메뉴 건너뛰기




Volumn 209, Issue 1-2, 1998, Pages 47-86

Non-commutative arithmetic circuits: Depth reduction and size lower bounds

Author keywords

Arithmetic circuits; AuxPDAs; Depth Complexity; Non commutative computation; Skew circuits

Indexed keywords


EID: 0002507290     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(97)00227-2     Document Type: Article
Times cited : (75)

References (57)
  • 1
    • 0030264127 scopus 로고    scopus 로고
    • On the isomorphism problem for weak reducibilities
    • M. Agrawal, On the isomorphism problem for weak reducibilities, J. Computer Sys. Sci. 53 (1996) 267-282. Preliminary version in Proc. 9th Structure in Complexity Theory Conference (1994) 338-355.
    • (1996) J. Computer Sys. Sci. , vol.53 , pp. 267-282
    • Agrawal, M.1
  • 4
    • 0024754676 scopus 로고
    • P-uniform circuit complexity
    • E. Allender, P-uniform circuit complexity, J. ACM 36 (1989) 912-928.
    • (1989) J. ACM , vol.36 , pp. 912-928
    • Allender, E.1
  • 5
  • 6
    • 0028516587 scopus 로고
    • A uniform circuit lower bound for the permanent
    • E. Allender, V. Gore, A uniform circuit lower bound for the permanent, SIAM J. Comput. 23 (1994) 1026-1049.
    • (1994) SIAM J. Comput. , vol.23 , pp. 1026-1049
    • Allender, E.1    Gore, V.2
  • 7
    • 0008651743 scopus 로고
    • Depth reduction for circuits of unbounded fanin
    • E. Allender, U. Hertrampf, Depth reduction for circuits of unbounded fanin, Inform. and Comput. 112 (1994) 217-238.
    • (1994) Inform. and Comput. , vol.112 , pp. 217-238
    • Allender, E.1    Hertrampf, U.2
  • 9
    • 38049091720 scopus 로고
    • A very hard log-space counting class
    • C. Álvarez, B, Jenner, A very hard log-space counting class, Theoret. Comput. Sci. 107 (1993) 3-30.
    • (1993) Theoret. Comput. Sci. , vol.107 , pp. 3-30
    • Álvarez, C.1    Jenner, B.2
  • 10
    • 21844492438 scopus 로고
    • A note on log space optimization
    • C. Álvarez, B. Jenner, A note on log space optimization, Comput. Complexity 5 (1995) 155-166.
    • (1995) Comput. Complexity , vol.5 , pp. 155-166
    • Álvarez, C.1    Jenner, B.2
  • 11
    • 0030554482 scopus 로고    scopus 로고
    • Relationships among PL, #L, and the determinant
    • E. Allender, M. Ogihara, Relationships among PL, #L, and the determinant, RAIRO Theoretical Information and Applications 30 (1996), 1-21. Preliminary version in Proc. 9th IEEE Structure in Complexity Theory Conference, 1994, 267-278.
    • (1996) RAIRO Theoretical Information and Applications , vol.30 , pp. 1-21
    • Allender, E.1    Ogihara, M.2
  • 13
    • 0022811668 scopus 로고
    • Log depth circuits for division and related problems
    • P.W. Beame, S.A. Cook, H.J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput. 15 (1986) 994-1003.
    • (1986) SIAM J. Comput. , vol.15 , pp. 994-1003
    • Beame, P.W.1    Cook, S.A.2    Hoover, H.J.3
  • 14
    • 0001455609 scopus 로고
    • On ACC
    • R. Beigel, J. Tarui, On ACC, Computational Complexity 4 (1994) 350-366. Special issue devoted to the 4th Annual McGill Workshop on Complexity Theory. Preliminary version in Proc. 32nd FOCS (1991) 783-792.
    • (1994) Computational Complexity , vol.4 , pp. 350-366
    • Beigel, R.1    Tarui, J.2
  • 15
    • 0001455609 scopus 로고
    • 4th Annual McGill Workshop on Complexity Theory
    • R. Beigel, J. Tarui, On ACC, Computational Complexity 4 (1994) 350-366. Special issue devoted to the 4th Annual McGill Workshop on Complexity Theory. Preliminary version in Proc. 32nd FOCS (1991) 783-792.
    • (1991) Preliminary Version in Proc. 32nd FOCS , pp. 783-792
  • 16
    • 0026813292 scopus 로고
    • An arithmetical model of computation equivalent to threshold circuits
    • J. Boyar, G. Frandsen, G. Sturtivant, An arithmetical model of computation equivalent to threshold circuits, Theoret. Comput. Sci. 93 (1992) 303-319.
    • (1992) Theoret. Comput. Sci. , vol.93 , pp. 303-319
