On zero error algorithms having oracle access to one query

Abstract

It is known that ${\rm S}_{2}^{p} \subseteq {\rm ZPP}^{NP}$ (Cai, 2001). The reverse direction of whether ZPP^NP is contained in ${\rm S}_{2}^{p}$ remains open. We show that if the zero-error algorithm is allowed to ask only one query to the NP oracle (for any input and random string), then it can be simulated in ${\rm S}_{2}^{p}$. That is, we prove that ${\rm S}_{2}^{p}$. Next we consider whether the above result can be improved as ${\rm ZPP}^{NP[1]} \subseteq {\rm P}^{NP}$ and point out a difficulty in doing so. Via a simple proof, we observe that BPP ⊆ ZPP^NP[1] (a result implicitly proven in some prior work). Thus, achieving the above improvement would imply BPP ⊆ P^NP, settling a long standing open problem.

We then argue that the above mentioned improvement can be obtained for the next level of the polynomial time hierarchy. Namely, we prove that ${\rm ZPP}^{\Sigma_{2}^{p}[1]} \subseteq {\rm P}^{\Sigma_{2}^{p}[2]}$. On the other hand, by adapting our proof of our main result it can be shown that ${\rm ZPP}^{\Sigma_{2}^{p}[1]} \subseteq {\rm S}_{2}^{\rm NP[1]}$. For the purpose of comparing these two results, we prove that ${\rm P}^{\Sigma_{2}^{p}} \subseteq {\rm S}_{2}^{\rm NP[1]}$. We conclude by observing that the above claims extend to the higher levels of the hierarchy: for k ≥ 2,

${\rm ZPP}^{\Sigma_{k}^{p}[1]} \subseteq {\rm P}^{\Sigma_{k}^{p}[2]}$ and ${\rm P}^{\Sigma_{k}^{p}} \subseteq {\rm S}_{2}^{\Sigma_{k-1}^{p}[1]}$.