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Circuit complexity before the dawn of the new millennium

  • Eric Allender
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1180)

Abstract

The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.

Keywords

Boolean Function Circuit Complexity Information Processing Letter Majority Gate Exponential Size 
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.

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References

  1. [AA96]
    M. Agrawal and E. Allender. An isomorphism theorem for circuit complexity. In IEEE Conference on Computational Complexity, pages 2–11, 1996.Google Scholar
  2. [AAR]
    M. Agrawal, E. Allender, and S. Rudich. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. submitted; preliminary version appeared as [AA96].Google Scholar
  3. [ABFR94]
    J. Aspnes, R. Beigel, M. Furst, and S. Rudich. The expressive power of voting polynomials. Combinatorica, 14:135–148, 1994.Google Scholar
  4. [AG91]
    E. Allender and V. Gore. Rudimentary reductions revisited. Information Processing Letters, 40:89–95, 1991.Google Scholar
  5. [AG94]
    E. Allender and V. Gore. A uniform circuit lower bound for the permanent. SIAM Journal on Computing, 23:1026–1049, 1994.Google Scholar
  6. [AH94]
    E. Allender and U. Hertrampf. Depth reductions for circuits of unbounded fan-in. Information and Computation, 112:217–238, 1994.Google Scholar
  7. [Ajt83]
    M. Ajtai. σ 11-formulae on finite structures. Annals of Pure and Applied Logic, 24:1–48, 1983.Google Scholar
  8. [All]
    E. Allender. The permanent requires large uniform threshold circuits. Submitted. A preliminary version of this paper appeared as [All96].Google Scholar
  9. [All96]
    E. Allender. A note on uniform circuit lower bounds for the counting hierarchy. In International Conference on Computing and Combinatorics Conference (COCOON), volume 1090 of Lecture Notes in Computer Science, pages 127–135. Springer-Verlag, 1996.Google Scholar
  10. [AS94]
    E. Allender and M. Strauss. Measure on small complexity classes, with applications for BPP. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 807–818, 1994.Google Scholar
  11. [Bar89]
    D. A. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. Journal of Computer and System Sciences, 38:150–164, 1989.Google Scholar
  12. [Bar92]
    D. A. Mix Barrington. Quasipolynomial size circuit classes. In IEEE Structure in Complexity Theory Conference, pages 86–93, 1992.Google Scholar
  13. [BBR94]
    D. A. Mix Barrington, R. Beigel, and S. Rudich. Representing Boolean functions as polynomials modulo composite numbers. Computational Complexity, 4:367–382, 1994.Google Scholar
  14. [BC89]
    D. A. Mix Barrington and J. Corbett. On the relative complexity of some languages in NC1. Information Processing Letters, 32:251–256, 1989.Google Scholar
  15. [BC91]
    D. A. Mix Barrington and J. Corbett. A note on some languages in uniform ACC0. Theoretical Computer Science, 78:357–362, 1991.Google Scholar
  16. [Bea]
    P. Beame. A switching lemma primer. manuscript, available from http://www.cs.washington.edu/homes/beame/papers.html.Google Scholar
  17. [Bei93]
    R. Beigel. The polynomial method in circuit complexity. In IEEE Structure in Complexity Theory Conference, pages 82–95, 1993.Google Scholar
  18. [Bei94]
    R. Beigel. When do extra majority gates help? polylog(n) majority gates are equivalent to one. Computational Complexity, 4:314–324, 1994.Google Scholar
  19. [BF]
    H. Buhrman and L. Fortnow. Personal Communication, Dagstuhl workshop on Structure and Complexity, 1996.Google Scholar
  20. [BFNW93]
    L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307–318, 1993.Google Scholar
  21. [BH92]
