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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Literaturverzeichnis
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
L. Adleman, K. Manders Reducibility, randomness, and intractibility. in: Proc. 9th ACM Symposium on Theory of Computing, ACM Press, 1977, S. 151–163
L.M. Adleman, J. DeMarrais, M.-D.A. Huang Quantum computability. SIAM Journal on Computing, 1997, 26(5):1524–1540
M. Agrawal, N. Kayal, N. Saxena PRIMES is in P. Annals of Mathematics, 2004, 160(2):781–793
M. Ajtai σ 11 -formulae on finite structures. Annals of Pure and Applied Logic, 1983, 24(1):1–48
M. Ajtai The complexity of the pigeonhole-principle. Combinatorica, 1994, 14(4):417–433
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
L. Babai Trading group theory for randomness. Proc. 17th ACM Symposium on Theory of Computing, ACM Press, 1985, S. 421–429
L. Babai E-mail and the unexpected power of interaction. Proc. 5th Structure in Complexity Theory Conference, 1990, S. 30–44
L. Babai, L. Fortnow, C. Lund Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1991, 1:1–40
T. Baker, J. Gill, R. Solovay Relativizations of the P=?NP question. SIAM Journal on Computing, 1975, 4:431–442
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
E. Ben-Sasson, A. Wigderson Short proofs are narrow — resolution made simple. Journal of the ACM, 2001, 48(2):149–169
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
E. Bernstein, U. Vazirani Quantum complexity theory. SIAM Journal on Computing, 1997, 26(5):1411–1473
M. Blum, S. Micali How to generate cryptographically strong sequences of pseudorandom bits. SIAM Journal on Computing, 1984, 13(4):850–864
R. Boppana, J. Håstad, S. Zachos Does co-NP have short interactive proofs? Information Processing Letters, 1987, 25(2):27–32
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
A. Church A note on the Entscheidungsproblem. The Journal of Symbolic Logic, 1936, 1:345–363
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
S.A. Cook The complexity of theorem proving procedures. Proc. 3rd Annual ACM Symposium on Theory of Computing, 1971, S. 151–158
S.A. Cook A taxonomy of problems with fast parallel algorithms. Information and Control, 1985, 64:2–22
S.A. Cook, P. McKenzie Problems complete for deterministic logarithmic space. Journal of Algorithms, 1987, 8:385–394
S.A. Cook, R.A. Reckhow The relative efficiency of propositional proof systems. The Journal of Symbolic Logic, 1979, 44:36–50
M. Davis, H. Putnam, J. Robertson The decision problem for exponential Diophantine equations. Ann. Math., 1961, 74:425–436
D. Deutsch Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. of the Royal Society, 1985, 400:97–117
D. Deutsch, R. Jozsa Rapid solutions of problems by quantum computations. Proc. of the Royal Society, 1992, 439:553–558
J. Edmonds Paths, trees, and flowers. Canadian Journal of Mathematics, 1965, 17(3):449–467
R. Feynman Simulating physics with computers. International Journal of Theoretical Physics, 1982, 21:467–488
L. Fortnow, S. Homer A short history of computational complexity. Bulletin of the EATCS, 2003, 80:95–133
L. Fortnow, J. Rogers Complexity limitations on quantum computation. Journal of Computer and System Sciences, 1999, 59(2):240–252
L. Fortnow, M. Sipser Are there interactive protocols for co-NP languages. Information Processing Letters, 1988, 28:249–251
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
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
M.L. Furst, J.B. Saxe, M. Sipser Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 1984, 17(1):13–27
M. Garey, D. Johnson Computers and Intractability — A Guide to the Theory of NP-Completeness. Freeman and Company, 1979
J. Gill Computational complexity of probabilistic complexity classes. SIAM Journal on Computing, 1977, 6:675–695
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
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
O. Goldreich, S. Micali, C. Rackoff The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 1989, 18(2):186–208
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
L. Grover A fast quantum mechanical algorithm for database search. Proc. 28th ACM Symposium on Theory of Computing, 1996, S. 212–219
A. Haken The intractability of resolution. Theoretical Computer Science, 1985, 39:297–308
J. Hartmanis, R.E. Stearns On the computational complexity of algorithms. Transactions of the American Mathematical Society, 1965, 117:285–306
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
J. Håstad Some optimal inapproximability results. Journal of the ACM, 2001, 48(4):798–859
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
F.C. Hennie, R.E. Stearns Two-tape simulation of multitape turing machines. Journal of the ACM, 1966, 13(4):533–546
N. Immerman Nondeterministic space is closed under complementation. SIAM Journal on Computing, 1988, 17(5):935–938
