Advertisement

My favorite ten complexity theorems of the past decade

  • Lance Fortnow
Invited Talk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 880)

Abstract

We review the past ten years in computational complexity theory by focusing on ten theorems that the author enjoyed the most. We use each of the theorems as a springboard to discuss work done in various areas of complexity theory.

Keywords

Complexity Theory Proof System GapP Function Pseudorandom Generator Random Coin 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AB87]
    N. Alon and R. Boppana. The monotone complexity of Boolean functions. Combinatorica, 7(1): 1–22, 1987.Google Scholar
  2. [Ajt83]
    M. Ajtai. ∑11-formulae on unite structures. Annals of Pure and Applied Logic, 24: 1–48, 1983.Google Scholar
  3. [AKL+79]
    R. Aleliunas, R. Karp, R. Lipton, L. Lovász, and C. Rackoff. Random walks, universal traversal sequences, and the complexity of maze problems. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, pages 218–223. IEEE, New York, 1979.Google Scholar
  4. [All89]
    E. Allender. A note on the power of threshold circuits. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 580–584. IEEE, New York, 1989.Google Scholar
  5. [ALM+92]
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 14–23. IEEE, New York, 1992.Google Scholar
  6. [AS92]
    S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 2–13. IEEE, New York, 1992.Google Scholar
  7. [Bab85]
    L. Babai. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on the Theory of Computing, pages 421–429. ACM, New York, 1985.Google Scholar
  8. [Bar89]
    D. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences, 38(1): 150–164, 1989.Google Scholar
  9. [BCD+89a]
    A. Borodin, S. Cook, P. Dymond, L. Ruzzo, and M. Tompa. Erratum: Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(6): 1283, 1989.Google Scholar
  10. [BCD+89b]
    A. Borodin, S. Cook, P. Dymond, L. Ruzzo, and M. Tompa. Two applications of inductive counting for complementation problems. SIAM Journal on Computing, 18(3): 559–578, 1989. See also Erratum [BCD+89a].Google Scholar
  11. [BCS92]
    D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104: 263–283, 1992.Google Scholar
  12. [Bei92]
    R. Beigel. Perceptrons, PP and the polynomial hierarchy. In Proceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 14–19. IEEE, New York, 1992.Google Scholar
  13. [BFL91]
    L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1): 3–40, 1991.Google Scholar
  14. [BG92]
    R. Beigel and J. Gill. Counting classes: Thresholds, parity, mods, and fewness. Theoretical Computer Science, 103: 3–23, 1992.Google Scholar
  15. [BGKW88]
    M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson. Multi-prover interactive proofs: How to remove intractability assumptions. In Proceedings of the 20th ACM Symposium on the Theory of Computing, pages 113–131. ACM, New York, 1988.Google Scholar
  16. [BH77]
    L. Berman and J. Hartmanis. On isomorphism and density of NP and other complete sets. SIAM. Journal on Computing, 1: 305–322, 1977.Google Scholar
  17. [BHZ87]
    R. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2): 127–132, 1987.Google Scholar
  18. [BM84]
    M. Blum and S. Micali. How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing, 13: 850–864, 1984.Google Scholar
  19. [BM88]
    L. Babai and S. Moran, Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2): 254–276, 1988.Google Scholar
  20. [BRS94]
    R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. Journal of Computer and System Sciences, 1994. To appear. Paper also appeared in Proceedings of 23rd STOC conference, 1991, pages 1–9.Google Scholar
  21. [BT91]
    R. Beigel and J. Tarui. On ACC. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 783–792. IEEE, 1991.Google Scholar
  22. [CF91]
    J. Cai and M. Furst. PSPACE survives constant-width bottlenecks. International Journal of Foundations of Computer Science, 2: 67–76, 1991.Google Scholar
  23. [CKS81]
    A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 28(1): 114–133, 1981.Google Scholar
  24. [Edm65]
    J. Edmonds. Paths, trees and flowers. Canadian Journal of Mathematics, 17: 449–467, 1965.Google Scholar
  25. [Fel86]
    Feldman. The optimum prover lives in PSPACE. Manuscript, 1986.Google Scholar
  26. [FFK92]
    S. Fenner, L. Fortnow, and S. Kurtz. The isomorphism conjecture holds relative to an oracle. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 30–39. IEEE, New York, 1992.Google Scholar
  27. [FFK94]
    S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1): 116–148, 1994.Google Scholar
  28. [FL92]
    U. Feige and L. Lovász. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th ACM Symposium on the Theory of Computing, pages 733–744. ACM, New York, 1992.Google Scholar
  29. [FR91]
    L. Fortnow and N. Reingold. PP is closed under truth-table reductions. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, pages 13–15. IEEE, New York, 1991.Google Scholar
  30. [FRS94]
