On TC0, AC0, and arithmetic circuits

Proceedings of the Annual IEEE Conference on Computational Complexity

Volumn , Issue , 1997, Pages 134-148

  Agrawal, M ^a   Allender, E ^b   Datta, S ^b  

^a INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; FUNCTIONS; LOGIC CIRCUITS; TIMING CIRCUITS;

ARITHMETIC CIRCUIT; CIRCUIT LOWER BOUNDS; CLOSURE PROPERTY; FINITE FIELDS; FUNCTION CLASS; LOWER BOUNDS; POLYNOMIAL SIZE; THRESHOLD GATES;

COMPUTER CIRCUITS;

EID: 78649883139     PISSN: 10930159     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/CCC.1997.612309     Document Type: Conference Paper

References (48)

1. E. Allender and U. Hertrampf, Depth Reduction for Circuits of Unbounded Fan-In. Information and Computation, 112 (2): 217-238, 1994.
2. E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. In Proc. 28th ACM Symposium on Theory of Computing (STOC), pages 161-167, 1996.
3. E. Allender and J. Jiao. Depth reduction for noncommutative arithmetic circuits. In Proc. 25th ACM Symposium on Theory of Computing (STOC), pages 515-522, 1993.
4. C. Alvarez and B. Jenner. A very hard log-space counting class. Theoretical Computer Science, 107: 3-30, 1993.
5. E. Allender, J. Jiao, M. Mahajan, and V. Vinay. Non-commutative arithmetic circuits: depth reduction and size lower bounds. To appear in Theoretical Computer Science. Preliminary versions appeared as [AJ93a, MV94].
6. E. Allender. The permanent requires large uni-form threshold circuits. Submitted. A preliminary version of this paper appeared as [A1196].
7. E. Allender. P-uniform circuit complexity. J. ACM, 36: 912-928, 1989.
8. 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.
9. E. Allender and M. Ogihara. Relationships among PL, #L, and the determinant. In Proc. 9th IEEE Structure in Complexity Theory Conference, pages 267-278, 1994.
10. P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAMJ. Comput., 15: 994-1003, 1986.
11. J. Boyar, G. Frandsen, and G. Sturtivant. An arithmetical model of computation equivalent to threshold circuits. Theoretical Computer Science, 93: 303-319, 1992.
12. D. A. Mix Barrington, Personal Communication, 1997.
13. D. A. Mix Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41: 274306, 1990.
14. D. A. Mix Barrington and D. Therien. Finite monoids and the fine structure of NC1. Journal of the ACM, 35: 941-952, 1988.
15. H. Caussinus, P. McKenzie, D. Therien, and H. Vollmer. Nondeterministic NC1 computation. In Proceedings, 11th Annual IEEE Conference on Computational Complexity, pages 12-21, 1996.
16. A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13: 423-439, 1984.
17. L. Denenberg, Y. Gurevich, and S. Shelah. Definability by constant-depth polynomial-size circuits. Information and Control, 70: 216-240, 1986.
18. George I. Davida and Bruce Litow. Fast parallel arithmetic via modular representation. SIAM Journal on Computing, 20 (4): 756-765, August 1991.
19. Paul F. Dietz, loan I. Macarie, and Joel I. Seiferas. Bits and relative order from residues, space efficiently. Information Processing Letters, 50 (3): 123-127, 9May 1994.
20. K. Etessami. Counting quantifiers, successor relations, and logarithmic space. In Proc. 10th IEEE Structure in Complexity Theory Conference, pages 2-11, 1995.
21. Stephen A. Fenner, Lance J. Fortnow, and Stuart A. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48 (1): 116-148, February 1994.
22. R. Fagin, M. Klawe, N. Pippenger, and L. Stockmeyer. Bounded-depth, polynomial-size circuits for symmetric functions. Theoretical Computer Science, 36: 239-250, 1985.
23. G. Frandsen, M. Valence, and D. Mix Barrington. Some results on uniform arithmetic circuit complexity. Mathematical Systems Theory, 27, 1994.
24. J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6: 675-695, 1977.
25. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford press, USA, 1979.
26. N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116: 103-116, 1995.
27. B. Litow. On iterated integer product. Information Processing Letters, 42 (5): 269-272, 03 July 1992.
28. I. Macarie. Space-efficient deterministic simulation of probabilistic automata. To appear in SIAM J. Comput. Preliminary version appeared as [Mac94].
29. I. Macarie. Space-efficient deterministic simulation of probabilistic automata. In 11th Symposium on Theoretical Aspects of Computing (STACS), volume 775 of Lecture Notes in Computer Science, pages 109-122. Springer-Verlag, 1994.
30. M. Mahajan and V. Vinay. A combinatorial algorithm for the determinant. To appear in Proc. SODA '97.
31. M. Mahajan and V. Vinay. Non-commutative computation, depth reduction and skew circuits. In Proc. 14th FST&TCS, volume 880 of Lecture Notes in Computer Science, pages 4859. Springer-Verlag, 1994.
32. I. Parberry and G. Schnitger. Parallel computation with threshold functions. Journal of Computer and System Sciences, 36: 278-302, 1988.
33. A. A. Razborov. Lower bounds on the size of bounded depth networks over a complete basis with logical addition. Mathematicheskie Za-metki, 41: 598-607, 1987.
34. English translation in Mathematical Notes of the Academy of Sciences of the USSR 41: 333-338, 1987.
35. A. A. Razborov and S. Rudich. Natural proofs. In Proceedings, 26th ACM Symposium on Theory of Computing, pages 204-213, 1994.
36. J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput, 21: 896-908, 1992.
37. W. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, 21: 365-383, 1981.
38. 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.
39. M. Santha and S. Tan. Verifying the determi-nant in parallel. To appear in Computational Complexity. A preliminary version of this paper appeared in Proceedings of the 5th ACM-SIGSAM International Symposium on Symbolic and Algebraic Computation, 1994.
40. S. Toda. Counting problems computationally equivalent to the determinant. Manuscript.
41. S. Toda. Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Trans. Inf. And Syst., E75-D: 116-124, 1992.
42. L. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing (STOC), pages 249-261, 1979.
43. L. Valiant. The complexity of computing the Permanent. Theoretical Computer Science, 8: 189-201, 1979.
44. L. Valiant. Why is boolean complexity theory difficult? In M. S. Paterson, editor, Boolean Function Complexity, volume 169 of London Mathematical Society Lecture Notes Series, pages 84-94. Cambridge University Press, 1992.
45. H. Venkateswaran. Circuit definitions of non-deterministic complexity classes. SIAM Journal on Computing, 21: 655-670, 1992.
46. V. Vinay. Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proc. 6th IEEE Structure in Complexity Theory Conference, pages 270-284, 1991.
47. K. W. Wagner. The complexity of combinatorial problems with succinct input representation. Acta Informatica, 23: 325-356, 1986.
48. T. Yamakami. Uniform AC0 Counting Circuits. Manuscript, 1996.