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
N.D. Jones Space-bounded reducibility among combinatorial problems. Journal of Computer and System Sciences, 1975, 11:68–85
R. Karp, A. Wigderson A fast parallel algorithm for the maximal independent set problem. Journal of the ACM, 1985, 32:762–773
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
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
R.W. Keyes Miniaturization of electronics and its limits. IBM Journal of Research and Development, 1988, 32:24–28
J. Köbler, O. Watanabe New collapse consequences of NP having small circuits. SIAM Journal on Computing, 1998, 28(1):311–324
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
J. Krajíček Tautologies from pseudo-random generators. Bulletin of Symbolic Logic, 2001, 7(2):197–212
J. Krajíček, P. Pudlák Some consequences of cryptographical conjectures for S 1 2 and EF. Information and Computation, 1998, 140(1):82–94
R.E. Ladner On the structure of polynomial-time reducibility. Journal of the ACM, 1975, 22:155–171
C. Lautemann BPP and the polynomial hierarchy. Information Processing Letters, 1983, 17:215–217
T. Lengauer, K.W. Wagner The binary network flow problem is logspace complete for P. Theoretical Computer Science, 1990, 75(3):357–363
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
D. Lichtenstein, M. Sipser GO is polynomial-space hard. Journal of the ACM, 1980, 27(2):393–401
C. Lund, L. Fortnow, H. Karloff, N. Nisan Algebraic methods for interactive proof systems. Journal of the ACM, 1992, 39(4):859–868
J.V. Matijasevič Enumerable sets are Diophantine. Dokl. Akad. Nauk, 1970, 191:27–282 (russisch).
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
G.L. Miller Riemann’s hypothesis and tests for primality. Journal of Computer and System Sciences, 1976, 13:300–317
N. Nisan, A. Wigderson Hardness vs randomness. Journal of Computer and System Sciences, 1994, 49(2):149–167
C.H. Papadimitriou, M. Yannakakis Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 1991, 43(3):425–440
N. Pippenger On simultaneous resource bounds. Proc. 20th IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society Press, 1979, S. 307–311
M.O. Rabin Probabilistic algorithm for testing primality. Journal of Number Theory, 1980, 12(1):128–138
M.O. Rabin, D. Scott Finite automata and their decision problems. IBM Journal of Research and Development, 1959, 3:114–125
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.
A.A. Razborov, S. Rudich Natural proofs. Proc. 26th ACM Symposium on Theory of Computing, 1994, S. 204–213
O. Reingold Undirected st-connectivity in log-space. Proc. 37th ACM Sympo-sium on Theory of Computing, ACM Press, 2005, S. 376–385
H.G. Rice Classes of recursively enumerable sets and their decision problems. Trans. Am. Math. Soc., 1953, 74:358–366
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
D. Rolf Improved bound for the PPSZ/Schöning-algorithm for 3-SAT. Technical Report TR05-159, Electronic Colloquium on Computational Complexity, 2005.
J.E. Savage Computational work and time of finite machines. Journal of the ACM, 1972, 19:660–674
W. Savitch Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 1970, 4(2):177–192
U. Schöning Complexity and Structure, Band 211 von Lecture Notes in Computer Science. Berlin Heidelberg, Springer-Verlag, 1986.
U. Schöning Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 1988, 37:312–323
A. Shamir IP=PSPACE. Journal of the ACM, 1992, 39(4):869–877
C. Shannon Communication theory of secrecy systems. Bell Systems Technical Journal, 1949, 28:657–715
J.C. Sheperdson, H.E. Sturgis Computability of recursive functions. Journal Ass. Comp. Mach., 1963, 10:217–255
P. Shor Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 1997, 26(5):1484–1509
M. Sipser A complexity theoretic approach to randomness. Proc. 15th ACM Symposium on Theory of Computing, ACM Press, 1983, S. 330–335
R. Solovay, V. Strassen A fast Monte-Carlo test for primality. SIAM Journal on Computing, 1977, 6:84–85
L.J. Stockmeyer Arithmetic versus Boolean operations in idealized register machines. Technical Report RC 5954, Math. Dept., IBM Thomas J. Watson Research Center, 1976.
L.J. Stockmeyer, A.R. Meyer Word problems requiring exponential time. Proc. 5th ACM Symposium on Theory of Computing, 1973, S. 1–9
R. Szelepcsényi The method of forced enumeration for nondeterministic automata. Acta Informatica, 1988, 26(3):279–284
S. Toda PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 1991, 20:865–877
A.M. Turing On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc., 1936, 42:230–265
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
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/3-540-32743-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32742-4
Online ISBN: 978-3-540-32743-1
eBook Packages: Computer Science and Engineering (German Language)