can it be said that NL^NL = NL? I was wondering if it can be said that $\mbox{NL}^\mbox{NL}=\mbox{NL}$. That is, a decision problem  $S$ can be decided using a nondeterministic TM using logarithmic space with oracle calls to $S'\in \mbox{NL}$ iff $S\in \mbox{NL}$.
My intuition is yes, because if $S'\in \mbox{NL}$ then there is a NDTM $M$ that decides it in logarithmic space. And so one can use the NDTM $M'$ for $S$ but every oracle call will be simulated on $M$, preserving the logarithmic space constraint.
I am not sure if this approach is correct, and if it is, how to formally explain such an algorithm for the simulation.
It's not quite as simple. Calls to oracle have a &quot;magic&quot; quality of taking only one unit of time. Even if NL can simulate the oracle calls, these simulated invocations of the oracle can take more resources than is needed by the oracle machine. To prove these classes equal you need to prove that calls to the oracle do not improve performance in the computational complexity sense because producing the input to the oracle already requires sufficient resources that the oracle machine cannot use this &quot;magic&quot; to get improved performance.