Advertisement

Interactive proof systems: Provers, rounds, and error bounds

  • Ulrich Hertrampf
  • Klaus Wagner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 533)

Abstract

We introduce generalized multi-prover interactive proof systems and the associated polynomial time complexity classes IP(m, r, 1/h), which depend on the number m of provers, number r of rounds and the value 1/h by which the error is bounded away from one half. In this denotation the class IP(m, r) of languages accepted by ordinary IP-systems with m provers and r rounds appears as IP(m, r, 1/6), whereas we define IP'(m, r) to be the union of all IP(m, r, 1/h) with an arbitrary polynomial h. We prove several simulation theorems that enable us to prove most of the known relations between different IP-classes and a collapse of the IP' hierarchy to essentially only four classes, namely
$$\begin{gathered}IP'(1,1) = IP(1,1) \subseteq IP'(1,poly) = IP(1,poly) = PSPACE \hfill \\\subseteq IP'(2,1) = IP(poly,1) \hfill \\\subseteq IP'(2,2) = IP(poly,poly) = NEXPTIME \hfill \\\end{gathered}$$
Finally we show how to reduce the space needed by an interactive proof system introducing one additional prover.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ba85]
    L. Babai. Trading group theory for randomness. 17th Annual ACM Symposium on Theory of Computing (STOC), 421–429, 1985.Google Scholar
  2. [Ba90]
    L. Babai. E-mail and the unexpected power of interaction. 5th Structure in Complexity Theory (IEEE), 30–44, 1990.Google Scholar
  3. [BFL90]
    L. Babai, L. Fortnow, C. Lund. Non-deterministic exponential time has two-prover interactive protocols. University of Chicago Technical Report 90-03, 1990.Google Scholar
  4. [BGKW88]
    M. Ben-Or, S. Goldwasser, J. Kilian, A. Wigderson. Multi-prover interactive proofs: How to remove the intractability assumptions. 20th Annual ACM Symposium on Theory of Computing (STOC), 113–131, 1988.Google Scholar
  5. [BHZ87]
    R. Boppana, J. Håstad, S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters 25, 127–132, 1987.CrossRefGoogle Scholar
  6. [BM88]
    L. Babai, S. Moran. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Science 36 2, 254–276, 1988.CrossRefGoogle Scholar
  7. [CCL90]
    J. Cai, A. Condon, R. Lipton. PSPACE is provable by two provers in one round. Manuscript, 1990.Google Scholar
  8. [FRS88]
    L. Fortnow, J. Rompel, M. Sipser. On the power of multi-prover interactive protocols. 3rd Structure in Complexity Theory (IEEE), 156–161, 1988.Google Scholar
  9. [FRS90]
    L. Fortnow, J. Rompel, M. Sipser. Errata for on the power of multi-prover interactive protocols. 5th Structure in Complexity Theory (IEEE), 318–319, 1990.Google Scholar
  10. [FS88]
    L. Fortnow, M. Sipser. Are there interactive protocols for co-NP languages? Information Processing Letters 28, 249–251, 1988.CrossRefGoogle Scholar
  11. [GMR85]
    S. Goldwasser, S. Micali, C. Rackoff. The knowledge complexity of interactive proof systems. 17th Annual ACM Symposium on Theory of Computing (STOC), 291–304, 1985.Google Scholar
  12. [GS86]
    S. Goldwasser, M. Sipser. Private coins versus public coins in interactive proof systems. 18th Annual ACM Symposium on Theory of Computing (STOC), 59–68, 1986.Google Scholar
  13. [LFKN89]
    C. Lund, L. Fortnow, H. Karloff, N. Nisan. The polynomial time hierarchy has interactive proofs. E-mail announcement, 1989.Google Scholar
  14. [Sc86]
    U. Schöning. Complexity and Structure. Springer-Verlag Lecture Notes in Computer Science 211, 1986.Google Scholar
  15. [Sh89]
    A. Shamir. IP = PSPACE. E-mail announcement, 1989, also in 31st Annual Symposium on Foundations of Computer Science (FOCS), 11–15, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Ulrich Hertrampf
    • 1
  • Klaus Wagner
    • 1
  1. 1.Institut für InformatikUniversität WürzburgWürzburgGermany

Personalised recommendations