    • Boyar, J.1    Frandsen, G.2    Sturtivant, G.3
  • 17
    • 85035362546 scopus 로고
    • On one-way auxiliary pushdown automata
    • Proc. 3rd GI Conf., Springer, Berlin
    • F.-J. Brandenberg, On one-way auxiliary pushdown automata, in: Proc. 3rd GI Conf., Lecture notes in Computer Science, vol. 48, Springer, Berlin, 1977, pp. 133-144.
    • (1977) Lecture Notes in Computer Science , vol.48 , pp. 133-144
    • Brandenberg, F.-J.1
  • 19
    • 0029499709 scopus 로고
    • Randomness-optimal unique element isolation, with applications to perfect matching and related problems
    • S. Chari, P. Rohatgi, A. Srinivasan, Randomness-optimal unique element isolation, with applications to perfect matching and related problems, SIAM Journal on Computing 24(5) (1995) 1036-1050. Preliminary version in Proc. 25th STOC (1993) 458-467.
    • (1995) SIAM Journal on Computing , vol.24 , Issue.5 , pp. 1036-1050
    • Chari, S.1    Rohatgi, P.2    Srinivasan, A.3
  • 20
    • 0029499709 scopus 로고
    • S. Chari, P. Rohatgi, A. Srinivasan, Randomness-optimal unique element isolation, with applications to perfect matching and related problems, SIAM Journal on Computing 24(5) (1995) 1036-1050. Preliminary version in Proc. 25th STOC (1993) 458-467.
    • (1993) Preliminary Version in Proc. 25th STOC , pp. 458-467
  • 21
    • 0014979424 scopus 로고
    • Characterization of pushdown machines in terms of time-bounded computers
    • S. Cook, Characterization of pushdown machines in terms of time-bounded computers, J. ACM 18 (1971) 4-18.
    • (1971) J. ACM , vol.18 , pp. 4-18
    • Cook, S.1
  • 22
    • 0346648729 scopus 로고
    • #L?
    • Fachbereich Informatik der Humboldt-Universität zu Berlin
    • #L?, Informatik-Preprint 8, Fachbereich Informatik der Humboldt-Universität zu Berlin, 1991.
    • (1991) Informatik-Preprint 8
    • Damm, C.1
  • 23
    • 84945710332 scopus 로고
    • Explicit constructions of linear size superconcentrators
    • O. Gabber, Z. Galil, Explicit constructions of linear size superconcentrators, in: Proc. 20th FOCS, 1979, pp. 364-370.
    • (1979) Proc. 20th FOCS , pp. 364-370
    • Gabber, O.1    Galil, Z.2
  • 24
    • 0032330819 scopus 로고    scopus 로고
    • Isolating an odd number of elements and applications in complexity theory
    • to appear (formerly Math. Systems Theory)
    • S. Gupta, Isolating an odd number of elements and applications in complexity theory, Theory of Computing Systems, to appear (formerly Math. Systems Theory).
    • Theory of Computing Systems
    • Gupta, S.1
  • 25
    • 0038773392 scopus 로고
    • Languages simultaneously complete for one-way and two-way log-tape automata
    • J. Hartmanis, S. Mahaney, Languages simultaneously complete for one-way and two-way log-tape automata, SIAM J. Comput. 10 (1981) 383-390.
    • (1981) SIAM J. Comput. , vol.10 , pp. 383-390
    • Hartmanis, J.1    Mahaney, S.2
  • 26
    • 0002783587 scopus 로고
    • The complexity of ranking simple languages
    • D. Huynh, The complexity of ranking simple languages, Mathematical Systems Theory 23 (1990) 1-19.
    • (1990) Mathematical Systems Theory , vol.23 , pp. 1-19
    • Huynh, D.1
  • 27
    • 0343298010 scopus 로고
    • The complexity of iterated multiplication
    • N. Immerman, S. Landau, The complexity of iterated multiplication, Inform. and Comput. 116 (1995) 103-116.
    • (1995) Inform. and Comput. , vol.116 , pp. 103-116
    • Immerman, N.1    Landau, S.2
  • 29
    • 0347909151 scopus 로고    scopus 로고
    • personal communication
    • B. Jenner, personal communication.
    • Jenner, B.1
  • 30
    • 0020160366 scopus 로고
    • Some exact complexity results for straight-line computations over semirings
    • M. Jerrum, M. Snir, Some exact complexity results for straight-line computations over semirings, J. ACM 29 (1982) 874-897.
    • (1982) J. ACM , vol.29 , pp. 874-897
    • Jerrum, M.1    Snir, M.2
  • 32
    • 24244461766 scopus 로고