    H. Buhrman and S. Homer. Superpolynomial circuits, almost sparse oracles and the exponential hierarchy. In Foundations of Software Technology and Theoretical Computer Science (FST&TCS), volume 652 of Lecture Notes in Computer Science, pages 116–127. Springer-Verlag, 1992.Google Scholar
  22. [BIS90]
    D. A. Mix Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41:274–306, 1990.Google Scholar
  23. [BS90]
    R. Boppana and M. Sipser. The complexity of finite functions. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science (Vol. A: Algorithms and Complexity). Elsevier and MIT Press, 1990.Google Scholar
  24. [BS94]
    D. A. Mix Barrington and H. Straubing. Complex polynomials and circuit lower bounds for modular counting. Computational Complexity, 4:325–338, 1994.Google Scholar
  25. [BS95]
    D. A. Mix Barrington and H. Straubing. Superlinear lower bounds for bounded-width branching programs. Journal of Computer and System Sciences, 50:374–381, 1995.Google Scholar
  26. [BST90]
    D. A. Mix Barrington, H. Straubing, and D. Thérien. Non-uniform automata over groups. Information and Computation, 89:109–132, 1990.Google Scholar
  27. [BT88]
    D. A. Mix Barrington and D. Thérien. Finite monoids and the fine structure of NC1. Journal of the ACM, 35:941–952, 1988.Google Scholar
  28. [BT94]
    R. Beigel and J. Tarui. On ACC. Computational Complexity, 4:350–367, 1994.Google Scholar
  29. [BTT92]
    R. Beigel, J. Tarui, and S. Toda. On probabilistic ACC circuits with an exact-threshold output gate. In Proceedings of the 3rd ACM-SIGSAM International Symposium on Symbolic and Algebraic Computation (ISAAC), volume 650 of Lecture Notes in Computer Science, pages 420–429. Springer-Verlag, 1992.Google Scholar
  30. [Bus93]
    S. Buss. Algorithm for Boolean formula evaluation and for tree contraction. In P. Clote and J. Krajíček, editors, Arithmetic, Proof Theory, and Computational Complexity, volume 25 of Oxford Logic Guides, pages 96–115. Clarendon Press, 1993.Google Scholar
  31. [Cai89]
    J. Cai. With probability 1, a random oracle separates PSPACE from the polynomial-time hierarchy. J. Computer and System Science, 38:68–85, 1989.Google Scholar
  32. [Cau]
    H. Caussinus. A note on a theorem of Barrington, Straubing, and Thérien. To appear in Information Processing Letters.Google Scholar
  33. [Cau96]
    H. Caussinus. Contributions à l'Etude du Non-déterminisme Restreint. PhD thesis, Université de Montréal, 1996.Google Scholar
  34. [CKS81]
    A. Chandra, D. Kozen, and L. Stockmeyer. Alternation. Journal of the ACM, 28:114–133, 1981.Google Scholar
  35. [CMTV96]
    H. Caussinus, P. McKenzie, D. Thérien, and H. Vollmer. Nondeterministic NC1 computation. In Proceedings, 11th Annual IEEE Conference on Computational Complexity, pages 12–21, 1996.Google Scholar
  36. [CR96]
    S. Chaudhuri and J. Radhakrishnan. Deterministic restrictions in circuit complexity. In ACM Symposium on Theory of Computing (STOC), pages 30–36, 1996.Google Scholar
  37. [CSS]
    J.-Y. Cai, D. Sivakumar, and M. Strauss. Constant-depth circuits and the Lutz hypothesis. Manuscript.Google Scholar
  38. [CSV84]
    A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423–439, 1984.Google Scholar
  39. [ES]
    P. Enflo and M. Sitharam. Stable basis families and complexity lower bounds. Submitted.Google Scholar
  40. [Ete]
    K. Etessami. Counting quantifiers, successor relations, and logarithmic space. To appear in Journal of Computer and System Sciences. Preliminary version appeared in IEEE Structure in Complexity Theory Conference, 1995, pp. 2–11.Google Scholar
  41. [FL95]
    L. Fortnow and S. Laplante. Circuit lower bounds à la Kolmogorov. Information and Computation, 123:121–126, 1995.Google Scholar
  42. [FSS84]
    M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17:13–27, 1984.Google Scholar
  43. [GHR92]
