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Non-deterministic exponential time has two-prover interactive protocols

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Abstract

We determine the exact power of two-prover interactive proof systems introduced by Ben-Or, Goldwasser, Kilian, and Wigderson (1988). In this system, two all-powerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the inputx belongs to the languageL. We show that the class of languages having tow-prover interactive proof systems is nondeterministic exponential time.

We also show that to prove membership in languages inEXP, the honest provers need the power ofEXP only.

The first part of the proof of the main result extends recent techniques of polynomial extrapolation used in the single prover case by Lund, Fortnow, Karloff, Nisan, and Shamir.

The second part is averification scheme for multilinearity of a function in several variables held by an oracle and can be viewed as an independent result onprogram verification. Its proof rests on combinatorial techniques employing a simple isoperimetric inequality for certain graphs:

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References

  1. A. V. Aho, J. E. Hopcroft, andJ. D. Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading MA, 1974.

    Google Scholar 

  2. D. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing.Probability in the Engineering and Informational Sciences 1 (1987), 33–46.

    Google Scholar 

  3. L. Babai, Trading group theory for randomness, inProc. 17th Ann. ACM Symp. Theory of Computing, 1985, 421–429.

  4. L. Babai, E-mail and the unexpected power of interaction, in: Proc. 5th Ann. IEEE Structures in Complexity Theory Conf., 1990, 30–44.

  5. L. Babai andP. Erdős, Representation of group elements as short products,Annals of Discrete Mathematics 12 (1982), 27–30.

    Google Scholar 

  6. L. Babai andL. Fortnow, Arithmetization: a new method in structural complexity theory,Computational Complexity 1 (1991), 41–66. (Preliminary version appeared as: A characterization of #P by arithmetic straight line programs, inProc. 31st Ann. IEEE Symp. Foundations of Comp. Sci., 1990, 26–34.)

    Google Scholar 

  7. L. Babai, L. Fortnow, L. Levin, and M. Szegedy, Checking computations in polylogarithmic time, in:Proc 23rd ACM Symp. Theory of Computing, 1991, to appear.

  8. L. Babai, L. Fortnow, and C. Lund, Non-deterministic exponential time has two-prover interactive protocols (extended abstract),Proc. 31st Ann. IEEE Symp. Found. Comp. Sci., 1990, 16–25.

  9. L. Babai and P. Frankl,Linear Algebra Methods in Combinatorics, I, Preliminary Version, University of Chicago, Dept. C. S. 1988.

  10. D. Beaver and J. Feigenbaum, Hiding instances in multioracle queries, inProc. 7th Symp. on Theoretical Aspects of Comp. Sci. Lecture Notes in Comp. Sci. 415 (1990), 37–48.

  11. M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, Multiprover interactive proofs: How to remove the intractability assumptions, inProc. 20th Ann. ACM Symp. Theory of Computing, 1988, 113–131.

  12. M. Blum and S. Kannan, Designing programs that check their work, inProc. 21st Ann. ACM Symp. Theory of Computing, 1989, 86–97.

  13. M. Blum, M. Luby, and R. Rubinfeld, Self-testing and self-correcting programs, with applications to numerical programs, inProc. 22nd Ann. ACM Symp. Theory of Computing, 1990, 73–83.

  14. L. Babai andS. Moran, Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes.J. Comp. Sys. Sci. 36 (1988), 254–276.

    Google Scholar 

  15. J. Cai, PSPACE is provable by two provers in one round, manuscript, 1990.

  16. S. A. Cook, The complexity of theorem proving procedures, inProc. 3rd Ann. ACM Symp. Theory of Computing 1971, 151–158.

  17. U. Feige, S. Goldwasser, L. Lovász, and S. Safra, On the complexity of clique approximation, in preparation.

  18. P. Feldman, The Optimum Prover lives inPSPACE, manuscript, 1986.

  19. L. Fortnow, The Complexity of Perfect Zero-Knowledge, In S. Micali, ed.,Randomness and Computation, Advances in Computing Research 5 (1989), 327–343.

  20. L. Fortnow, Complexity-Theoretic Aspects of Interactive Proof Systems, Ph.D. Thesis,Massachusetts Institute of Technology, Laboratory for Computer Science, Tech. Report MIT/LCS/TR-447 1989.

  21. L. Fortnow, J. Rompel, and M. Sipser, On the power of multiprover interactive protocols,Proc. 3rd Structure in Complexity Theory Conf., 1988, 156–161.

  22. L. Fortnow andM. Sipser, Are there interactive protocols for co-NP languages?,Inf. Process. Letters 28 (1988), 249–251.