    • Extension of Toda's theorem to middle bit classes
    • 11-13 December Institute of Mathematical Sciences, Madras, Technical Report IMSc-94/51
    • J. Köbler, Extension of Toda's theorem to middle bit classes, in: Proc. Workshop on Algebraic Methods in Complexity Theory (AMCoT), 11-13 December 1994, Institute of Mathematical Sciences, Madras, Technical Report IMSc-94/51.
    • (1994) Proc. Workshop on Algebraic Methods in Complexity Theory (AMCoT)
    • Köbler, J.1
  • 33
    • 0344179997 scopus 로고
    • On the parallel evaluation of classes of circuits
    • Proc. 10th FST&TCS, Springer, Berlin
    • S.R. Kosaraju, On the parallel evaluation of classes of circuits, in: Proc. 10th FST&TCS, Lecture Notes in Computer Science, vol. 472, Springer, Berlin, 1990, pp. 232-237.
    • (1990) Lecture Notes in Computer Science , vol.472 , pp. 232-237
    • Kosaraju, S.R.1
  • 34
    • 0024020188 scopus 로고
    • The complexity of optimization problems
    • M. Krentel, The complexity of optimization problems, J. Comput. System. Sci. 36 (1988) 490-509.
    • (1988) J. Comput. System. Sci. , vol.36 , pp. 490-509
    • Krentel, M.1
  • 35
    • 0024054539 scopus 로고
    • Efficient parallel evaluation of straight-line code and arithmetic circuits
    • G. Miller, V. Ramachandran, E. Kaltofen, Efficient parallel evaluation of straight-line code and arithmetic circuits, SIAM J. Comput. 17 (1988) 687-695.
    • (1988) SIAM J. Comput. , vol.17 , pp. 687-695
    • Miller, G.1    Ramachandran, V.2    Kaltofen, E.3
  • 36
    • 0346017689 scopus 로고
    • Dynamic parallel complexity of computational circuits
    • G. Miller, S.-H. Teng, Dynamic parallel complexity of computational circuits, in: Proc. 19th STOC, 1987, pp. 478-489.
    • (1987) Proc. 19th STOC , pp. 478-489
    • Miller, G.1    Teng, S.-H.2
  • 37
    • 85000788527 scopus 로고
    • Non-commutative computation, depth reduction and skew circuits
    • Proc. 14th FST&TCS
    • M. Mahajan, V. Vinay, Non-commutative computation, depth reduction and skew circuits, in: Proc. 14th FST&TCS, Lecture Notes in Computer Science, vol. 880, 1994, pp. 48-59.
    • (1994) Lecture Notes in Computer Science , vol.880 , pp. 48-59
    • Mahajan, M.1    Vinay, V.2
  • 38
    • 84939715814 scopus 로고
    • Las Vegas is better than determinism in VLSI and distributed computing
    • K. Mehlhorn, E. Schmidt, Las Vegas is better than determinism in VLSI and distributed computing, in: Proc. 14th STOC, 1982, pp. 330-337.
    • (1982) Proc. 14th STOC , pp. 330-337
    • Mehlhorn, K.1    Schmidt, E.2
  • 39
    • 0029635470 scopus 로고
    • On quasilinear time complexity theory
    • A. Naik, K. Regan, D. Sivakumar, On quasilinear time complexity theory, Theoretical Computer Science 148(2) (1995) 325-329. Preliminary version in Proc. 11th STACS, Lecture Notes in Computer Science 778 (1994) 289-300.
    • (1995) Theoretical Computer Science , vol.148 , Issue.2 , pp. 325-329
    • Naik, A.1    Regan, K.2    Sivakumar, D.3
  • 40
    • 0029635470 scopus 로고
    • Preliminary version in Proc. 11th STACS
    • A. Naik, K. Regan, D. Sivakumar, On quasilinear time complexity theory, Theoretical Computer Science 148(2) (1995) 325-329. Preliminary version in Proc. 11th STACS, Lecture Notes in Computer Science 778 (1994) 289-300.
    • (1994) Lecture Notes in Computer Science , vol.778 , pp. 289-300
  • 41
    • 0008431594 scopus 로고
    • Unambiguous auxiliary pushdown automata and semi-unbounded fanin circuits
    • R. Niedermeier, P. Rossmanith, Unambiguous auxiliary pushdown automata and semi-unbounded fanin circuits, Inform. and Comput. 118 (1995) 227-245.
    • (1995) Inform. and Comput. , vol.118 , pp. 227-245
    • Niedermeier, R.1    Rossmanith, P.2
  • 42
    • 85000547940 scopus 로고
    • Lower bounds for non-commutative computation
    • N. Nisan, Lower bounds for non-commutative computation, in: Proc. 23rd STOC, 1991, pp. 410-418.
    • (1991) Proc. 23rd STOC , pp. 410-418
