Logspace verifiers, NC, and NP

Extended abstract
  • Satyanarayana V. Lokam
  • Meena Mahajan
  • V. Vinay
Session 3A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1004)


We explore the connection between public-coin interactive proof systems with logspace verifiers and \(\mathcal{N}\mathcal{C}\)using two different approaches. In the first approach, we describe an interactive proof system for accepting any language in \(\mathcal{N}\mathcal{C}\)after a logspace reduction, where the verifier is logspace-bounded and the protocol requires polylog time. These results are proved by describing \(\mathcal{N}\mathcal{C}\)computations as computations over arithmetic circuits using maximum and average gates, and then translating the arithmetic circuits into interactive proof systems in a natural way. In the second approach, we give a characterization of \(\mathcal{N}\mathcal{C}\)in terms of interactive proof systems where the verifier is logspace-bounded and runs in polylog time. The equivalent interactive proof systems work with error-correcting encodings of inputs, using the polylogarithmically checkable codes introduced in the context of transparent proofs.

We also characterize \(\mathcal{N}\mathcal{C}\)and \(\mathcal{P}\mathcal{S}\mathcal{P}\mathcal{A}\mathcal{C}\mathcal{E}\)via public-coin interactive proof systems where the verifier is logspace-bounded, but has restricted access to auxiliary storage.


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  1. [Bab85]
    L. Babai. Trading group theory for randomness. In Proc. 17th STOC, pp 421–429, 1985.Google Scholar
  2. [BF93]
    L. Babai and K. Friedl. On slightly superlinear transparent proofs. Technical Report CS-93-13, Department of Computer Science, University of Chicago, 1993.Google Scholar
  3. [BFLS91]
    L. Babai, L. Fortnow, L.A. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In Proc. 23rd STOC, pp 21–31, 1991.Google Scholar
  4. [BM88]
    L. Babai and S. Moran. Arthur-Merlin games: a randomized proof system and a hierarchy of complexity classes. J. Comp. Syst. Sci., 36:254–276, 1988.Google Scholar
  5. [CL88]
    A. Condon and R. Ladner. Probabilistic game automata. J. Comp. Syst. Sci., 36(3):452–489, 1988.Google Scholar
  6. [CL89]
    A. Condon and R. Lipton. On the complexity of space-bounded interactive proofs. In Proc. 30th FOCS, pp 462–467, 1989.Google Scholar
  7. [CL92]
    A. Condon and R. Ladner. Interactive proof systems with polynomially bounded strategies. In Proc. 7th Conference on Structure in Complexity Theory, pp 282–294, 1992.Google Scholar
  8. [Con91]
    A. Condon. The complexity of the max word problem and the power of one-way interactive proof systems. In Proc. 8th STACS, pp 456–465, 1991. LNCS 480.Google Scholar
  9. [FL93]
    L. Fortnow and C. Lund. Interactive proof systems and alternating timespace complexity. Theoretical Computer Science, 113:55–73, 1993. also in Proc. 8th STACS 1991, LNCS 480.Google Scholar
  10. [For89]
    L. Fortnow. Complexity-theoretic aspects of interactive proof systems. PhD thesis, M. I. T., May 1989. Tech. Rep. MIT/LCS/TR-447.Google Scholar
  11. [GMR85]
    S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proof systems. In Proc. 17th STOC, pp 291–304, 1985. full version in SIAM J. Comput., Vol 18(1), pp 186–208.Google Scholar
  12. [GS86]
    S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems. In Proc. 18th STOC, 1986. also in Advances in Computing Research 5: Randomness and Computation, JAI Press, Greenwich, CT, 1989.Google Scholar
  13. [JK89]
    B. Jenner and B. Kersig. Characterizing the polynomial hierarchy by alternating auxiliary pushdown automata. RAIRO Theoretical Informatics and Applications, 23:93–99, 1989. also in Proc. STACS(1988), LNCS 294 118–125.Google Scholar
  14. [LFKN92]
    C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859–868, October 1992. also in Proc. 31st FOCS 1990, pp 1–10.Google Scholar
  15. [Lip90]
    R. J. Lipton. Efficient checking of computations. In Proc. 7th STACS, pp 207–215, 1990. LNCS 415.Google Scholar
  16. [Ruz81]
    W.L. Ruzzo. On uniform circuit complexity. J. Comput. Syst. Sci., 22:365–383, 1981.Google Scholar
  17. [Sha92]
    A. Shamir. IP=PSPACE. J. ACM, 39(4):869–877, 1992. also in Proc. 31st FOCS 1990, pp 11–15.Google Scholar
  18. [She92]
    A. Shen. IP=PSPACE: Simplified proof. J. ACM, 39(4):878–880, 1992.Google Scholar
  19. [VC90]
    V. Vinay and V. Chandru. The expressibility of nondeterministic auxiliary stack automata and its relation to treesize bounded alternating auxiliary pushdown automata. In Proc. 10th FST & TCS, pp 104–114, 1990. LNCS 472.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Satyanarayana V. Lokam
    • 1
  • Meena Mahajan
    • 2
  • V. Vinay
    • 3
  1. 1.Department of Computer ScienceThe University of ChicagoChicagoUSA
  2. 2.C.I.T. CampusThe Institute of Mathematical SciencesMadrasIndia
  3. 3.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia

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