Separating AC 0 from depth-2 majority circuits

SIAM Journal on Computing

Volumn 38, Issue 6, 2009, Pages 2113-2129

  Sherstov, Alexander A ^a  

^a University of Texas at Austin   (United States)

Author keywords

Communication complexity; Constant depth and or not circuits; Degree Discrepancy theorem; Discrepancy; Majority circuits; Threshold degree of Boolean functions

Indexed keywords

COMMUNICATION COMPLEXITY; CONSTANT-DEPTH AND/OR/NOT CIRCUITS; DEGREE/DISCREPANCY THEOREM; DISCREPANCY; MAJORITY CIRCUITS; THRESHOLD DEGREE OF BOOLEAN FUNCTIONS;

COMMUNICATION; COMPUTER CIRCUITS; CONTROL THEORY; GATES (TRANSISTOR);

BOOLEAN FUNCTIONS;

EID: 65949113138     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/08071421X     Document Type: Article

References (45)

1. E. ALLENDER, A note on the power of threshold circuits, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Research Triangle Park, NC, 1989, pp. 580-584.
2. J. ASPNES, R. BEIGEL, M. L. FURST, AND S. RUDICH, The expressive power of voting polynomials, Combinatorica, 14 (1994), pp. 135-148.
3. L. BABAI, P. FRANKL, AND J. SIMON, Complexity classes in communication complexity theory, in Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Toronto, 1986, pp. 337-347.
4. L. BABAI, N. NISAN, AND M. SZEGEDY, Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs, J. Comput. System Sci., 45 (1992), pp. 204-232.
5. H. BUHRMAN, N. K. VERESHCHAGIN, AND R. DE WOLF, On computation and communication with small bias, in Proceedings of the 22nd Annual IEEE Conference on Computational Complexity (CCC), San Diego, CA, 2007, pp. 24-32.
6. A. CHATTOPADHYAY, Discrepancy and the power of bottom fan-in in depth-three circuits, in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Providence, RI, 2007, pp. 449-458.
7. A. CHATTOPADHYAY AND A. ADA, Multiparty communication complexity of disjointness, in Electronic Colloquium on Computational Complexity (ECCC), 2008, Report TR08-002 (electronic).
8. F. R. K. CHUNG AND P. TETALI, Communication complexity and quasi randomness, SIAM J. Discrete Math., 6 (1993), pp. 110-123.
9. M. DAVID AND T. PITASSI, Separating NOF communication complexity classes RP and NP, in Electronic Colloquium on Computational Complexity (ECCC), 2008, Report TR08-014 (electronic).
10. M. DAVID, T. PITASSI, AND E. VIOLA, Improved separations between nondeterministic and randomized multiparty communication, in Proceedings of the 12th International Workshop on Randomization and Computation (RANDOM), 2008, pp. 371-384.
11. J. FORD AND A. GÁL, Hadamard tensors and lower bounds on multiparty communication complexity, in Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP), Springer, Berlin, 2005, pp. 1163-1175.
12. J. FORSTER, A linear lower bound on the unbounded error probabilistic communication complexity, J. Comput. System Sci., 65 (2002), pp. 612-625.
13. J. FORSTER, M. KRAUSE, S. V. LOKAM, R. MUBARAKZJANOV, N. SCHMITT, AND H.-U. SIMON, Relations between communication complexity, linear arrangements, and computational complexity, in Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science (FST TCS), Springer, Berlin, 2001, pp. 171-182.
14. M. GOLDMANN, J. HÅSTAD, AND A. A. RAZBOROV, Majority gates vs. general weighted threshold gates, Comput. Complex., 2 (1992), pp. 277-300.
15. A. HAJNAL, W. MAASS, P. PUDLÁK, M. SZEGEDY, AND G. TURÁN, Threshold circuits of bounded depth, J. Comput. System Sci., 46 (1993), pp. 129-154.
16. H. KLAUCK, Lower bounds for quantum communication complexity, SIAM J. Comput., 37 (2007), pp. 20-46.
17. 1/3), J. Comput. System Sci., 68 (2004), pp. 303-318.
18. A. R. KLIVANS AND R. A. SERVEDIO, Learning intersections of halfspaces with a margin, in Proceedings of the 17th Conference on Learning Theory (COLT), Springer, Berlin, 2004, pp. 348-362.
