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Informatik pp 165–195Cite as

Von der Turingmaschine zum Quantencomputer — ein Gang durch die Geschichte der Komplexitätstheorie

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Zusammenfassung

Die Komplexitätstheorie beschäftigt sich mit der Abschätzung des Aufwandes, welcher zur Lösung algorithmischer Probleme nötig ist. In diesem Aufsatz verfolgen wir die spannende Entwicklung dieses Teilgebiets der Theoretischen Informatik von ihren Wurzeln in den 30er Jahren des 20. Jahrhunderts bis in die heutige Zeit.

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Literaturverzeichnis

  1. L. Adleman, M. Huang Recognizing primes in random polynomial time. in: Proc. 19th ACM Symposium on Theory of Computing, ACM Press, 1987, S. 462–469

    Google Scholar 

  2. L. Adleman, K. Manders Reducibility, randomness, and intractibility. in: Proc. 9th ACM Symposium on Theory of Computing, ACM Press, 1977, S. 151–163

    Google Scholar 

  3. L.M. Adleman, J. DeMarrais, M.-D.A. Huang Quantum computability. SIAM Journal on Computing, 1997, 26(5):1524–1540

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Agrawal, N. Kayal, N. Saxena PRIMES is in P. Annals of Mathematics, 2004, 160(2):781–793

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Ajtai σ 11 -formulae on finite structures. Annals of Pure and Applied Logic, 1983, 24(1):1–48

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Ajtai The complexity of the pigeonhole-principle. Combinatorica, 1994, 14(4):417–433

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy Proof verification and the hardness of approximation problems. Journal of the ACM, 1998, 45(3):501–555

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Babai Trading group theory for randomness. Proc. 17th ACM Symposium on Theory of Computing, ACM Press, 1985, S. 421–429

    Google Scholar 

  9. L. Babai E-mail and the unexpected power of interaction. Proc. 5th Structure in Complexity Theory Conference, 1990, S. 30–44

    Google Scholar 

  10. L. Babai, L. Fortnow, C. Lund Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1991, 1:1–40

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Baker, J. Gill, R. Solovay Relativizations of the P=?NP question. SIAM Journal on Computing, 1975, 4:431–442

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Ben-Or, S. Goldwasser, J. Kilian, A. Wigderson Multiprover interactive proofs: How to remove intractability assumptions. Proc. 20th ACM Symposium on Theory of Computing, 1988, S. 113–131

    Google Scholar 

  13. E. Ben-Sasson, A. Wigderson Short proofs are narrow — resolution made simple. Journal of the ACM, 2001, 48(2):149–169

    Article  MATH  MathSciNet  Google Scholar 

  14. C.H. Bennett, J. Gill Relative to a random oracle A, P A ≠ NP A ≠ co-NP A with probability 1. SIAM Journal on Computing, 1981, 10:96–113

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Bernstein, U. Vazirani Quantum complexity theory. SIAM Journal on Computing, 1997, 26(5):1411–1473

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Blum, S. Micali How to generate cryptographically strong sequences of pseudorandom bits. SIAM Journal on Computing, 1984, 13(4):850–864

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Boppana, J. Håstad, S. Zachos Does co-NP have short interactive proofs? Information Processing Letters, 1987, 25(2):27–32

    Article  MathSciNet  Google Scholar 

  18. S.R. Buss, A. Kechris, A. Pillay, R.A. Shore The prospects for mathematical logic in the twenty-first century. Bulletin of Symbolic Logic, 2001, 7(2):169–196

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Church A note on the Entscheidungsproblem. The Journal of Symbolic Logic, 1936, 1:345–363

    MathSciNet  Google Scholar 

  20. A. Cobham The intrinsic computational difficulty of functions. Proc. of the 1964 International Congress for Logic, Methodology, and Philosophy of Science, 1964, S. 24–30

    Google Scholar 

  21. S.A. Cook The complexity of theorem proving procedures. Proc. 3rd Annual ACM Symposium on Theory of Computing, 1971, S. 151–158

    Google Scholar 

  22. S.A. Cook A taxonomy of problems with fast parallel algorithms. Information and Control, 1985, 64:2–22

    Article  MATH  MathSciNet  Google Scholar 

  23. S.A. Cook, P. McKenzie Problems complete for deterministic logarithmic space. Journal of Algorithms, 1987, 8:385–394

    Article  MATH  MathSciNet  Google Scholar 

  24. S.A. Cook, R.A. Reckhow The relative efficiency of propositional proof systems. The Journal of Symbolic Logic, 1979, 44:36–50

