Theory of Computing Systems

, Volume 46, Issue 2, pp 193–221 | Cite as

The 1-Versus-2 Queries Problem Revisited

Article

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 Σkp-complete language Lk has an efficient simulation that makes at most 1 query to Lk. We obtain solutions to this problem for hypotheses weaker than previously considered. We prove that:
  1. (I)

    For each k≥2, \(\mathrm{P}^{\Sigma^{p}_{k}[2]}_{tt}\subseteq \mathrm{ZPP}^{\Sigma^{p}_{k}[1]}\Rightarrow \mathrm{PH}=\Sigma^{p}_{k}\) , and

     
  2. (II)

    PttNP[2]⊆ZPPNP[1]PH=S2p.

     
Here, for any complexity class \(\mathcal{C}\) and integer j≥1, we define \(\mathrm{ZPP}^{\mathcal{C}[j]}\) to be the class of problems solvable by zero-error randomized algorithms that run in polynomial time, make at most j queries to \(\mathcal{C}\) , and succeed with probability at least 1/2+1/poly(⋅). This same definition of \(\mathrm{ZPP}^{\mathcal{C}[j]}\) , also considered in Cai and Chakaravarthy (J. Comb. Optim. 11(2):189–202, 2006), subsumes the class of problems solvable by randomized algorithms that always answer correctly in expected polynomial time and make at most j queries to \(\mathcal{C}\) .

Hemaspaandra, Hemaspaandra, and Hempel (SIAM J. Comput. 28(2):383–393, 1998), for k>2, and Buhrman and Fortnow (J. Comput. Syst. Sci. 59(2):182–194, 1999), for k=2, had obtained the same consequence as ours in (I) using the stronger hypothesis \(\mathrm{P}^{\Sigma^{p}_{k}[2]}_{tt}\subseteq \mathrm{P}^{\Sigma^{p}_{k}[1]}\) . Fortnow, Pavan, and Sengupta (J. Comput. Syst. Sci. 74(3):358–363, 2008) had obtained the same consequence as ours in (II) using the stronger hypothesis PttNP[2]⊆PNP[1].

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

Keywords

Computational complexity Complexity classes Bounded queries Zero-error algorithms 

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of South FloridaTampaUSA

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