CFL were contained in space o(log^2(n)), then NL would also be contained in space o(log^2(n)). The question whether NL is contained in space o(log^2(n)) is surprisingly still an open question in complexity theory.

  • $\begingroup$ Thanks so much for your answer! Is this because there is an NL-complete problem in CFL? $\endgroup$ Apr 28 at 14:23
  • $\begingroup$ Oh, maybe this follows because NL is a subset of logCFL and CFL in space o(log^2(n)) would imply that logCFL in o(log^2(n)). Cool! Thank you again for pointing this out! :) $\endgroup$ Apr 30 at 5:15
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    $\begingroup$ Ah, I see. My original question had to do with nspace, not dspace. This is great, but doesn't seem to lead to conclusions about the original question. $\endgroup$ Apr 30 at 5:18

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