# Karp-like reductions vs Cook-like reductions for Functional Complexity Classes.

Assume we have two counting (functional) complexity classes $A$ and $B$. Suppose that

1. Under Karp-like reductions $A$ is strictly inside $B$.
2. Under Cook-like reductions $P^A=P^B$.

What does this tell us about the relationship between these two counting complexity classes?

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I'm not sure I understand. Are you stating that A and B satisfy the given properties and then asking what this might mean ? – Suresh Venkat Oct 28 '11 at 3:01
@suresh exactly! – Tayfun Pay Oct 28 '11 at 12:27
Some motivation might help – Suresh Venkat Oct 28 '11 at 17:21
So if A is Karp reducible to B then A is Turing reducible to B. If A is not Karp reducible to B, then it doesn't necessarily hold that A is not Turing reducible to B. – Tayfun Pay Nov 25 '11 at 3:05

If $A$ is closed under Cook-like reductions, then (2) would imply $B \subseteq A$, contradicting (1), so it tells you that $A$ is not closed under Cook-like reductions.
One can then ask about other intermediate types of reductions to get a better sense of just how many queries are needed and in what way e.g. are the classes equivalent under truth-table (nonadaptive) reductions? What's the best bound we can put on the number of queries ($2$, $O(1)$, $O(\log n)$?) Since these are counting classes, one could also ask about parsimonious reductions.