On the power of many one-bit provers

  • Authors:

  • Per Austrin;Johan Håstad;Rafael Pass

  • Affiliations:

  • Aalto Science Institute and KTH Royal Institute of Technology, Helsinki and Stockholm, Sweden;KTH Royal Institute of Technology, Stockholm, Sweden;Cornell University, Ithaca, NY, USA

  • Venue:
  • Proceedings of the 4th conference on Innovations in Theoretical Computer Science
  • Year:
  • 2013

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Abstract

We study the class of languages, denoted by MIP[k, 1-ε, s], which have k-prover games where each prover just sends a single bit, with completeness 1-ε and soundness error s. For the case that k=1 (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson (Computational Complexity'02) demonstrate that SZK exactly characterizes languages having 1-bit proof systems with "non-trivial" soundness (i.e., 1/2 k - ε, MIP[k, 1-ε, s] = BPP; When 1/2k + ε ≤ s k -ε, MIP[k, 1-ε, s] = SZK; When s ≥ 2/2k + ε, AM ⊆ MIP[k, 1-ε, s]; For s ≤ 0.62 k/2k and sufficiently large k, MIP[k, 1-ε, s] ⊆ EXP; For s ≥ 2k/2k, MIP[k, 1, 1-ε, s] = NEXP. As such, 1-bit k-prover games yield a natural "quantitative" approach to relating complexity classes such as BPP, SZK, AM, EXP, and NEXP. We leave open the question of whether a more fine-grained hierarchy (between AM and NEXP) can be established for the case when s ≥ 2/2k + ε.