    M. Goldmann, J. Håstad, and A. A. Razborov. Majority gates vs. general weighted threshold gates. Computational Complexity, 2:277–300, 1992.Google Scholar
  44. [GK93]
    M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In ACM Symposium on Theory of Computing (STOC), pages 551–560, 1993.Google Scholar
  45. [GKR+95]
    F. Green, J. Köbler, K. Regan, T. Schwentick, and J. Torán. The power of the middle bit of a #P function. Journal of Computer and System Sciences, 50:456–467, 1995.Google Scholar
  46. [Gol95]
    M. Goldmann. A note on the power of majority gates and modular gates. Information Processing Letters, 53:321–327, 1995.Google Scholar
  47. [Gre]
    F. Green. Complex Fourier technique for lower bounds on the Mod-m degree. Submitted. An earlier version appeared as [Gre95].Google Scholar
  48. [Gre95]
    F. Green. Lower bounds for depth-three circuits with equals and modgates. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 900 of Lecture Notes in Computer Science, pages 71–82. Springer-Verlag, 1995.Google Scholar
  49. [Gro94]
    V. Grolmusz. A weight-size trade-off for circuits with MOD m gates. In ACM Symposium on Theory of Computing (STOC), 1994.Google Scholar
  50. [Gro95a]
    V. Grolmusz. Separating the communication complexities of MOD m and MOD p circuits. Journal of Computer and System Sciences, 51:307–313, 1995.Google Scholar
  51. [Gro95b]
    Vince Grolmusz. On the weak mod m representation of Boolean functions. Chicago Journal of Theoretical Computer Science, 1995(2), July 1995.Google Scholar
  52. [Hås87]
    J. Håstad. Computational Limitations for Small Depth Circuits. MIT Press, Cambridge, MA, 1987.Google Scholar
  53. [Hel86]
    H. Heller. On relativized exponential and probabilistic complexity classes. Information and Computation, 71:231–243, 1986.Google Scholar
  54. [HG91]
    J. Håstad and M. Goldmann. On the power of small-depth threshold circuits. Computational Complexity, 1:113–129, 1991.Google Scholar
  55. [HJP93]
    J. Håstad, S. Jukna, and P. Pudlák. Top-down lower bounds for depth 3 circuits. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 124–129, 1993.Google Scholar
  56. [HMP+93]
    A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán. Threshold circuits of bounded depth. Journal of Computer and System Sciences, 46:129–154, 1993.Google Scholar
  57. [HNW93]
    R. Heiman, I. Newman, and A. Wigderson. On read-once threshold formulae and their randomized decision tree complexity. Theoretical Computer Science, 107:63–76, 1993.Google Scholar
  58. [Hof96]
    T. Hofmeister. A note on the simulation of exponential threshold weights. In International Conference on Computing and Combinatorics (COCOON), volume 1090 of Lecture Notes in Computer Science, pages 136–141. Springer-Verlag, 1996.Google Scholar
  59. [II96]
    K. Iwama and C. Iwamoto. Parallel complexity hierarchies based on PRAMs and DLOGTIME-uniform circuits. In IEEE Conference on Computational Complexity, pages 24–32, 1996.Google Scholar
  60. [IL95]
    N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116:103–116, 1995.Google Scholar
  61. [Imm87]
    N. Immerman. Languages which capture complexity classes. SIAM J. Comput., 4:760–778, 1987.Google Scholar
  62. [Imm89]
    N. Immerman. Expressibility and parallel complexity. SIAM Journal on Computing, 18:625–638, 1989.Google Scholar
  63. [IN89]
    R. Impagliazzo and M. Naor. Efficient cryptographic schemes provably as secure as subset sum. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 236–243, 1989.Google Scholar
  64. [Jon75]
    N. Jones. Space-bounded reducibility among combinatorial problems. Journal of Computer and System Sciences, 11:68–85, 1975. Corrigendum: Journal of Computer and System Sciences 15:241, 1977.Google Scholar
  65. [Juk95]
