메뉴 건너뛰기




Volumn 6, Issue 1, 1996, Pages 3-26

Complexity and real computation: A manifesto

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0029681225     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127496001818     Document Type: Article
Times cited : (31)

References (115)
  • 8
    • 0011356638 scopus 로고
    • Lectures on a theory of computation and complexity over the reals (or an arbitrary ring)
    • ed., E. Jen (Addison-Wesley)
    • Blum, L. [1990] "Lectures on a theory of computation and complexity over the reals (or an arbitrary ring)," in Lectures in the Sciences of Complexity II, ed., E. Jen (Addison-Wesley), pp. 1-47.
    • (1990) Lectures in the Sciences of Complexity II , vol.2 , pp. 1-47
    • Blum, L.1
  • 9
    • 0011318379 scopus 로고
    • A theory of computation and complexity over the real numbers
    • Springer-Verlag
    • Blum, L. [1991] "A theory of computation and complexity over the real numbers," in Proc. of the Int. Congress of Mathematicians (Springer-Verlag), pp. 1491-1507.
    • (1991) Proc. of the Int. Congress of Mathematicians , pp. 1491-1507
    • Blum, L.1
  • 10
    • 84968516134 scopus 로고
    • On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
    • Blum, L., Shub, M. & Smale, S. [1989] "On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines," Bulletin of the Amer. Math. Soc. 21, 1-46.
    • (1989) Bulletin of the Amer. Math. Soc. , vol.21 , pp. 1-46
    • Blum, L.1    Shub, M.2    Smale, S.3
  • 11
    • 0003389241 scopus 로고
    • The Gödel incompleteness theorem and decidability over a ring
    • eds. Hirsch, M., Marsden, J. & Shub, M. (Springer-Verlag)
    • Blum, L. & Smale, S. [1993] "The Gödel incompleteness theorem and decidability over a ring," in From Topology to Computation: Proc. of the Smalefest, eds. Hirsch, M., Marsden, J. & Shub, M. (Springer-Verlag), pp. 321-339.
    • (1993) From Topology to Computation: Proc. of the Smalefest , pp. 321-339
    • Blum, L.1    Smale, S.2
  • 12
    • 84926396434 scopus 로고
    • A machine-independent theory of the complexity of recursive functions
    • Blum, M. [1967] "A machine-independent theory of the complexity of recursive functions," J. ACM 14, 322-336.
    • (1967) J. ACM , vol.14 , pp. 322-336
    • Blum, M.1
  • 15
    • 0001259424 scopus 로고
    • Bounds for the degrees in the Nullstellensatz
    • Brownawell, W. [1987] "Bounds for the degrees in the Nullstellensatz," Annals Math. 126, 577-591.
    • (1987) Annals Math. , vol.126 , pp. 577-591
    • Brownawell, W.1
  • 16
    • 0001250424 scopus 로고
    • Borne simple exponentielle pour les degrés dans les théorèmes de zéros sur un corps de caractéristique quelconque
    • Caniglia, L., Galligo, A. & Heintz, J. [1988] "Borne simple exponentielle pour les degrés dans les théorèmes de zéros sur un corps de caractéristique quelconque," C. R. Acad. Sci. Paris 307, 255-258.
    • (1988) C. R. Acad. Sci. Paris , vol.307 , pp. 255-258
    • Caniglia, L.1    Galligo, A.2    Heintz, J.3
  • 18
    • 0000954072 scopus 로고
    • An unsolvable problem of elementary number theory
    • Church, A. [1936] "An unsolvable problem of elementary number theory," Amer. J. of Math. 58, 354-363.
    • (1936) Amer. J. of Math. , vol.58 , pp. 354-363
    • Church, A.1
  • 22
    • 0011352652 scopus 로고
    • The arithmetical hierarchy over the reals
    • Cucker, F. [1992a] "The arithmetical hierarchy over the reals," J. of Logic and Computation 2, 375-395.