    Google Scholar 

  23. S. Goldwasser, S. Micali, andC. Rackoff, The knowledge complexity of interactive proofs,SIAM J. Comput. 18 (1989), 186–208. (Preliminary version appeared inProc. 18th Ann. ACM Symp. Theory of Computing, 1985, 291–304.)

    Google Scholar 

  24. J. Hartmanis, N. Immerman, andV. Sewelson, Sparse sets inNP-P: EXPTIME versusNEXPTIME, Inf. and Control 65 (1985), 158–181.

    Google Scholar 

  25. H. Heller On Relativized Exponential and Probabilistic Complexity Classes,Information and Computation 71 (1986), 231–243.

    Google Scholar 

  26. R. Karp, R. Lipton, Some Connections between Nonuniform and Uniform Complexity Classes,Proc. 12th Ann. ACM Symp. Theory of Computing, 1980, 302–309.

  27. L. Levin, Universal'nyîe perebornyîe zadachi (Universal search problems, in Russian),Problemy Peredachi Informatsii 9 (1973), 265–266. A corrected English translation appears in an appendix to Trakhtenbrot [39].

    Google Scholar 

  28. R. Lipton, New directions in testing, inProceedings of the DIMACS Workshop on Distributed Computing and Cryptography, 1989, to appear.

  29. C. Lund, L. Fortnow, H. Karloff, and N. Nisan, Algebraic methods for interactive proof systems, inProc. 31st Ann. IEEE Symp. Foundations of Comp. Sci., 1990, 1–10.

  30. P. Orponen, Complexity Classes of Alternating Machines with Oracles,Proc. 10th ICALP, Lecture Notes in Comp. Sci. 154 (1983), 573–584.

    Google Scholar 

  31. C. Papadimitriou, Games against Nature,Proc. 24th Ann. IEEE Symp. Foundations of Comp. Sci., 1983, 446–450.

  32. G. Peterson and J. Reif, Multiple-person alternation,Proc. 20th Ann. IEEE Symp. Foundations of Comp. Sci., 1979, 348–363.

  33. M. Santha, Relativized Arthur-Merlin versus Merlin-Arthur games,Inf. and Computation 80 (1989), 44–49.

    Google Scholar 

  34. J. Seiferas, M. Fischer, andA. Meyer, Separating Nondeterministic Time Complexity ClassesJ. Assoc. Comput. Mach. 25 (1978), 146–167.

    Google Scholar 

  35. A. Shamir, IP=PSPACE, inProc. 31st Ann. IEEE Symp. Foundations of Comp. Sci., 1990, 11–15.

  36. J. Simon, On Some Central Problems in Computational Complexity, Ph.D. Thesis,Cornell University, Computer Science, Tech. Report TR 74-224, 1975.

  37. J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities,J. Assoc. Comput. Mach. 27 (1980), 701–717.

    Google Scholar 

  38. M. Szegedy, EfficientMIP protocol and a stronger condition on clique approximation, in preparation.

  39. B. A. Trakhtenbrot, A survey of Russian approaches toPerebor (brute-force search) algorithms,Annals of the History of Computing 6 (1984), 384–400.

    Google Scholar 

  40. S. Toda, On the computational power ofPP and ⊕P, inProc. 30th Ann. IEEE Symp. Foundations of Comp. Sci., 1989, 514–519.

  41. L. Valiant, The complexity of computing the permanent,Theoretical Computer Science 8 (1979), 189–201.

    Google Scholar 

  42. L. Valiant, V. Vazirani,N P is as Easy as Detecting Unique Solutions,Theoretical Computer Science 47 (1986), 85–93.

    Google Scholar 

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Author notes

  1. Carsten Lund

    Present address: DIMACS, Rutgers University, P.O. Box 1179, 08855-1179, Piscataway, NJ

Authors and Affiliations

  1. University of Chicago, 60637, Chicago, IL

    László Babai

  2. Eötvös University, Budapest, Hungary

    László Babai

  3. Department of Computer Science, University of Chicago, 1100 E. 58th St., 60637, Chicago, IL

    Carsten Lund

  4. Department of Computer Science, University of Chicago, 1100 E. 58th St., 60637, Chicago, IL

    Lance Fortnow

Authors
  1. László Babai
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  2. Lance Fortnow
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  3. Carsten Lund
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Babai, L., Fortnow, L. & Lund, C. Non-deterministic exponential time has two-prover interactive protocols. Comput Complexity 1, 3–40 (1991). https://doi.org/10.1007/BF01200056

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