    L. Fortnow, J. Rompel, and M. Sipser. On the power of multi-prover interactive protocols. Theoretical Computer Science A, 1994. To appear.Google Scholar
  31. [FS88]
    L. Fortnow and M. Sipser. Are there interactive protocols for co-NP languages? Information Processing Letters, 28: 249–251, 1988.Google Scholar
  32. [FSS84]
    M. Furst, J. Saxe, and M. Sipser. Parity, circuits and the polynomial-time hierarchy. Mathematical Systems Theory, 17: 13–27, 1984.Google Scholar
  33. [Gil77]
    J. Gill. Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 6: 675–695, 1977.Google Scholar
  34. [GKL93]
    O. Goldreich, H. Krawczyk, and M. Luby. On the existence of pseudorandom generators. SIAM Journal on Computing, 22(6): 1163–1175, December 1993.Google Scholar
  35. [GL89]
    O. Goldreich and L. Levin. A hard-core predicate for all one-way functions. In Proceedings of the 21st ACM Symposium on the Theory of Computing, pages 25–32. ACM, New York, 1989.Google Scholar
  36. [GMR89]
    S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proof-systems. SIAM Journal on Computing, 18(1): 186–208, 1989.Google Scholar
  37. [GMW91]
    O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3): 691–729, 1991.Google Scholar
  38. [GS89]
    S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 73–90. JAI Press, Greenwich, 1989.Google Scholar
  39. [Hås89]
    J. Håstad. Almost optimal lower bounds for small depth circuits. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 143–170. JAI Press, Greenwich, 1989.Google Scholar
  40. [Hås90]
    J. Håstad. Pseudo-random generators under uniform assumptions. In Proceedings of the 22nd ACM Symposium on the Theory of Computing, pages 395–404. ACM, New York, 1990.Google Scholar
  41. [Her90]
    U. Hertrampf. Relations among MOD classes (note). Theoretical Computer Science, 74: 325–328, 1990.Google Scholar
  42. [HH91]
    J. Hartmanis and L. Hemachandra. One-way functions and the nonisomorphism of NP-complete sets. Theoretical Computer Science, 81(1): 155–163, 1991.Google Scholar
  43. [HL94]
    S. Homer and L. Longpré. On reductions of NP sets to sparse sets. Journal of Computer and System Sciences, 48(2): 324–336, 1994.Google Scholar
  44. [HLS+93]
    U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, and K. Wagner. On the power of polynomial time bit-reductions. In Proceedings of the 8th IEEE Structure in Complexity Theory Conference, pages 200–207. IEEE, New York, 1993.Google Scholar
  45. [HS65]
    J. Hartmanis and R. Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117: 285–306, 1965.Google Scholar
  46. [HU79]
    J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, Mass., 1979.Google Scholar
  47. [ILL89]
    R. Impagliazzo, L. Levin, and M. Luby. Pseudo-random number generation from one-way functions. In Proceedings of the 21st ACM Symposium on the Theory of Computing, pages 12–24. ACM, New York, 1989.Google Scholar
  48. [Imm88]
    N. Immerman. Nondeterministic space is closed under complementation. SIAM Journal on Computing, 17(5): 935–938, 1988.Google Scholar
  49. [IR89]
    R. Impagliazzo and S. Rudich. Limits on the provable consequences of oneway permutations. In Proceedings of the 21st ACM Symposium on the Theory of Computing, pages 44–61. ACM, New York, 1989.Google Scholar
  50. [Ist88]
    S. Istrail. Polynomial, universal traversing sequences for cycles are constructible. In Proceedings of the 20th ACM Symposium on the Theory of Computing, pages 491–503. ACM, New York, 1988.Google Scholar
  51. [IZ89]
    R. Impagliazzo and D. Zuckerman. How to recycle random bits. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 248–253. IEEE, New York, 1989.Google Scholar
  52. [JY85]
    D. Joseph and P. Young. Some remakrs on witness functions for poynomial reducibilities in NP. Theoretical Computer Science, 39: 225–237, 1985.Google Scholar
  53. [KL82]
    R. Karp and R. Lipton. Turing machines that take advice. L'Enseignement Mathematique, 28: 191–209, 1982.Google Scholar
  54. [KLD87]
    K. Ko, T. Long, and D. Du. A note on one-way functions and polynomial-time isomorphisms. Theoretical Computer Science, 47: 263–276, 1987.Google Scholar
  55. [KMR88]
    S. Kurtz, S. Mahaney, and J. Royer. Collapsing degrees. Journal of Computer and System Sciences, 37(2): 247–268, 1988.Google Scholar
  56. [KMR89]
    S. Kurtz, S. Mahaney, and J. Royer. The isomorphism conjecture fails relative to a random oracle. In Proceedings of the 21st ACM Symposium on the Theory of Computing, pages 157–166. ACM, New York, 1989.Google Scholar
  57. [KW90]
    M. Karchmer and A. Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3: 255–265, 1990.Google Scholar
  58. [Lan63]
    P. Landweber. Three theorems on phase structure grammars of type 1. Information and Control, 6(2): 131–136, 1963.Google Scholar
  59. [Lev87]
    L. Levin. One-way functions and pseudo-random generators. Combinatorica, 7: 357–363, 1987.Google Scholar
  60. [LFKN92]
    C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4): 859–868, 1992.Google Scholar
  61. [LMN93]
