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It is a well-known fact that $ \mathsf{NL} = \cup_{k>0} \mathsf{2NFA[k]} $, where $ \mathsf{2NFA[k]} $ is the class of languages recognized by two-way nondeterministic finite automata with $ k>0 $ input heads, shortly 2nfa(k).

I have two but similar questions:

Is there any known language in $ \mathsf{NL} $ requiring super-linear time, where the model is a standard space-bounded NTM having a read-only input tape and a read/write work tape?

Is there any known language recognized by some 2nfa(k) in super-linear time but not recognized by any 2nfa(k) in linear time?

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  • $\begingroup$ I think both your questions are the same question. Do you have a reason for them being different, or are you just unsure? $\endgroup$ Jul 25, 2012 at 13:50
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    $\begingroup$ @LukeMathieson: I am not sure whether for a given $ t(n) $ time bounded NTM, there exists an equivalent $ O(t(n)) $ time bounded 2nfa(k) for some $ k>0 $, and vice versa. $\endgroup$ Jul 25, 2012 at 14:00
  • $\begingroup$ From the equivalence though, you know that every language in NL has a 2nfa(k) and that every language that has a 2nfa(k) is in NL. $\endgroup$ Jul 25, 2012 at 14:17
  • $\begingroup$ @LukeMathieson: Yes! $\endgroup$ Jul 25, 2012 at 14:25
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    $\begingroup$ In the first question, you wrote "requiring super-linear time," but in which computational model? $\endgroup$ Jul 26, 2012 at 13:19

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I believe that the "state of the art" is that it is extremely hard to prove superlinear lower bounds even for NP-complete problems. If you're fine with conditional lower bounds, then the field of fine-grained complexity gives you many problems that are in P but require superlinear time under some assumptions. In this survey of fine-grained complexity there are several such problems and some of them are in NL (for example there is an obvious NL algorithm for "Detecting if an edge-weighted graph has a triangle of negative total edge weight").

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