Theory of Computing Systems

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

The 1-Versus-2 Queries Problem Revisited



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)


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 \(\mathrm{BPP}^{\Sigma^{p}_{k}[1]}\) .


Computational complexity Complexity classes Bounded queries Zero-error algorithms 


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  1. 1.
    Amir, A., Beigel, R., Gasarch, W.: Some connections between bounded query classes and non-uniform complexity. Inf. Comput. 186(1), 104–139 (2003) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bshouty, N., Cleve, R., Gavaldà, R., Kannan, S., Tamon, C.: Oracles and queries that are sufficient for exact learning. J. Comput. Syst. Sci. 52(3), 421–433 (1996) MATHCrossRefGoogle Scholar
  3. 3.
    Beigel, R., Chang, R., Ogiwara, M.: A relationship between difference hierarchies and relativized polynomial hierarchies. Math. Syst. Theory 26(3), 293–310 (1993) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buhrman, H., Fortnow, L.: Two queries. J. Comput. Syst. Sci. 59(2), 182–194 (1999) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cai, J.: S2p⊆ZPPNP. J. Comput. Syst. Sci. 73(1), 25–35 (2007) MATHCrossRefGoogle Scholar
  6. 6.
    Canetti, R.: More on BPP and the polynomial-time hierarchy. Inf. Process. Lett. 57(5), 237–241 (1996) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cai, J., Chakaravarthy, V.: On zero error algorithms having oracle access to one query. J. Comb. Optim. 11(2), 189–202 (2006) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cai, J., Chakaravarthy, V., Hemaspaandra, L., Ogihara, M.: Competing provers yield improved Karp-Lipton collapse results. Inf. Comput. 198(1), 1–23 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The boolean hierarchy I: Structural properties. SIAM J. Comput. 17(6), 1232–1252 (1988) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The boolean hierarchy II: Applications. SIAM J. Comput. 18(1), 95–111 (1989) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chang, R., Kadin, J.: On computing boolean connectives of characteristic functions. Math. Syst. Theory 28(3), 173–198 (1995) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chang, R., Kadin, J.: The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM J. Comput. 25(2), 340–354 (1996) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chang, R., Purini, S.: Bounded queries and the NP machine hypothesis. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, pp. 52–59. IEEE Comput. Soc., Los Alamitos (2007) Google Scholar
  14. 14.
    Chakaravarthy, V., Roy, S.: Oblivious symmetric alternation. In: Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 3884, pp. 230–241. Springer, Berlin (2006) Google Scholar
  15. 15.
    Fortnow, L., Impagliazzo, R., Kabanets, V., Umans, C.: On the complexity of succinct zero-sum games. In Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 323–332 (2005). To appear in Comput. Complex. Google Scholar
  16. 16.
    Fortnow, L., Pavan, A., Sengupta, S.: Proving SAT does not have small circuits with an application to the two queries problem. J. Comput. Syst. Sci. 74(3), 358–363 (2008) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hemaspaandra, E., Hemaspaandra, L., Hempel, H.: A downward collapse within the polynomial hierarchy. SIAM J. Comput. 28(2), 383–393 (1998) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hemaspaandra, E., Hemaspaandra, L., Hempel, H.: Extending downward collapse from 1-versus-2 queries to m-versus-m+1 queries. SIAM J. Comput. 34(6), 1352–1369 (2005) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hemaspaandra, L., Jain, S., Vereshchagin, N.: Banishing robust Turing completeness. Int. J. Found. Comput. Sci. 4(3), 245–265 (1993) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kadin, J.: The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM J. Comput. 17(6), 1263–1282 (1988). Erratum appears in the same journal, 20(2), 404 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Karp, R., Lipton, R.: Some connections between nonuniform and uniform complexity classes. In: Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 302–309. Assoc. Comput. Mach., New York (1980) Google Scholar
  22. 22.
    Krentel, M.: The complexity of optimization problems. J. Comput. Syst. Sci. 36(3), 490–509 (1988) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pavan, A., Santhanam, R., Vinodchandran, N.: Some results on average-case hardness within the polynomial hierarchy. In: Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 188–199 (2006) Google Scholar
  24. 24.
    Russell, A., Sundaram, R.: Symmetric alternation captures BPP. Comput. Complex. 7(2), 152–162 (1998) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sipser, M.: On relativization and the existence of complete sets. In: Proceedings of the 9th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 140, pp. 523–531. Springer, Berlin (1982) CrossRefGoogle Scholar
  26. 26.
    Tripathi, R.: The 1-versus-2 queries problem revisited. In: Proceedings of the 18th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 4835, pp. 137–147. Springer, Berlin (2007) Google Scholar
  27. 27.
    Wagner, K.: Number-of-query hierarchies. Technical Report 4, Institut für Informatik, Universität Würzburg, Würzburg, Germany, February 1989 Google Scholar
  28. 28.
    Yap, C.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26(3), 287–300 (1983) MATHCrossRefMathSciNetGoogle Scholar

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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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