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Davis, H. Putnam, J. Robertson The decision problem for exponential Diophantine equations. Ann. Math., 1961, 74:425–436

    Article  MATH  Google Scholar 

  26. D. Deutsch Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. of the Royal Society, 1985, 400:97–117

    MATH  MathSciNet  Google Scholar 

  27. D. Deutsch, R. Jozsa Rapid solutions of problems by quantum computations. Proc. of the Royal Society, 1992, 439:553–558

    Article  MATH  MathSciNet  Google Scholar 

  28. J. Edmonds Paths, trees, and flowers. Canadian Journal of Mathematics, 1965, 17(3):449–467

    MATH  MathSciNet  Google Scholar 

  29. R. Feynman Simulating physics with computers. International Journal of Theoretical Physics, 1982, 21:467–488

    MathSciNet  Google Scholar 

  30. L. Fortnow, S. Homer A short history of computational complexity. Bulletin of the EATCS, 2003, 80:95–133

    MathSciNet  MATH  Google Scholar 

  31. L. Fortnow, J. Rogers Complexity limitations on quantum computation. Journal of Computer and System Sciences, 1999, 59(2):240–252

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Fortnow, M. Sipser Are there interactive protocols for co-NP languages. Information Processing Letters, 1988, 28:249–251

    Article  MATH  MathSciNet  Google Scholar 

  33. A.S. Fraenkel, M.R. Garey, D.S. Johnson, T. Schäfer, Y. Yesha The complexity of checkers on an n x n board-preliminary report. Proc. 19th IEEE Symposium on the Foundations of Computer Science, 1978, S. 55–64

    Google Scholar 

  34. A.S. Fraenkel, D. Lichtenstein Computing a perfect strategy for n x n chess requires time exponential in n. J. Comb. Theory, Ser. A, 1981, 31(2):199–214

    Article  MATH  MathSciNet  Google Scholar 

  35. M.L. Furst, J.B. Saxe, M. Sipser Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 1984, 17(1):13–27

    Article  MATH  MathSciNet  Google Scholar 

  36. M. Garey, D. Johnson Computers and Intractability — A Guide to the Theory of NP-Completeness. Freeman and Company, 1979

    Google Scholar 

  37. J. Gill Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 1977, 6:675–695

    Article  MATH  MathSciNet  Google Scholar 

  38. K. Gödel Ein Brief an Johann von Neumann, 20. März, 1956. in: P. Clote, J. Krajíček (Hrsg.) Arithmetic, Proof Theory, and Computational Complexity, Oxford University Press, 1993, S. 7–9

    Google Scholar 

  39. A. Goldberg, M. Sipser Private coins versus public coins in interactive proof systems. in: S. Micali (Hrsg.) Randomness and Computation, Advances in Computing Reasearch, Vol 5, JAI Press, 1989, S. 73–90

    Google Scholar 

  40. O. Goldreich, S. Micali, C. Rackoff The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 1989, 18(2):186–208

    MathSciNet  Google Scholar 

  41. O. Goldreich, S. Micali, A. Wigderson Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 1991, 38:691–729

    Article  MATH  MathSciNet  Google Scholar 

  42. L. Grover A fast quantum mechanical algorithm for database search. Proc. 28th ACM Symposium on Theory of Computing, 1996, S. 212–219

    Google Scholar 

  43. A. Haken The intractability of resolution. Theoretical Computer Science, 1985, 39:297–308

    Article  MATH  MathSciNet  Google Scholar 

  44. J. Hartmanis, R.E. Stearns On the computational complexity of algorithms. Transactions of the American Mathematical Society, 1965, 117:285–306

    Article  MATH  MathSciNet  Google Scholar 

  45. J. Håstad Almost optimal lower bounds for small depth circuits. in: S. Micali (Hrsg.) Randomness and Computation, Advances in Computing Reasearch, Vol 5, JAI Press, 1989, S. 143–170

    Google Scholar 

  46. J. Håstad Some optimal inapproximability results. Journal of the ACM, 2001, 48(4):798–859

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Håstad, R. Impagliazzo, L.A. Levin, M. Luby Construction of a pseudorandom generator from any one-way function. SIAM Journal on Computing, 1999, 28(4):1364–1396

