Constant Depth Circuits, Fourier Transform, and Learnability

Journal of the ACM (JACM)

Volumn 40, Issue 3, 1993, Pages 607-620

  Linial, Nathan ^a   Mansour, Yishay ^b   Nisan, Noam ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

AC0 circuits; approximation; Boolean functions; complexity; harmonic analysis learning

Indexed keywords

ALGORITHMS; BOOLEAN FUNCTIONS; FOURIER TRANSFORMS;

CONSTANT DEPTH CIRCUITS; HASTAD'S SWITCHING LEMMA; PSEUDORANDOM FUNCTION GENERATORS;

COMPUTER SCIENCE;

EID: 0027627287     PISSN: 00045411     EISSN: 1557735X     Source Type: Journal    
DOI: 10.1145/174130.174138     Document Type: Article

References (20)

1. AJTAI. M. Σ1,1-formulae on finite structure. Ann. Pure Appl. Logic 24 (1983), 1-48.
2. AJTAI, M., AND WIGDERSON, A. Deterministic simulation of probabilistic constant depth circuits. In Advances in computing research. Vol. 5. S. Micali, ed. JAI Press, Greenwich, Ct., 1989, pp. 199-222.
3. BRANDMAN, Y., HENNESSY, J., AND ORLITSKY, A. A spectral lower bound technique for the size of decision trees and two level circuits. IEEE Trans. Comput. 39, 2 (1990), 282-287.
4. DYM, H., AND MCKEAN, H. P. Fourier Series and Integrals. Academic Press, Orlando, Fla., 1972.
5. FURST, M., SAXE, J., AND SIPSER, M. Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17 (1984), 13-27.
6. GOLDREICH, O., GOLDWASSER, S., AND MICALI, S. HOW to construct random functions. J. ACM 33, 4 (Oct. 1986), 792-807.
7. HAGERUP, T., AND RUB, C. A guided tour to Chernoff bounds. Inf. Proc. Lett. 33 (1989), 305-308.
8. HASTAD, J. AND BOPPANA, Computational limitations for small depth circuits. Ph.D. dissertation, MIT Press, Cambridge, Mass., 1986.
9. KAHN, J., KALAI, G., AND LINIAL, N. The influence of variables on Boolean functions. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science (White Plains, N.Y., Oct.). IEEE, New York, 1988, pp. 68-80.
10. KEARNS, M., AND VALIANT, L. Cryptographic limitations on learning Boolean formulae and finite automata. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (Seattle, Wash., May). ACM, New York, 1989, pp. 433-444.
11. LINIAL, N., AND NISAN, N. Approximate inclusion-exclusion. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing. (Baltimore, Md., May 12-14). ACM, New York, 1990, pp. 260-270.
12. MANSOUR, Y., NISAN, N., AND TIWARI, P. The computational complexity of universal hashing. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing. (Baltimore, Md., May 12-14). ACM, New York, 1990, pp. 235-243.
13. NISAN, N., AND WIGDERSON, A. Hardness vs. randomness. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science (White Plains, N.Y., Oct.) IEEE, New York, 1988, pp. 2-12.
14. RAZBOROV, A. A. Lower bounds for the size of circuits of bounded depth with basis AND, XOR. Math. Zametski 41 (1987), 598-607 (in Russian). English translation in Math Notes 41 (1987), 333-338.
15. RIVEST, R. L. Learning decision lists. Machine Learning 2, 3 (1987), 229-246.
16. SANTHA, M., AND WILSON, C. Polynomial size circuits with a limited number of negations. In Proceedings of the 8th Annual Symposium on Aspects of Theoretical Computer Science. IEEE, New York, 1991, pp. 228-237.
17. SMOLENSKY, R. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (New York City, N.Y., May 25-27). ACM, New York, 1987, pp. 77-82.
18. VALIANT, L. G. A theory of the learnable. Commun ACM 27, 11 (Nov. 1984), 1134-1142.
19. YAO, A. C. Theory and applications of trapdoor functions. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science. IEEE, New York, 1982, pp. 80-91.
20. YAO, A. C. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th Annual Symposium on Foundations of Computer Science (Portland, Ore. Oct.). IEEE, New York, 1985, pp. 1-10.