    • (1992) J. of Logic and Computation , vol.2 , pp. 375-395
    • Cucker, F.1
  • 24
    • 0000466371 scopus 로고
    • On the complexity of quantifier elimination: The structural approach
    • Cucker, F. [1993] "On the complexity of quantifier elimination: The structural approach," The Computer Journal 36, 400-408.
    • (1993) The Computer Journal , vol.36 , pp. 400-408
    • Cucker, F.1
  • 25
    • 0003408347 scopus 로고
    • Cambridge University Press
    • Cutland, N. [1980] Computability (Cambridge University Press).
    • (1980) Computability
    • Cutland, N.1
  • 28
    • 0000927535 scopus 로고
    • Paths, trees, and flowers
    • Edmonds, J. [1965] "Paths, trees, and flowers," Canadian J. of Math. 17, 449-467.
    • (1965) Canadian J. of Math. , vol.17 , pp. 449-467
    • Edmonds, J.1
  • 29
    • 0001281016 scopus 로고
    • Algorithmic properties of structures
    • Engeler, E. [1967] "Algorithmic properties of structures," Maths. Systems Theory 1, 183-195.
    • (1967) Maths. Systems Theory , vol.1 , pp. 183-195
    • Engeler, E.1
  • 32
    • 0002406565 scopus 로고
    • Algorithmic procedures, generalized turing algorithms, and elementary recursion theory
    • eds. Gandy, R. & Yates, C. M. E. (North-Holland)
    • Friedman, H. [1971] "Algorithmic procedures, generalized turing algorithms, and elementary recursion theory," in Logic Colloquium 1969, eds. Gandy, R. & Yates, C. M. E. (North-Holland), pp. 361-390.
    • (1971) Logic Colloquium 1969 , pp. 361-390
    • Friedman, H.1
  • 33
    • 85012630262 scopus 로고
    • Computational complexity of real functions
    • Friedman, H. & Ko, K. [1982] "Computational complexity of real functions," Theoretical Computer Sci. 20, 323-352.
    • (1982) Theoretical Computer Sci. , vol.20 , pp. 323-352
    • Friedman, H.1    Ko, K.2
  • 37
    • 85025493187 scopus 로고
    • Solving systems of polynomial inequalities in subexponential time
    • Grigoriev, D. & Vorobjov, N. [1988] "Solving systems of polynomial inequalities in subexponential time," J. of Symbolic Computation 5, 37-64.
    • (1988) J. of Symbolic Computation , vol.5 , pp. 37-64
    • Grigoriev, D.1    Vorobjov, N.2
  • 39
  • 42
    • 0002984225 scopus 로고
    • Computability over arbitrary fields
    • Herman, G. & Isard, S. [1970] "Computability over arbitrary fields," J. London Math. Soc. 2, 73-79.
    • (1970) J. London Math. Soc. , vol.2 , pp. 73-79
    • Herman, G.1    Isard, S.2
  • 43
    • 34250954644 scopus 로고
    • Die Frage der endlich vielen Schritte in der Theorie der Polynomideale
    • Hermann, G. [1926] "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale," Math. Ann. 95, 736-788.
    • (1926) Math. Ann. , vol.95 , pp. 736-788
    • Hermann, G.1
  • 45
    • 0004211773 scopus 로고
    • Ph.D. Thesis, Department of Computer Science, University of Toronto
    • Hoover, H. [1987] "Feasibly constructive analysis," Ph.D. Thesis, Department of Computer Science, University of Toronto.
    • (1987) Feasibly Constructive Analysis
    • Hoover, H.1
  • 46
    • 0003037529 scopus 로고
    • Reducibility among combinatorial problems
    • eds. Miller, R. & Thatcher, J. (Plenum Press)
    • Karp, R. [1972] "Reducibility among combinatorial problems," in Complexity of Computer Computations, eds. Miller, R. & Thatcher, J. (Plenum Press), pp. 85-103.
    • (1972) Complexity of Computer Computations , pp. 85-103
    • Karp, R.1
  • 48
    • 0001481675 scopus 로고
    • General recursive functions of natural numbers
    • Kleene, S. [1936] "General recursive functions of natural numbers," Math. Annalen 112, 727-742.