    Article  MATH  MathSciNet  Google Scholar 

  48. F.C. Hennie, R.E. Stearns Two-tape simulation of multitape turing machines. Journal of the ACM, 1966, 13(4):533–546

    Article  MATH  MathSciNet  Google Scholar 

  49. N. Immerman Nondeterministic space is closed under complementation. SIAM Journal on Computing, 1988, 17(5):935–938

    Article  MATH  MathSciNet  Google Scholar 

  50. R. Impagliazzo, A. Wigderson P=BPP unless E has sub-exponential circuits: derandomizing the XOR lemma. Proc. 29th ACM Symposium on Theory of Computing, ACM Press, 1997, S. 220–229

    Google Scholar 

  51. N.D. Jones Space-bounded reducibility among combinatorial problems. Journal of Computer and System Sciences, 1975, 11:68–85

    MATH  MathSciNet  Google Scholar 

  52. R. Karp, A. Wigderson A fast parallel algorithm for the maximal independent set problem. Journal of the ACM, 1985, 32:762–773

    Article  MATH  MathSciNet  Google Scholar 

  53. R.M. Karp Reducibility among combinatorial problems. in R.E. Miller, J.W. Thatcher (Hrsg.) Complexity of Computer Computations, Plenum Press, 1972, S. 85–103

    Google Scholar 

  54. R.M. Karp, R.J. Lipton Some connections between nonuniform and uniform complexity classes. Proc. 12th ACM Symposium on Theory of Computing, ACM Press, 1980, S. 302–309

    Google Scholar 

  55. R.W. Keyes Miniaturization of electronics and its limits. IBM Journal of Research and Development, 1988, 32:24–28

    Google Scholar 

  56. J. Köbler, O. Watanabe New collapse consequences of NP having small circuits. SIAM Journal on Computing, 1998, 28(1):311–324

    MATH  Google Scholar 

  57. J. Krajíček Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. The Journal of Symbolic Logic, 1997, 62(2):457–486

    Article  MathSciNet  MATH  Google Scholar 

  58. J. Krajíček Tautologies from pseudo-random generators. Bulletin of Symbolic Logic, 2001, 7(2):197–212

    Article  MathSciNet  MATH  Google Scholar 

  59. J. Krajíček, P. Pudlák Some consequences of cryptographical conjectures for S 1 2 and EF. Information and Computation, 1998, 140(1):82–94

    Article  MathSciNet  MATH  Google Scholar 

  60. R.E. Ladner On the structure of polynomial-time reducibility. Journal of the ACM, 1975, 22:155–171

    Article  MATH  MathSciNet  Google Scholar 

  61. C. Lautemann BPP and the polynomial hierarchy. Information Processing Letters, 1983, 17:215–217

    Article  MATH  MathSciNet  Google Scholar 

  62. T. Lengauer, K.W. Wagner The binary network flow problem is logspace complete for P. Theoretical Computer Science, 1990, 75(3):357–363

    Article  MATH  MathSciNet  Google Scholar 

  63. L. Levin Universal sequential search problems. Problems of Information Trans-mission, 1975, 9(3):265–266, Englische Übersetzung des 1973 in russischer Spra-che veröffentlichten Originalartikels

    Google Scholar 

  64. D. Lichtenstein, M. Sipser GO is polynomial-space hard. Journal of the ACM, 1980, 27(2):393–401

    Article  MATH  MathSciNet  Google Scholar 

  65. C. Lund, L. Fortnow, H. Karloff, N. Nisan Algebraic methods for interactive proof systems. Journal of the ACM, 1992, 39(4):859–868

    Article  MATH  MathSciNet  Google Scholar 

  66. J.V. Matijasevič Enumerable sets are Diophantine. Dokl. Akad. Nauk, 1970, 191:27–282 (russisch).

    Google Scholar 

  67. A.R. Meyer, L.J. Stockmeyer The equivalence problem for regular expressions with squaring requires exponential space. Proc. 13th IEEE Symposium on Switching and Automata Theory, 1972, S. 125–129

    Google Scholar 

  68. G.L. Miller Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences, 1976, 13:300–317

    MATH  MathSciNet  Google Scholar 

  69. N. Nisan, A. Wigderson Hardness vs randomness. Journal of Computer and System Sciences, 1994, 49(2):149–167

    Article  MATH  MathSciNet  Google Scholar 

  70. C.H. Papadimitriou, M. Yannakakis Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 1991, 43(3):425–440

    Article  MATH  MathSciNet  Google Scholar 

  71. N. Pippenger On simultaneous resource bounds. Proc. 20th IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1979, S. 307–311

    Google Scholar 

  72. M.O. Rabin Probabilistic algorithm for testing primality. Journal of Number Theory, 1980, 12(1):128–138

    Article  MATH  MathSciNet  Google Scholar 

  73. M.O. Rabin, D. Scott Finite automata and their decision problems. IBM Journal of Research and Development, 1959, 3:114–125

    Article  MathSciNet  Google Scholar 

  74. A.A. Razborov Lower bounds on the monotone complexity of boolean functions. Doklady Akademii Nauk SSSR, 1985, 282:1033–1037, Englische Übersetzung in: Soviet Math. Doklady, 31, S. 354–357.

