Trading group theory for randomness

ABSTRACT

In a previous paper [BS] we proved, using the elements of the theory of nilpotent groups, that some of the fundamental computational problems in matriz groups belong to NP. These problems were also shown to belong to coNP, assuming an unproven hypothesis concerning finite simple groups.

The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments. The result we prove is that relative to a random oracle B, the mentioned matrix group problems belong to (NP∩coNP)^B.

The problems we consider are membership in and order of a matrix group given by a list of generators. These problems can be viewed as multidimensional versions of a close relative of the discrete logarithm problem. Hence NP∩coNP might be the lowest natural complexity class they may fit in.

We remark that the results remain valid for black box groups where group operations are performed by an oracle.

The tools we introduce seem interesting in their own right. We define a new hierarchy of complexity classes AM(k) “just above NP”, introducing Arthur vs. Merlin games, the bounded-away version of Papdimitriou's Games against Nature. We prove that in spite of their analogy with the polynomial time hierarchy, the finite levels of this hierarchy collapse to AM=AM(2). Using a combinatorial lemma on finite groups [BE], we construct a game by which the nondeterministic player (Merlin) is able to convince the random player (Arthur) about the relation [G]=N provided Arthur trusts conclusions based on statistical evidence (such as a Slowly-Strassen type “proof” of primality).

One can prove that AM consists precisely of those languages which belong to NP^B for almost every oracle B.

Our hierarchy has an interesting, still unclarified relation to another hierarchy, obtained by removing the central ingredient from the User vs. Expert games of Goldwasser, Micali and Rackoff.

References

1. Ad.L. Adleman, Two theorems on random polynomial time, in' Proc. 19th IEEE Syrup. Found. Comp. Sei., 1078, pp. 75-8,3
2. Ba.L. Bahai, On the eomplexity of matrix group problems 1I, in preparation
3. BE.L. Babai and P. Erdbs, Representation of gro,p elements as short products, in: Theory and Practice of Combinatorics (A. Rosa et al. eds.), Annals of Distr. Math. 12 (1982), pp. 27-30
4. BS.L. Babai and E. Szemerddi, On the comp}exity of matrix group problems I, in: Proc. 25th IEEE Syrup. Found. Comp. Sei., Palm Beach, FL 1084, pp. 220-240
5. BG.C. l{. Bennett and J. Gill, Relative to a random oracle A, Pa~NP~y~coNp'4 with probability 1, SIAM J. Comp. 10 (1981), 96-113
6. BKL.L. Bahai, W. M. Kantor and E. M. Luks, Con~putational complexity and the classification of finite simple gro, ps, in: Proc. 24th IEEE Syrup. Found. Comp. Sci., Tucson AZ 1983, pp. 162-171
7. CW.J. 1,. C~rter and M. N. Wegman, Universal classes of hash functions, JCSS 18, no.2 (1079), 143-154
8. FHL.M. L. Furst, J. Hoperoft and E. M. Luks, Polynomial-time algorithms for permutation groups, in' Proc. 21st IEEE Symp. Found. Comp. Sci., Syracuse, N. Y. 1980, pp. 36-41
9. GG.O. Gaber and Z. Galil, Explicit construction of linear sized superconeentrators, j. Comp. Syst. Sei. 22 (1981), 40%420
10. Gi.J. Gill, Computational complexity of probabilistic Turing machines, SiAM J. Comp. 6 (1077), 675-605
11. GMR.S. Goldwasser, S. Mieali and C. Rackoff, The knowledge complexity of interactive protocols, in: Proe. 17th ACM Symp. Theory of Comp., Providence, R. I. 1985 (this volume)
12. Ku.S. A. Kurtz, Randomness and genericity in thc degrees of unsolvability, Ph.D. Thesis, Univ. of Illinois at Urbana, 1981
13. Lau.C. l,autemann, Bi'P and the polynomial hierarchy, Info. Pro~. Letters 17, no.4 (1~183) 215-217
14. Mar.G. A. Margulis, Explicit constructions of concentrators, Probl. Pored. lnfo. 9 (1973}, 71-80 {English transl, in Probi. lnfo. Transm. (~975), a2a-aa2)
15. Mih.K. A. Mihailova, The occurrence problem for direct products of groups {Russian), Dokl. Akad. Nauk SSSR 119 (1958), 1103-1105 and M~t. Sb. (N. S.) 7O (tl2)(~g66), 24~-281
16. Mil.G. L. Miller, Riemann's hypothesis and tt~ts for primMity, J. Comput. System Sci. 13- (1076), 300-317
17. Pa.C. H. Papadimitriou, Games against Nature, in: Proc. 24th IEEE Syrup. Found. Comp. Sci., Tucson AZ, 1983, pp. 446-450
18. Pin.M. Pinsker, On the complexity of a concentrator, 7th Internat. Teletraffic Conf., Stockholm 107a, als/1-4
19. Pip.N. Pippenger, Superconcentrators, SIAM J. Comp. 6 (1977), 298-304
20. Sac.Gerald E. Sacks, Degrees of unsolvability, Annals of Math. Studies 55, Princeton Univ. Press, Princeton N.J. 1066 (2nd ed.)
21. Sim.C. C. Sims. Some group theoretic algorithms, in: Lect. Notes in Math., Springer, N. Y. 1078, pp. 108-124
22. Sip.M. Sipser, A complexity theoretic approach to randomness, in: Proc. 15th ACM Syrup. on Theory of Comp.: Boston 1983, 330-335
23. SS.R. Solovay and V. Strassen, A fast Monte Carlo test for primality, SIAM J. Comp. 6 (1977) 84-85
24. St.L. Stoekmeyer, The polynomial time hiararchy, Theor. Comp. Sci. 3, no. I (1976}, 1-22

Index Terms

• Computer systems organization

  • Embedded and cyber-physical systems

    • Robotics

• Computing methodologies

  • Artificial intelligence

    • Control methods

      • Robotic planning

    • Planning and scheduling

      • Robotic planning

• Theory of computation

  • Computational complexity and cryptography

    • Complexity classes

  • Models of computation

    • Computability

      • Turing machines