    • (1936) Math. Annalen , vol.112 , pp. 727-742
    • Kleene, S.1
  • 51
    • 84968474102 scopus 로고
    • Sharp effective Nullstellensatz
    • Kollár, J. [1988] "Sharp effective Nullstellensatz," J. of Amer. Math. Soc. 1, 963-975.
    • (1988) J. of Amer. Math. Soc. , vol.1 , pp. 963-975
    • Kollár, J.1
  • 52
    • 0020219723 scopus 로고    scopus 로고
    • Complexity theory of real numbers and functions
    • Kreitz, C. & Weihrauch, K. [1982] "Complexity theory of real numbers and functions," in Theoretical Computer Science, eds. Cremers, A. & Kreigel, H. vol. 145, Lecture Notes in Computer Science (Springer-Verlag), pp. 165-174.
    • (1982) Theoretical Computer Science
    • Kreitz, C.1    Weihrauch, K.2
  • 53
    • 0020219723 scopus 로고    scopus 로고
    • Springer-Verlag
    • Kreitz, C. & Weihrauch, K. [1982] "Complexity theory of real numbers and functions," in Theoretical Computer Science, eds. Cremers, A. & Kreigel, H. vol. 145, Lecture Notes in Computer Science (Springer-Verlag), pp. 165-174.
    • Lecture Notes in Computer Science , vol.145 , pp. 165-174
    • Cremers, A.1    Kreigel, H.2
  • 56
    • 0000202647 scopus 로고
    • Universal sequential search problems
    • Levin, L. [1973] "Universal sequential search problems," Probl. Pered. Inform. IX 3, 265-266. (in Russian, English translation in Problems of Information Trans. 9(3); corrected translation in Trakhtenbrot [1984]).
    • (1973) Probl. Pered. Inform. IX , vol.9 , Issue.3 , pp. 265-266
    • Levin, L.1
  • 57
    • 0011278919 scopus 로고
    • corrected translation in Trakhtenbrot
    • Levin, L. [1973] "Universal sequential search problems," Probl. Pered. Inform. IX 3, 265-266. (in Russian, English translation in Problems of Information Trans. 9(3); corrected translation in Trakhtenbrot [1984]).
    • (1984) Problems of Information Trans. , vol.9 , Issue.3
  • 59
    • 38249017395 scopus 로고
    • Computations over ℤ and ℝ: A comparison
    • Meer, K. [1990] "Computations over ℤ and ℝ: A comparison," J. of Complexity 6, 256-263.
    • (1990) J. of Complexity , vol.6 , pp. 256-263
    • Meer, K.1
  • 60
    • 0001074993 scopus 로고
    • A note on a P ≠ NP result for a restricted class of real machines
    • Meer, K. [1992] "A note on a P ≠ NP result for a restricted class of real machines," J. of Complexity 8, 451-453.
    • (1992) J. of Complexity , vol.8 , pp. 451-453
    • Meer, K.1
  • 61
    • 0001304670 scopus 로고
    • Real number models under various sets of operations
    • Meer, K. [1993] "Real number models under various sets of operations," J. of Complexity 9, 366-372.
    • (1993) J. of Complexity , vol.9 , pp. 366-372
    • Meer, K.1
  • 62
    • 0028518386 scopus 로고
    • On the complexity of quadratic programming in real number models of computation
    • Meer, K. [1994] "On the complexity of quadratic programming in real number models of computation," Theoretical Computer Sci. 133, 85-94.
    • (1994) Theoretical Computer Sci. , vol.133 , pp. 85-94
    • Meer, K.1
  • 63
    • 0011284549 scopus 로고
    • Une remarque à propos des machines sur ℝ introduites par Blum, Shub et Smale
    • Série I
    • Michaux, C. [1989] "Une remarque à propos des machines sur ℝ introduites par Blum, Shub et Smale," C. R. Acad. Sci. Paris 309, Série I, 435-437.
