On the Structure of Bounded Queries to Arbitrary NP Sets

In [Kad87b], Kadin showed that if the Polynomial Hierarchy (PH) has infinitely many levels, then for all $k$, $P^{SAT[k]} \subseteq P^{SAT[k+1]}$. In this paper, we extend Kadin''s technique to show that a proper query hierarchy is not an exclusive property of SAT. In fact, for any $A \in NP \overbrace{low_{3}}$, if PH is infinite, then $P^{A[k]} \subseteq P^{A[k+1]}$. Moreover, for the case of parallel queries, we show that $P^{A||[k+1]}$ is not contained in $P^{SAT||[k]}$. We claim that having a proper query hierarchy is a consequence of the oracle access mechanism and not a result of the ``hardness'''' of a set. To support this claim, we show that assuming PH is infinite, one can construct an intermediate set $B \in NP$ so that $P^{B[k+1]} \subseteq P^{SAT[k]}$. That is, the query hierarchy for $B$ grows as ``tall'''' as the query hierarchy for SAT. In addition, $B$ is intermediate, so it is not ``hard'''' in any sense (e.g., not NP hard under many-one, Turing, or strong nondeterministic reductions). Using these same techniques, we explore some other questions about query hierarchies. For example, we show that is there exists any $A$ such that $P^{A[2]} = P^{SAT[1]}$ then PH collapses to $\Delta^{P}_{3}$.