    N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, fourier transform, and learnability. Journal of the ACM, 40(3): 607–620, 1993.Google Scholar
  62. [Mah82]
    S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25: 130–143, 1982.Google Scholar
  63. [Nis92a]
    N. Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4): 449–461, 1992.Google Scholar
  64. [Nis92b]
    N. Nisan. RL is contained in SC. In Proceedings of the 24th ACM Symposium on the Theory of Computing, pages 619–623. ACM, New York, 1992.Google Scholar
  65. [NSW92]
    N. Nisan, E. Szemerédi, and A. Wigderson. Undirected connectivity in O(log1.5 n) space. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, pages 24–29. IEEE, New York, 1992.Google Scholar
  66. [NZ93]
    N. Nisan and D. Zuckerman. More deterministic simulation in logspace. In Proceedings of the 25th ACM Symposium on the Theory of Computing, pages 235–244. ACM, New York, 1993.Google Scholar
  67. [OW91]
    M. Ogiwara and O. Watanabe. On polynomial-time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3): 471–483, 1991.Google Scholar
  68. [PY91]
    C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43: 425–440, 1991.Google Scholar
  69. [PZ83]
    C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings of the 6th GI Conference on Theoretical Computer Science, volume 145 of Lecture Notes in Computer Science, pages 269–276. Springer, Berlin, 1983.Google Scholar
  70. [Raz85a]
    A. Razborov. Lower bounds of monotone complexity of the logical permanent function. Mathematical Notes of the Academy of Sciences of the USSR, 37: 485–493, 1985.Google Scholar
  71. [Raz85b]
    A. Razborov. Lower bounds on the monotone complexity of some boolean functions. Dokl. Akad. Nauk SSSR, 281(4): 798–801, 1985. In Russian. English Translation in [Raz85c].Google Scholar
  72. [Raz85c]
    A. Razborov. Lower bounds on the monotone complexity of some boolean functions. Soviet Math. dokl., 31: 485–493, 1985.Google Scholar
  73. [Raz87]
    A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4): 333–338, 1987.Google Scholar
  74. [Raz89]
    A. Razborov. On the method of approximations. In Proceedings of the 21st ACM Symposium on the Theory of Computing, pages 167–176. ACM, New York, 1989.Google Scholar
  75. [Raz90]
    A. Razborov. Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica, 10(1): 81–93, 1990.Google Scholar
  76. [Raz94]
    A. Razborov. Bounded Arithmetic and lower bounds in Boolean complexity. Feasible Mathematics II, 1994. To appear.Google Scholar
  77. [RR94]
    A. Razborov and S. Rudich. Natural proofs. In Proceedings of the 26th ACM Symposium on the Theory of Computing, pages 204–213. ACM, New York, 1994.Google Scholar
  78. [RW92]
    R. Raz and A. Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3): 736–744, 1992.Google Scholar
  79. [Sav70]
    W., Savitch. Relationship between nondeterministic and deterministic tape classes. Journal of Computer and System Sciences, 4: 177–192, 1970.Google Scholar
  80. [Sha92]
    A. Shamir. IP = PSPACE. Journal of the ACM, 39(4): 869–877, 1992.Google Scholar
  81. [Smo87]
    R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on the Theory of Computing, pages 77–82. ACM, New York, 1987.Google Scholar
  82. [Sze88]
    R. Szelepcsényi. The method of forced enumeration for nondeterministic automata. Acta Informatica, 26: 279–284, 1988.Google Scholar
  83. [Tar88]
    É. Tardos. The gap between monotone and nonmonotone circuit complexity is exponential. Combinatorica, 8: 141–142, 1988.Google Scholar
  84. [Tar93]
    J. Tarui. Probabilistic polynomials, AC0 functions and the polynomialtime hierarchy. Theoretical Computer Science A, 113: 167–183, 1993.Google Scholar
  85. [TO92]
    S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2): 316–328, 1992.Google Scholar
  86. [Tod91]
    S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5): 865–877, 1991.Google Scholar
  87. [Val79a]
    L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8: 189–201, 1979.Google Scholar
  88. [Val79b]
    L. Valiant. The complexity of reliability and enumeration problems. SIAM Journal on Computing, 8: 410–421, 1979.Google Scholar
  89. [VV86]
    L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47: 85–93, 1986.Google Scholar
  90. [Yao82]
    A. Yao. Theory and applications of trapdoor functions. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, pages 80–91. IEEE, New York, 1982.Google Scholar
  91. [Yao83]
    A. Yao. Lower bounds by probabilistic arguments. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 420–428. IEEE, New York, 1983.Google Scholar
  92. [Yao85]
    A. Yao. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, pages 1–10. IEEE, New York, 1985.Google Scholar
  93. [Yao90]
    A. Yao. On ACC and threshold circuits. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 619–631. IEEE, New York, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Lance Fortnow
    • 1
  1. 1.Department of Computer ScienceThe University of ChicagoChicago

Personalised recommendations