The 1-Versus-2 Queries Problem Revisited

Abstract

The 1-versus-2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a $\Sigma^{p}_{k}$-complete set L _k has an efficient simulation that makes at most 1 query to L _k . We obtain solutions to this problem under hypotheses weaker than previously considered. We prove that:

• 1 For each k ≥ 2, ${\rm P}^{\Sigma^{p}_{k}[2]}_{tt} \subseteq {\rm ZPP}^{\Sigma^{p}_{k}[1]} \Longrightarrow {\rm PH} = \Sigma^{p}_{k}$, and

• 1 ${\rm P}^{{\rm NP}[2]}_{tt} \subseteq {\rm ZPP}^{{\rm NP}[1]} \Longrightarrow {\rm PH} = {\rm S}^{p}_{2}$.

Here, for a complexity class ${\rm \mathcal{C}}$ and integer j ≥ 1, we model ${\rm ZPP}^{{\rm \mathcal{C}}[j]}$ to be the class of problems solvable by zero-error randomized algorithms that always run in polynomial time, make at most j queries to ${\rm \mathcal{C}}$, and succeed with probability only 1/2 + 1/poly(·). This same model of ${\rm ZPP}^{{\rm \mathcal{C}}[j]}$, also considered in [CC06], subsumes the class of problems solvable by randomized algorithms that always answer correctly in expected polynomial time and make at most j queries to ${\rm \mathcal{C}}$.

Hemaspaandra, Hemaspaandra, and Hempel [HHH98], for k > 2, and Buhrman and Fortnow [BF99], for k = 2, had obtained the same consequence as of ours in (1) using the stronger hypothesis ${\rm P}^{\Sigma^{p}_{k}[2]}_{tt} \subseteq {\rm P}^{\Sigma^{p}_{k}[1]}$. Fortnow, Pavan, and Sengupta [FPS] had obtained the same consequence as of ours in (2) using the stronger hypothesis ${\rm P}^{{\rm NP}[2]}_{tt} \subseteq {\rm P}^{{\rm NP}[1]}$.

Our results may also be viewed as steps towards obtaining solutions to the most general form of the 1-versus-2 queries problem: For any k ≥ 1, whether ${\rm P}^{\Sigma^{p}_{k}[2]}_{tt}$ can be simulated in ${\rm BPP}^{\Sigma^{p}_{k}[1]}$.

Research supported by the New Researcher Grant of the University of South Florida.