    • (1989) C. R. Acad. Sci. Paris , vol.309 , pp. 435-437
    • Michaux, C.1
  • 64
    • 84968504310 scopus 로고
    • Ordered rings over which output sets are recursively enumerable
    • Michaux, C. [1991] "Ordered rings over which output sets are recursively enumerable," Proc. Amer. Math. Soc. 112, 569-575.
    • (1991) Proc. Amer. Math. Soc. , vol.112 , pp. 569-575
    • Michaux, C.1
  • 67
    • 0000238829 scopus 로고
    • Methods of computing values of polynomials
    • Pan, V. [1966] "Methods of computing values of polynomials," Russian Math. Surveys 21, 105-136.
    • (1966) Russian Math. Surveys , vol.21 , pp. 105-136
    • Pan, V.1
  • 68
    • 0023592849 scopus 로고
    • Sequential and parallel complexity of approximate evaluation of polynomial zeros
    • Pan, V. [1987] "Sequential and parallel complexity of approximate evaluation of polynomial zeros," Comput. Math. Appl. 14, 591-622.
    • (1987) Comput. Math. Appl. , vol.14 , pp. 591-622
    • Pan, V.1
  • 69
    • 0002333206 scopus 로고
    • Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros
    • Pan, V. [1995] "Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros," in 27th Annual ACM Symp. on the Theory of Computing, pp. 741-750.
    • (1995) 27th Annual ACM Symp. on the Theory of Computing , pp. 741-750
    • Pan, V.1
  • 74
    • 0000556615 scopus 로고
    • Formal reductions of the general combinatorial decision problem
    • Post, E. [1943] "Formal reductions of the general combinatorial decision problem," Amer. J. Math. 65, 197-268.
    • (1943) Amer. J. Math. , vol.65 , pp. 197-268
    • Post, E.1
  • 75
    • 0000664148 scopus 로고
    • Computability and noncomputability in classical analysis
    • Pour-El, M. & Richards, I. [1983] "Computability and noncomputability in classical analysis," Trans. of the Amer. Math. Soc. 275, 539-560.
    • (1983) Trans. of the Amer. Math. Soc. , vol.275 , pp. 539-560
    • Pour-El, M.1    Richards, I.2
  • 77
    • 84968513637 scopus 로고
    • Computable algebra, general theory and theory of computable fields
    • Rabin, M. [1960a] "Computable algebra, general theory and theory of computable fields," Trans. of the Amer. Math. Soc. 95, 341-360.
    • (1960) Trans. of the Amer. Math. Soc. , vol.95 , pp. 341-360
    • Rabin, M.1
  • 80
    • 0023292476 scopus 로고
    • On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials
    • Renegar, J. [1987] "On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials," Math. of Oper. Research 12, 121-148.
    • (1987) Math. of Oper. Research , vol.12 , pp. 121-148
    • Renegar, J.1
  • 81
    • 85025505234 scopus 로고
    • On the computational complexity and geometry of the first-order theory of the reals. Part I
    • Renegar, J. [1992] "On the computational complexity and geometry of the first-order theory of the reals. Part I," J. of Symbolic Computation 13, 255-299.
    • (1992) J. of Symbolic Computation , vol.13 , pp. 255-299
    • Renegar, J.1
  • 87
    • 0001637995 scopus 로고
    • Computability of recursive functions
    • Shepherdson, J. & Sturgis, H. [1963] "Computability of recursive functions," J. of the ACM 10, 217-255.
    • (1963) J. of the ACM , vol.10 , pp. 217-255
    • Shepherdson, J.1    Sturgis, H.2
  • 88
    • 84968505377 scopus 로고
    • Complexity of Bezout's theorem I: Geometric aspects
    • Shub, M. & Smale, S. [1993a] "Complexity of Bezout's theorem I: Geometric aspects," J. of the Amer. Math. Soc. 6, 459-501.