    MathSciNet  Google Scholar 

  75. A.A. Razborov, S. Rudich Natural proofs. Proc. 26th ACM Symposium on Theory of Computing, 1994, S. 204–213

    Google Scholar 

  76. O. Reingold Undirected st-connectivity in log-space. Proc. 37th ACM Sympo-sium on Theory of Computing, ACM Press, 2005, S. 376–385

    Google Scholar 

  77. H.G. Rice Classes of recursively enumerable sets and their decision problems. Trans. Am. Math. Soc., 1953, 74:358–366

    Article  MATH  MathSciNet  Google Scholar 

  78. R.L. Rivest, A. Shamir, L.M. Adleman A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, Feb. 1978, 21(2):120–126

    Article  MATH  MathSciNet  Google Scholar 

  79. D. Rolf Improved bound for the PPSZ/Schöning-algorithm for 3-SAT. Technical Report TR05-159, Electronic Colloquium on Computational Complexity, 2005.

    Google Scholar 

  80. J.E. Savage Computational work and time of finite machines. Journal of the ACM, 1972, 19:660–674

    Article  MATH  MathSciNet  Google Scholar 

  81. W. Savitch Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 1970, 4(2):177–192

    MATH  MathSciNet  Google Scholar 

  82. U. Schöning Complexity and Structure, Band 211 von Lecture Notes in Computer Science. Berlin Heidelberg, Springer-Verlag, 1986.

    MATH  Google Scholar 

  83. U. Schöning Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 1988, 37:312–323

    Article  MATH  MathSciNet  Google Scholar 

  84. A. Shamir IP=PSPACE. Journal of the ACM, 1992, 39(4):869–877

    Article  MATH  MathSciNet  Google Scholar 

  85. C. Shannon Communication theory of secrecy systems. Bell Systems Technical Journal, 1949, 28:657–715

    Google Scholar 

  86. J.C. Sheperdson, H.E. Sturgis Computability of recursive functions. Journal Ass. Comp. Mach., 1963, 10:217–255

    Google Scholar 

  87. P. Shor Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 1997, 26(5):1484–1509

    Article  MATH  MathSciNet  Google Scholar 

  88. M. Sipser A complexity theoretic approach to randomness. Proc. 15th ACM Symposium on Theory of Computing, ACM Press, 1983, S. 330–335

    Google Scholar 

  89. R. Solovay, V. Strassen A fast Monte-Carlo test for primality. SIAM Journal on Computing, 1977, 6:84–85

    Article  MATH  MathSciNet  Google Scholar 

  90. L.J. Stockmeyer Arithmetic versus Boolean operations in idealized register machines. Technical Report RC 5954, Math. Dept., IBM Thomas J. Watson Research Center, 1976.

    Google Scholar 

  91. L.J. Stockmeyer, A.R. Meyer Word problems requiring exponential time. Proc. 5th ACM Symposium on Theory of Computing, 1973, S. 1–9

    Google Scholar 

  92. R. Szelepcsényi The method of forced enumeration for nondeterministic automata. Acta Informatica, 1988, 26(3):279–284

    Article  MATH  MathSciNet  Google Scholar 

  93. S. Toda PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 1991, 20:865–877

    Article  MATH  MathSciNet  Google Scholar 

  94. A.M. Turing On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc., 1936, 42:230–265

    MATH  Google Scholar 

  95. A.C. Yao Theory and applications of trapdoor functions. Proc. 23rd IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1982, S. 80–91

    Google Scholar 

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  1. Humboldt-Universität zu Berlin, Berlin

    Johannes Köbler & Olaf Beyersdorff

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  1. Johannes Köbler
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  2. Olaf Beyersdorff
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  1. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin

    Wolfgang Reisig  & Johann-Christoph Freytag  & 

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Köbler, J., Beyersdorff, O. (2006). Von der Turingmaschine zum Quantencomputer — ein Gang durch die Geschichte der Komplexitätstheorie. In: Reisig, W., Freytag, JC. (eds) Informatik. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-32743-6_8

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