computational complexity

, Volume 5, Issue 2, pp 99–112 | Cite as

Top-down lower bounds for depth-three circuits

  • J. Håstad
  • S. Jukna
  • P. Pudlák


We present a top-down lower bound method for depth-three ⋎, ⋏, ¬-circuits which is simpler than the previous methods and in some cases gives better lower bounds. In particular, we prove that depth-three ⋎, ⋏, ¬-circuits that compute parity (or majority) require size at least\(2^{0.618...\sqrt n } (or 2^{0.849...\sqrt n } \), respectively). This is the first simple proof of a strong lower bound by a top-down argument for non-monotone circuits.

Key words

Computational complexity small-depth circuits 

subject classifications



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  1. M. Ajtai, 111-1 on finite structures.Ann. Pure and Appl. Logic 24 (1983), 1–48.Google Scholar
  2. A. Borodin, A. Razborov andR. Smolensky, On lower bounds for reak-k-times branching programs.Computational Complexity 3 (1993), 1–18.Google Scholar
  3. P. Erdös andR. Rado, Intersection theorems for systems of sets.J. London Math. Soc. 35 (1960), 85–90.Google Scholar
  4. M. Furst, J. Saxe andM. Sipser, Parity, circuits and the polynomial time hierarchy.Math. Systems Theory 17 (1984), 13–27.Google Scholar
  5. J. Håstad,Almost Optimal Lower Bounds for Small Depth Circuits. Advances in Computing Research, ed.S. Micali, Vol 5 (1989), 143–170.Google Scholar
  6. S. Jukna,Finite limits and lower bounds for circuit size. Tech. Rep. 94-06, Informatik, University of Trier, 1994.Google Scholar
  7. M. Karchmer andA. Wigderson, Monotone circuits for connectivity require super-logarithmic depth.SIAM J. Disc. Math. 3 (1990), 255–265.Google Scholar
  8. M. Klawe, W. J. Paul, N. Pippenger, M. Yannakakis, On monotone formulae with restricted depth. InProc. Sixteenth Ann. ACM Symp. Theor. Comput., 1984, 480–487.Google Scholar
  9. R. Raz and A. Wigderson, Monotone circuits for matching require linear depth. InProc. Twenty-second Ann. ACM Symp. Theor. Comput., 1990, 287–292.Google Scholar
  10. A. A. Razborov, Lower bounds for the size of circuits of bounded depth with basis {⋏, ⊕}.Math. Notes of the Academy of Sciences of the USSR 41:4 (1987), 333–338.Google Scholar
  11. M. Sipser, Private communication, 1991.Google Scholar
  12. M. Sipser, A topological view of some problems in complexity theory. InColloq. math. Soc. János Bolyai 44 (1985), 387–391.Google Scholar
  13. R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity. InProc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 77–82.Google Scholar
  14. L.G. Valiant, Graph-theoretic arguments in low level complexity. InProc. Sixth Conf. Math. Foundations of Computer Science, Lecture Notes in Computer Science, 1977, Springer-Verlag, 162–176.Google Scholar
  15. A.C. Yao, Separating the polynomial time hierarchy by oracles. InProc. Twentysixth Ann. IEEE Symp. Found. Comput. Sci., 1985, 1–10.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • J. Håstad
    • 1
  • S. Jukna
    • 3
  • P. Pudlák
    • 2
  1. 1.Royal Institute of TechnologyStockholmSweden
  2. 2.Mathematical InstitutePragueCzech Republic
  3. 3.Institute of MathematicsVilniusLithuania

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