    • (1993) J. of the Amer. Math. Soc. , vol.6 , pp. 459-501
    • Shub, M.1    Smale, S.2
  • 89
    • 0000537756 scopus 로고
    • Complexity of Bezout's theorem II: Volumes and probabilities
    • Shub, M. & Smale, S. [1993b] "Complexity of Bezout's theorem II: Volumes and probabilities," in Computational Algebraic Geometry, eds. Eyssette, F. & Galligo, A., vol. 109 Progress in Mathematics (Birkhäuser), pp. 267-285.
    • (1993) Computational Algebraic Geometry
    • Shub, M.1    Smale, S.2
  • 90
    • 0011329525 scopus 로고    scopus 로고
    • Birkhäuser
    • Shub, M. & Smale, S. [1993b] "Complexity of Bezout's theorem II: Volumes and probabilities," in Computational Algebraic Geometry, eds. Eyssette, F. & Galligo, A., vol. 109 Progress in Mathematics (Birkhäuser), pp. 267-285.
    • Progress in Mathematics , vol.109 , pp. 267-285
    • Eyssette, F.1    Galligo, A.2
  • 91
    • 0002102930 scopus 로고
    • Complexity of Bezout's theorem III: Condition number and packing
    • Shub, M. & Smale, S. [1993c] "Complexity of Bezout's theorem III: Condition number and packing," J. of Complexity 9, 4-14.
    • (1993) J. of Complexity , vol.9 , pp. 4-14
    • Shub, M.1    Smale, S.2
  • 92
    • 85033305071 scopus 로고
    • Complexity of Bezout's theorem IV: Probability of success, extensions
    • Shub, M. & Smale, S. [1993d] "Complexity of Bezout's theorem IV: Probability of success, extensions," to appear in SIAM J. of Numer. Anal.
    • (1993) SIAM J. of Numer. Anal.
    • Shub, M.1    Smale, S.2
  • 93
    • 0028516294 scopus 로고
    • Complexity of Bezout's theorem V: Polynomial time
    • Shub, M. & Smale, S. [1994] "Complexity of Bezout's theorem V: Polynomial time," Theoretical Computer Sci. 133, 141-164.
    • (1994) Theoretical Computer Sci. , vol.133 , pp. 141-164
    • Shub, M.1    Smale, S.2
  • 95
    • 84967770163 scopus 로고
    • On the efficiency of algorithms of analysis
    • Smale, S. [1985] "On the efficiency of algorithms of analysis," Bulletin of the Amer. Math. Soc. 13, 87-121.
    • (1985) Bulletin of the Amer. Math. Soc. , vol.13 , pp. 87-121
    • Smale, S.1
  • 96
    • 45949121073 scopus 로고
    • On the topology of algorithms I
    • Smale, S. [1987] "On the topology of algorithms I," J. of Complexity 3, 81-89.
    • (1987) J. of Complexity , vol.3 , pp. 81-89
    • Smale, S.1
  • 97
    • 0011280332 scopus 로고
    • The Newtonian contribution to our understanding of the computer
    • ed. Stayer, M. (McGill-Queens University Press)
    • Smale, S. [1988] "The Newtonian contribution to our understanding of the computer," in Newton's Dream, ed. Stayer, M. (McGill-Queens University Press).
    • (1988) Newton's Dream
    • Smale, S.1
  • 98
    • 0025442789 scopus 로고
    • Some remarks on the foundations of numerical analysis
    • Smale, S. [1990] "Some remarks on the foundations of numerical analysis," SIAM Rev. 32, 211-220.
    • (1990) SIAM Rev. , vol.32 , pp. 211-220
    • Smale, S.1
  • 99
    • 0002943521 scopus 로고
    • Lower bounds for algebraic decision trees
    • Steele, J. & Yao, A. [1982] "Lower bounds for algebraic decision trees," J. of Algorithms 3, 1-8.
    • (1982) J. of Algorithms , vol.3 , pp. 1-8
    • Steele, J.1    Yao, A.2
  • 100
    • 34250487811 scopus 로고
    • Gaussian elimination is not optimal
    • Straßen, V. [1969] "Gaussian elimination is not optimal," Numer. Math. 13, 354-356.
