Skip to main content
SpringerLink
Log in

Theoretical Limits

  • Chapter
  • First Online:
Limits of AI - theoretical, practical, ethical

Part of the book series: Technik im Fokus ((TECHNIK))

  • 79 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Nevertheless, the idea underlying the sieve of Eratosthenes has an algorithmic added value, see the example below in the discussion of cryptographic protocols.

  2. 2.

    By now, we know algorithms for a direct calculation of the digit in hexadecimal or binary expansion [3, §1.2]. But this still does not give a regularity in the usual sense.

  3. 3.

    The following presentation of chaos theory follows the book: K. Mainzer (2016), Information, Algorithmus, Probability, Complexity, Quantum World, Life, Brain, Society, Berlin University Press [26, 67 ff.].

References

  1. Kahle, R. (2021), Primzahlen als Herausforderung, in: R. Reussner, A. Koziolek, and R. Heinrich (Eds.), INFORMATIK 2020, Lecture Notes in Informatics, pages 719–727. Gesellschaft für Informatik.

    Google Scholar 

  2. Hoche, R. (Ed.) (1866), Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II. Teubner.

    Google Scholar 

  3. Berggren, L.; Jonathan Borwein, J.; Peter Borwein (2004), A Pamphlet on Pi, in: L. Berggren, J. Borwein, P. Borwein (Eds.), Pi: A Source Book. 3rd edition, Springer, 721–739.

    Chapter  Google Scholar 

  4. Euler, L. (1771), Extrait d’un lettre de M. Euler le pere à M. Bernoulli concernant leM’emoire imprimé parmi ceux de 1771, 318. Nouveaux Mémoires de l’Académie royale des Sciences. Berlin, 1774, 35–36, 1772.

    Google Scholar 

  5. Weyl, H. (1971). Über den Symbolismus der Mathematik und mathematischen Physik, in: K. Reidemeister (ed.) Hilbert, 20–38. Springer.

    Chapter  Google Scholar 

  6. Homeister, M. (2018), Quantum Computing verstehen, Springer: Berlin 5. Aufl., 195–196.

    Google Scholar 

  7. Pomerance, C. (1982), Analysis and comparison of some integer factoring algorithms, in: Computational Methods in Number Theory, Part I, H.W. Lenstra, Jr. and R. Tijdeman, eds., Math. Centre Tract 154, Amsterdam, 89–139

    Google Scholar 

  8. R. Crandall, C. Pomerance (2001), Prime Numbers: A Computational Perspective. Springer, New York.

    Book  Google Scholar 

  9. Lenstra, A.K.; Lenstra (1993), H.W., The Development of the Number Field Sieve, Lecture Notes in Mathematics V, 1554.

    Google Scholar 

  10. Werner, A. (2002), Elliptische Kurven in der Kryptographie, Springer, Berlin.

    Book  Google Scholar 

  11. The case for Elliptic Curve Cryptography: https://www.nsa.gov/business/programs/elliptic_curve.shtml (abgerufen 06.052020).

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

    Article  MathSciNet  Google Scholar 

  13. Ekert, A.; Jozsa, R. (1996) Quantum computation and Shor’s factoring algorithm, in: Rev. mod. Phys. 68.

    Google Scholar 

  14. Benenti, G.; Casati, G.; Strini, G. (2008), Principles of Quantum Computation and Information. Vol. I: Basic Concepts, World Scientific Singapore, 161–162.

    Google Scholar 

  15. Quisquater, J.-J.; Guillou,L. (1990), How to explain zero-knowledge protocols to your children, in: Advances in Cryptology – CRYPTO ’89, Lecture Notes in computer Science 435, 628–631.

    Google Scholar 

  16. Köbler, J.; Beyersdorff, O. (2006), Von der Turingmaschine zum Quantencomputer – ein Gang durch die Geschichte der Komplexitätstheorie, in: W. Reisig, J.-C. Freytag (HRSG.), Informatik. Aktuelle Themen im historischen Kontext, Springer: Berlin.

    Google Scholar 

  17. Babai, L. (1985), Trading group theory for randomness, in: Proc. 17th ACM Symposium on Theory of Computing, ACM Press, 421–429.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Goldberg, A.; Sipser, M. (1989), Private coins versus public coins in interactive proof systems, in: S. Micli (Hrsg.), Randomness and Computation, Advances in Computing Research 5, JAI Press, 73–90.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Feige, U.; Goldwasser, S.; Lovasz, L.; Safra, S.; Szegedy, M. (1996), Interactive proofs and the hardness of approximating cliques, in: Journal of the ACM 43, 268–292.

    Article  MathSciNet  Google Scholar 

  22. Arora, S.; Safra, S. (1998), Probabilistic checking of proof: A new characterization of NP, in: Journal of ACM 45(1), 70–122.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Nissan, N.; Widgerson, A. (1994), Hardness vs. randomness, in: Journal of Computer and System Sciences 49(2), 149–167

    Google Scholar 

  25. Impagliazzo, R.; Widgerson, A. (1997), P=BPP unless E has sub-exponential circuits: derandomizing the XOR lemma, in: Proc. 29th ACM Symposium on Theory of Computing, ACM Press, 220–229.

    Google Scholar 

  26. Mainzer K (2016) Information: Algorithmus-Wahrscheinlichkeit-Komplexit¨at- Quantenwelt-Leben-Gehirn-Gesellschaft. Berlin.

    Google Scholar 

  27. Dershowitz, N. (2005). The four sons of Penrose. In G. Sutcliffe and A. Voronkov (Eds.), Proceedings of the Eleventh Conference on Logic Programming for Artificial Intelligence and Reasoning (LPAR) (Montego Bay, Jamaica), Volume 3835 of Lecture Notes in Artificial Intelligence, pp. 125–138. Springer.

    Google Scholar 

  28. Mainzer, K. (2018), The Digital and the Real World. Computational Foundations of Mathematics, Science, Technology, and Philosophy, World Scientific Singapore.

    Google Scholar 

  29. Hidary, J.D. (2019), Quantum Computing: An Applied Approach, Springer: Cham, 20–21.

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Adleman, L.; Huang, M. (1987), Recognizing primes in random polynomial time, in: Proc. 19th ACM Symposium on theory of computing, ACM Press, 462–469.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Mainzer, K. (2020), Quantencomputer. Von er Quantenwelt zur Künstlichen Intelligenz, Springer: Berlin.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TUM Senior Excellence Faculty, Technische Universität München, München, Germany

    Klaus Mainzer

  2. Carl Friedrich von Weizsäcker-Zentrum, Universität Tübingen, Tübingen, Germany

    Reinhard Kahle

Authors
  1. Klaus Mainzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhard Kahle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Mainzer .

Rights and permissions

Reprints and permissions

Copyright information

© 2024 Springer-Verlag GmbH Germany, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mainzer, K., Kahle, R. (2024). Theoretical Limits. In: Limits of AI - theoretical, practical, ethical . Technik im Fokus. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-68290-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-68290-6_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-68289-0

  • Online ISBN: 978-3-662-68290-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation