In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in the literature. I have been reading some papers on relativized small space classes, and one of the primary tools in this area is the Ruzzo-Simon-Tompa(RST) oracle access mechanism which demands that a non-deterministic space-bounded turing machine act deterministically while making queries to the oracle.

Now consider circuit families with oracle gates - say, $A^B$, where $A$ is a circuit complexity class containing logspace with oracle access to another class $B$, via oracle gates appended to the basis of $A$. Are there any pathological examples similar in spirit to the Ladner-Lynch paper, known for such classes? What would be a RST-like restriction necessary for such classes? In case there are indeed such examples, am I right in guessing that a RST analogue would be to insist that $A$ be a logspace-uniform circuit family?


2 Answers 2


A definition of oracle access that works for small circuit complexity classes (the $AC^k$ and $NC^k$ hierarchies) as well as for logarithmic space classes, with the property that all known inclusions relativize, can be found in this paper:

Klaus Aehlig, Stephen Cook and Phuong Nguyen: Relativizing Small Complexity Classes and their Theories, CSL 2007, Springer LNCS 4646

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    $\begingroup$ Thanks for the pointer. But I still don't have a clear idea as to how RST applies to circuits with oracle gates. I hope you don't mind if I wait for a while to see if there are any other interesting takes on this issue, before accepting your answer. $\endgroup$
    – Nikhil
    Jun 14, 2012 at 4:51

In this paper https://people.cs.rutgers.edu/~allender/papers/pl3.pdf Mitsu Ogihara and I show that the "oracle" #L hierarchy (and related classes) with the Ruzzo-Simon-Tompa access mechanism corresponds exactly to the AC^0 hierarchy over #L. Thus, in at least this instance, the RST access mechanism corresponds nicely to the natural "circuits-with-oracle-gates" formalism.


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