19. A. R. KLIVANS AND A. A. SHERSTOV, A lower bound for agnostically learning disjunctions, in Proceedings of the 20th Conference on Learning Theory (COLT), Springer, Berlin, 2007, pp. 409-423.
20. A. R. KLIVANS AND A. A. SHERSTOV, Unconditional lower bounds for learning intersections of halfspaces, Mach. Learn., 69 (2007), pp. 97-114.
21. M. KRAUSE AND P. PUDLÁAK, On the computational power of depth-2 circuits with threshold and modulo gates, Theoret. Comput. Sci., 174 (1997), pp. 137-156.
22. M. KRAUSE AND P. PUDLÁK, Computing Boolean functions by polynomials and threshold circuits, Comput. Complexity, 7 (1998), pp. 346-370.
23. E. KUSHILEVITZ AND N. NISAN, Communication Complexity, Cambridge University Press, Cambridge, UK, 1997.
24. T. LEE AND A. SHRAIBMAN, Disjointness is hard in the multi-party number-on-the-forehead model, in Proceedings of the 23rd IEEE Conference on Computational Complexity (CCC), College Park, MD, 2008, pp. 81-91.
25. N. LINIAL AND A. SHRAIBMAN, Learning complexity vs. communication complexity, in Proceedings of the 23rd IEEE Conference on Computational Complexity (CCC), College Park, MD, 2008, pp. 53-63.
26. M. L. MINSKY AND S. A. PAPERT, Perceptrons: Expanded Edition, MIT Press, Cambridge, MA, 1988.
27. N. NISAN, The communication complexity of threshold gates, in Combinatorics, Paul Erdos Is Eighty, Vol. 1, János Bolyai Math. Soc., Budapest, 1993, pp. 301-315.
28. A. B. J. NOVIKOFF, On convergence proofs for perceptrons, in Proceedings of the Symposium on the Mathematical Theory of Automata, Vol. XII, Polytechnic Press, Brooklyn, NY, 1962, pp. 615-622.
29. R. O'DONNELL AND R. A. SERVEDIO, New degree bounds for polynomial threshold functions, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), San Diego, CA, 2003, pp. 325-334.
30. R. O'DONNELL AND R. A. SERVEDIO, Extremal properties of polynomial threshold functions, J. Comput. System Sci., 74 (2008), pp. 298-312.
31. R. PATURI AND J. SIMON, Probabilistic communication complexity, J. Comput. System Sci., 33 (1986), pp. 106-123.
32. R. RAZ, Fourier analysis for probabilistic communication complexity, Comput. Complex., 5 (1995), pp. 205-221.
33. R. RAZ, The BNS-Chung criterion for multi-party communication complexity, Comput. Complex., 9 (2000), pp. 113-122.
34. A. A. RAZBOROV, On the distributional complexity of disjointness, Theoret. Comput. Sci., 106 (1992), pp. 385-390.
35. A. A. RAZBOROV, Quantum communication complexity of symmetric predicates, Izv. Math., 67 (2003), pp. 145-159.
36. 0, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Philadelphia, 2008, pp. 57-66.
37. A. SCHRIJVER, Theory of Linear and Integer Programming, John Wiley & Sons, New York, 1998.
38. A. A. SHERSTOV, Powering requires threshold depth 3, Inform. Process. Lett., 102 (2007), pp. 104-107.
39. A. A. SHERSTOV, Communication lower bounds using dual polynomials, Bull. EATCS, 95 (2008), pp. 59-93.
40. A. A. SHERSTOV, Halfspace matrices, Comput. Complex., 17 (2008), pp. 149-178.
41. A. A. SHERSTOV, The pattern matrix method for lower bounds on quantum communication, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), Victoria, British Columbia, 2008, pp. 85-94.
42. A. A. SHERSTOV, The unbounded-error communication complexity of symmetric functions, in Proceedings of the 49th Symposium on Foundations of Computer Science (FOCS), 2008, pp. 384-393.
43. K.-Y. SIU AND J. BRUCK, On the power of threshold circuits with small weights, SIAM J. Discrete Math., 4 (1991), pp. 423-435.
44. K.-Y. SIU, J. BRUCK, T. KAILATH, AND T. HOFMEISTER, Depth efficient neural networks for division and related problems, IEEE Trans. Inform. Theory, 39 (1993), pp. 946-956.
45. K.-Y. SIU AND V. P. ROYCHOWDHURY, On optimal depth threshold circuits for multiplication and related problems, SIAM J. Discrete Math., 7 (1994), pp. 284-292.