    S. Jukna. Computing threshold functions by depth-3 threshold circuits with smaller thresholds of their gates. Information Processing Letters, 56:147–150, 1995.Google Scholar
  66. [Kan82]
    R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55:40–56, 1982.Google Scholar
  67. [KP94]
    M. Krause and P. Pudlák. On the computational power of depth 2 circuits with threshold and modulo gates. In ACM Symposium on Theory of Computing (STOC), pages 48–57, 1994.Google Scholar
  68. [KP95]
    M. Krause and P. Pudlák. On computing Boolean functions by sparse real polynomials. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 682–691, 1995.Google Scholar
  69. [KP96]
    M. Krause and P. Pudlák. More on computing Boolean functions by sparse real polynomials and related types of threshold circuits. Technical Report 622, University of Dortmund, 1996. A preliminary version appeared as [KP95].Google Scholar
  70. [Kra91]
    M. Krause. Geometric arguments yield better bounds for threshold circuits and distributed computing. In Structure in Complexity Theory Conference, pages 314–321, 1991.Google Scholar
  71. [KVVY93]
    R. Kannan, H. Venkateswaran, V. Vinay, and A. Yao. A circuit-based proof of Toda's theorem. Information and Computation, 104:271–276, 1993.Google Scholar
  72. [KW95]
    M. Krause and S. Waack. Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in. Mathematical Systems Theory, 28:553–564, 1995.Google Scholar
  73. [Lem96]
    F. Lemieux. Finite Groupoids and their Applications to Computational Complexity. PhD thesis, McGill University, 1996.Google Scholar
  74. [LM94]
    J. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 775 of Lecture Notes in Computer Science, pages 415–426. Springer-Verlag, 1994.Google Scholar
  75. [LMN93]
    N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM, 40:607–620, 1993.Google Scholar
  76. [Lut92]
    J. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44:220–258, 1992.Google Scholar
  77. [Mac95]
    A. Maciel. Threshold Circuits of Small Majority-Depth. PhD thesis, McGill University, 1995.Google Scholar
  78. [MPT91]
    P. McKenzie, P. Péladeau, and D. Thérien. NC1: The automata-theoretic viewpoint. Computational Complexity, 1:330–359, 1991.Google Scholar
  79. [MT93]
    A. Maciel and D. Thérien. Threshold circuits for iterated multiplication: Using AC0 for free. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 665 of Lecture Notes in Computer Science, pages 545–554. Springer-Verlag, 1993.Google Scholar
  80. [NS94]
    N. Nisan and M. Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4:301–313, 1994.Google Scholar
  81. [NW94]
    N. Nisan and A. Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49:149–167, 1994.Google Scholar
  82. [PS88]
    I. Parberry and G. Schnitger. Parallel computation with threshold functions. Journal of Computer and System Sciences, 36:278–302, 1988.Google Scholar
  83. [PS89]
    I. Parberry and G. Schnitger. Relating Boltzmann machines to conventional models of computation. Neural Networks, 2:59–67, 1989.Google Scholar
  84. [Rad94]
    J. Radhakrishnan. σπσ threshold formulas. Combinatorica, 14:345–374, 1994.Google Scholar
  85. [Raz87]
    A. A. Razborov. Lower bounds on the size of bounded depth networks over a complete basis with logical addition. Mathematicheskie Zametki, 41:598–607, 1987. English translation in Mathematical Notes of the Academy of Sciences of the USSR 41:333–338, 1987.Google Scholar
  86. [Raz92]
    A. A. Razborov. On small depth threshold circuits. In Scandinavian Workshop on Algorithm Theory (SWAT), volume 621 of Lecture Notes in Computer Science, pages 42–52. Springer-Verlag, 1992.Google Scholar
  87. [Raz95]
    A. A. Razborov. Bounded arithmetic and lower bounds. In P. Clote and J. Remmel, editors, Feasible Mathematics II, volume 13 of Progress in Computer Science and Applied Logic, pages 344–386. BirkhÄuser, 1995.Google Scholar