    • Nisan, N.1
  • 43
    • 0026932814 scopus 로고
    • On threshold circuits and polynomial computation
    • J. Reif, S. Tate, On threshold circuits and polynomial computation, SIAM J. Comput. 21 (1992) 896-908.
    • (1992) SIAM J. Comput. , vol.21 , pp. 896-908
    • Reif, J.1    Tate, S.2
  • 44
    • 0019572555 scopus 로고
    • On Uniform Circuit Complexity
    • W. Ruzzo, On Uniform Circuit Complexity, J. Comput. System Sci. 21 (1981) 365-383.
    • (1981) J. Comput. System Sci. , vol.21 , pp. 365-383
    • Ruzzo, W.1
  • 45
    • 84976758731 scopus 로고
    • On the tape complexity of deterministic context-free languages
    • I.H. Sudborough, On the tape complexity of deterministic context-free languages, J. ACM 25 (1978) 405-414.
    • (1978) J. ACM , vol.25 , pp. 405-414
    • Sudborough, I.H.1
  • 46
    • 0026239342 scopus 로고
    • PP is as hard as the polynomial-time hierarchy
    • S. Toda, PP is as hard as the polynomial-time hierarchy SIAM J. Comput. 20 (1991) 865-877.
    • (1991) SIAM J. Comput. , vol.20 , pp. 865-877
    • Toda, S.1
  • 47
    • 0041887590 scopus 로고
    • Classes of arithmetic circuits capturing the complexity of computing the determinant
    • S. Toda, Classes of arithmetic circuits capturing the complexity of computing the determinant, IEICE Trans. Inform. Syst., E75-D (1992) 116-124.
    • (1992) IEICE Trans. Inform. Syst. , vol.E75-D , pp. 116-124
    • Toda, S.1
  • 49
    • 85034715651 scopus 로고
    • Completeness classes in algebra
    • L. Valiant, Completeness classes in algebra, in: Proc. 11th STOC, 1979, pp. 249-261.
    • (1979) Proc. 11th STOC , pp. 249-261
    • Valiant, L.1
  • 51
    • 0003906310 scopus 로고
    • Fast parallel computation of polynomials using few processors
    • L. Valiant, S. Skyum, S. Berkowitz, C. Rackoff, Fast parallel computation of polynomials using few processors, SIAM J. Comput. 12 (1983) 641-644.
    • (1983) SIAM J. Comput. , vol.12 , pp. 641-644
    • Valiant, L.1    Skyum, S.2    Berkowitz, S.3    Rackoff, C.4
  • 52
    • 0022911518 scopus 로고
    • V, NP is as easy as detecting unique solutions
    • L. Valiant, V. Vazirani, V, NP is as easy as detecting unique solutions, Theoret. Comput. Sci. 47 (1986) 85-93.
    • (1986) Theoret. Comput. Sci. , vol.47 , pp. 85-93
    • Valiant, L.1    Vazirani, V.2
  • 53
    • 0026237246 scopus 로고
    • Properties that characterize LOGCFL
    • H. Venkateswaran, Properties that characterize LOGCFL, J. Comput. System. Sci. 42 (1991) 380-404.
    • (1991) J. Comput. System. Sci. , vol.42 , pp. 380-404
    • Venkateswaran, H.1
  • 54
    • 0026902139 scopus 로고
    • Circuit definitions of non-deterministic complexity classes
    • H. Venkateswaran, Circuit definitions of non-deterministic complexity classes, SIAM J. Comput. 21 (1992) 655-670.
    • (1992) SIAM J. Comput. , vol.21 , pp. 655-670
    • Venkateswaran, H.1
  • 55
    • 0024681761 scopus 로고
    • A new pebble game that characterizes parallel complexity classes
    • H. Venkateswaran, M. Tompa, A new pebble game that characterizes parallel complexity classes, SIAM J. Comput. 18 (1989) 533-549.
    • (1989) SIAM J. Comput. , vol.18 , pp. 533-549
    • Venkateswaran, H.1    Tompa, M.2
  • 56
    • 0026375502 scopus 로고
    • Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits
    • V. Vinay, Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits, in: Proc. 6th IEEE Structure in Complexity Theory Conf., 1991, pp. 270-284.
    • (1991) Proc. 6th IEEE Structure in Complexity Theory Conf. , pp. 270-284
    • Vinay, V.1
  • 57
    • 0041887591 scopus 로고
    • Doctoral Dissertation, Indian Institute of Science, Bangalore, July
    • V. Vinay, Semi-unboundedness and complexity classes, Doctoral Dissertation, Indian Institute of Science, Bangalore, July 1991.
    • (1991) Semi-unboundedness and Complexity Classes
    • Vinay, V.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.