    • (1969) Numer. Math. , vol.13 , pp. 354-356
    • Straßen, V.1
  • 101
    • 0001052336 scopus 로고
    • Algebraic complexity theory
    • ed., van Leeuwen, J., MIT Press/Elsevier
    • Straßen, V. [1990] "Algebraic complexity theory," in Handbook of Theoretical Computer Science, ed., van Leeuwen, J., vol. A (MIT Press/Elsevier), pp. 633-672.
    • (1990) Handbook of Theoretical Computer Science , vol.A , pp. 633-672
    • Straßen, V.1
  • 103
    • 78951485210 scopus 로고
    • A survey of Russian approaches to perebor (brute-force search) algorithms
    • Trakhtenbrot, B. [1984] "A survey of Russian approaches to perebor (brute-force search) algorithms," Annals of the History of Computing 6, 384-400.
    • (1984) Annals of the History of Computing , vol.6 , pp. 384-400
    • Trakhtenbrot, B.1
  • 105
    • 0013485373 scopus 로고
    • Computing in algebraic systems
    • eds. Drake, F. & Wainer, S. London Math. Soc. (Cambridge University Press)
    • Tucker, J. [1980] "Computing in algebraic systems," in Recursion Theory, its Generalizations and Applications, eds. Drake, F. & Wainer, S. London Math. Soc. (Cambridge University Press).
    • (1980) Recursion Theory, its Generalizations and Applications
    • Tucker, J.1
  • 106
    • 77956705762 scopus 로고    scopus 로고
    • Examples of semicomputable sets of real and complex numbers
    • Tucker, J. & Zucker, J. [1992] "Examples of semicomputable sets of real and complex numbers," in Constructivity in Computer Science, eds. O'Donnell, M. & Myers Jr, J., vol. 613, Lecture Notes in Computer Science (Springer-Verlag), pp. 179-198.
    • (1992) Constructivity in Computer Science
    • Tucker, J.1    Zucker, J.2
  • 107
    • 77956705762 scopus 로고    scopus 로고
    • Springer-Verlag
    • Tucker, J. & Zucker, J. [1992] "Examples of semicomputable sets of real and complex numbers," in Constructivity in Computer Science, eds. O'Donnell, M. & Myers Jr, J., vol. 613, Lecture Notes in Computer Science (Springer-Verlag), pp. 179-198.
    • Lecture Notes in Computer Science , vol.613 , pp. 179-198
    • O'Donnell, M.1    Myers J., Jr.2
  • 108
    • 84960561455 scopus 로고
    • On computable numbers, with an application to the Entscheidungsproblem
    • Turing, A. [1936] "On computable numbers, with an application to the Entscheidungsproblem," Proc. London Math. Soc. Ser. 2, 42, 230-265.
    • (1936) Proc. London Math. Soc. Ser. 2 , vol.42 , pp. 230-265
    • Turing, A.1
  • 109
    • 0011352655 scopus 로고
    • A survey of the logic of effective definitions
    • Tyurin, J. [1979] "A survey of the logic of effective definitions," in Logic of Programs, ed. Engeler, E., vol. 125, Lecture Notes in Computer Science (Springer-Verlag), pp. 198-245.
    • (1979) Logic of Programs
    • Tyurin, J.1
  • 110
    • 84882451748 scopus 로고    scopus 로고
    • Springer-Verlag
    • Tyurin, J. [1979] "A survey of the logic of effective definitions," in Logic of Programs, ed. Engeler, E., vol. 125, Lecture Notes in Computer Science (Springer-Verlag), pp. 198-245.
    • Lecture Notes in Computer Science , vol.125 , pp. 198-245
    • Engeler, E.1
  • 114
    • 0000860484 scopus 로고
    • On the number of multiplications required to compute certain functions
    • Winograd, S. [1967] "On the number of multiplications required to compute certain functions," Proc. National Acad. Sci. 58, 1840-1842.
    • (1967) Proc. National Acad. Sci. , vol.58 , pp. 1840-1842
    • Winograd, S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.