  88. [RR94]
    A. A. Razborov and S. Rudich. Natural proofs. In Proceedings, 26th ACM Symposium on Theory of Computing, pages 204–213, 1994.Google Scholar
  89. [RSC95]
    K. Regan, D. Sivakumar, and J.-Y. Cai. Pseudorandom generators, measure theory, and natural proofs. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 26–35, 1995.Google Scholar
  90. [RSO94]
    V. Roychowdhury, K.-Y. Siu, and A. Orlitsky, editors. Theoretical Advances in Neural Computation and Learning. Kluwer, 1994.Google Scholar
  91. [RSOK95]
    V. Roychowdhury, K.-Y. Siu, A. Orlitsky, and T. Kailath. Vector analysis of threshold functions. Information and Computation, 120:22–31, 1995.Google Scholar
  92. [RW93]
    A. A. Razborov and A. Wigderson. n Ω(logn) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Information Processing Letters, 45:303–307, 1993.Google Scholar
  93. [Sit]
    M. Sitharam. Approximation from linear spaces, lower bounds, pseudorandomness, and learning. Submitted.Google Scholar
  94. [Sit95]
    M. Sitharam. Pseudorandom generators and learning algorithms for AC0. Computational Complexity, 5:248–266, 1995.Google Scholar
  95. [Siv96]
    D. Sivakumar. Probabilistic Techniques in Structural Complexity Theory. PhD thesis, SUNY Buffalo, 1996.Google Scholar
  96. [Smo87]
    R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings, 19th ACM Symposium on Theory of Computing, pages 77–82, 1987.Google Scholar
  97. [Smo93]
    R. Smolensky. On representations by low-degree polynomials. In IEEE Symposium on Foundations of Computer Science (FOCS), 1993.Google Scholar
  98. [Sto74]
    L. Stockmeyer. The complexity of decision problems in automata theory and logic. PhD thesis, Mass. Inst. of Technology, 1974.Google Scholar
  99. [Sto87]
    L. Stockmeyer. Classifying the computational complexity of problems. Journal of Symbolic Logic, 52:1–43, 1987.Google Scholar
  100. [Tar93]
    J. Tarui. Probabilistic polynomials, AC0 functions, and the polynomial-time hierarchy. Theoretical Computer Science, 113:167–183, 1993.Google Scholar
  101. [TB95]
    G. Tardos and D. A. Mix Barrington. A lower bound on the MOD 6 degree of the OR function. In Proc. 3rd Israel Symposium on the Theory of Computing and Systems, pages 52–56. IEEE Press, 1995.Google Scholar
  102. [Thé]
    D. Thérien. Circuits constructed with MODq gates cannot compute AND in sublinear size. Computational Complexity, 4:383–388, 1994.Google Scholar
  103. [Tod91]
    S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20:865–877, 1991.Google Scholar
  104. [Tsa96]
    S.-C. Tsai. Lower bounds on representing Boolean functions as polynomials in z m. SIAM Journal on Discrete Mathematics, 9:55–62, 1996.Google Scholar
  105. [Wil85]
    C. Wilson. Relativized circuit complexity. Journal of Computer and System Sciences, 31:169–181, 1985.Google Scholar
  106. [Yao85]
    A. Yao. Separating the polynomial-time hierarchy by oracles. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 1–10, 1985.Google Scholar
  107. [Yao90]
    A. Yao. On ACC and threshold circuits. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 619–627, 1990.Google Scholar
  108. [YP94]
    P.Y. Yan and I. Parberry. Exponential size lower bounds for some depth three circuits. Information and Computation (formerly Information and Control), 112:117–130, 1994.Google Scholar
  109. [ZBT93]
    Z.-L. Zhang, D. A. Mix Barrington, and J. Tarui. Computing symmetric functions with AND/OR circuits and a single MAJORITY gate. In 10th Annual Symposium on Theoretical Aspects of Computer Science, volume 665 of Lecture Notes in Computer Science, pages 535–544. Springer-Verlag, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Eric Allender
    • 1
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA

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