Advertisement

Fundamental Limits to Computing

  • Rajendra K. BeraEmail author
Chapter
  • 91 Downloads
Part of the Undergraduate Lecture Notes in Physics book series (ULNP)

Abstract

This chapter introduces certain fundamental limits that mathematics, thermodynamics, information theory, and computational complexity impose on algorithm development. Topics include Hilbert’s second and tenth problem, Turing’s halting problem, resolution of Maxwell’s demon paradox, classification of computational complexity, and a brief discussion on NP-complete problems. The aim is to provide an understanding of the deep issues involved in the development of quantum algorithms and the hurdles that lie ahead.

References

  1. 1.
    S. Arora, B. Barak, Computational Complexity: A Modern Approach (Cambridge University Press, 2009)Google Scholar
  2. 2.
    P. Bachmann, Analytische Zahlentheorie (Analytic Number Theory), pt. 2 (Leipzig: B. G. Teubner, 1894)Google Scholar
  3. 3.
    C.H. Bennett, Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973). https://www.math.ucsd.edu/~sbuss/CourseWeb/Math268_2013W/Bennett_Reversibiity.pdf
  4. 4.
    C.H. Bennett, The thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 (1982). https://www.cc.gatech.edu/computing/nano/documents/Bennett%20-%20The%20Thermodynamics%20Of%20Computation.pdf
  5. 5.
    C.H. Bennett, Time/space trade-offs for reversible computation. SIAM J. Comput. 18, 766–776 (1989)Google Scholar
  6. 6.
    S.L. Braunstein, Quantum Computation (A tutorial paper.) (1995). http://www-users.cs.york.ac.uk/~schmuel/comp/comp_best.pdf
  7. 7.
    A. Church, An unsolvable problem of elementary number theory. Am. J. Math. 58(2), 345–363 (1936). https://www.ics.uci.edu/~lopes/teaching/inf212W12/readings/church.pdf
  8. 8.
    S.A. Cook, The complexity of theorem-proving procedures, in Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (Association of Computing Machinery, New York, 1971), pp. 151–158Google Scholar
  9. 9.
    D. Coppersmith, S. Winograd, Matrix multiplication via arithmetic progressions, in Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pp. 1–6 (1987)Google Scholar
  10. 10.
    T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd edn. (MIT Press, Cambridge, MA, 2009)Google Scholar
  11. 11.
    D. Deutsch, Quantum Theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond.; Ser. A, Math. Phys. Sci. 400(1818), 97–117 (1985). http://www.ceid.upatras.gr/tech_news/papers/quantum_theory.pdf
  12. 12.
    J. Edmonds, Paths, trees, and flowers. Canad. J. Math. 17, 449–467 (1965)Google Scholar
  13. 13.
    R.P. Feynman, The Feynman Lectures on Computation (Westview, 1999)Google Scholar
  14. 14.
    J.H. Gallier, Logic for Computer Science: Foundations of Automatic Theorem Proving, 2nd edn. (Dover Publications, 2015). http://phil.gu.se/logic/books/Gallier:Logic_For_Computer_Science.pdf
  15. 15.
    M.R. Garey, D.S. Johnson, Computers and Intractability (Freeman, 1979)Google Scholar
  16. 16.
    K. Gödel, Über formal unentseheid-bare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198. (Leipzig: 1931). http://www.ddc.net/ygg/etext/godel/, (English translation: On formally undecidable propositions of Principia Mathematica and related systems, I. http://www.cs.colorado.edu/~hirzel/papers/canon00-goedel.pdf The theorem appears as Proposition VI of the paper. Part II of the paper was never published.)
  17. 17.
    D. Hilbert, Mathematical problems, in Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900. http://aleph0.clarku.edu/~djoyce/hilbert/problems.html Dr. Maby Winton Newson translated this address into English with the author’s permission for Bull. Am. Math. Soc. 8, 437–479 (1902). A reprint of appears in Mathematical Developments Arising from Hilbert Problems, ed. by Felix Brouder, American Mathematical Society, 1976. The original address “Mathematische Probleme” appeared in Göttinger Nachrichten, 1900, pp. 253–297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44–63 and 213–237. [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]
  18. 18.
    R.M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (Proceedings of a Symposium IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1972) (Plenum, New York, 1972), pp. 85–103Google Scholar
  19. 19.
    R.W. Keyes R. Landauer, Minimal energy dissipation in logic. IBM J. Res. Dev. 14, 152–157 (1970)Google Scholar
  20. 20.
    D. Knuth, The art of computer programming, in Fundamental Algorithms, vol. 1, 3rd edn. (Addison-Wesley, Reading, MA, 1997)Google Scholar
  21. 21.
    E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2 vols. (Leipzig: B. G. Teubner, 1909)Google Scholar
  22. 22.
    R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183 (1961). Reprinted in IBM Journal of Research and Development, 44(1/2) January/March 2000. https://www.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/thermo_computing_docs/Landauer_1961.pdf
  23. 23.
    L. Levin, Universal sorting problems, Probl. Peredaci Inf. 9:115–116 (1973). Original in Russian. English translation in Probl. Inf. Transm. USSR 9:265–266 (1973)Google Scholar
  24. 24.
    I.L. Markov, Limits on fundamental limits to computation. arXiv:1408.3821v2 [cs.ET] 8 Jan 2015. https://arxiv.org/pdf/1408.3821.pdf Also as: Nature 512, 147–154 (14 August 2014).  https://doi.org/10.1038/nature13570
  25. 25.
    Y. Matiyasevich, Enumerable sets are diophantine. Dokl. Akad. Nauk SSSR 191, 279–282 (1970); English translation with addendum, Soviet Math. Doklady 11, 354–357 (1970)Google Scholar
  26. 26.
    Y. Matiyasevich, Hilbert’s Tenth Problem (MIT Press, 1993)Google Scholar
  27. 27.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000). [For errata: http://www.squint.org/qci/]
  28. 28.
    R. Penrose, Shadows of the Mind (Oxford University Press, 1994) (Vintage paperback)Google Scholar
  29. 29.
    M.B. Plenio V. Vitelli, The physics of forgetting: Landauer’s erasure principle and information theory. Contemp. Phys. 42, 25–60 (2001). https://arxiv.org/pdf/quant-ph/0103108.pdf
  30. 30.
    K.R. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge (Routledge, London, 1963)Google Scholar
  31. 31.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, “Is matrix inversion an N3 process?” §2.11, in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, England, 1989)Google Scholar
  32. 32.
    B. Russell, History of Western Philosophy (Simon and Schuster, 2008)Google Scholar
  33. 33.
    V. Strassen, Gaussian elimination is not optimal. Numerische Mathematik 13, 354–356 (1969)Google Scholar
  34. 34.
    L. Szilard, D. Über, Entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen. Zeitschrift für Physik. 1929, 53, 840–856. (In German) Szilard, L. Z. Physik (1929) 53: 840.  https://doi.org/10.1007/BF01341281 Translation by A. Rapoport and M. Knoller “On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings” reprinted in Harvey S. Leff and Andrew F. Rex, Maxwell’s Demon: Entropy, Information, Computing (Princeton: Princeton University Press, 1990); Second edition: Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics Publishing, 2003; pp. 124–133 (1st edition) and pp. 110–119 (2nd edition)
  35. 35.
    A. Turing, On computable numbers, with an application to the Entscheidungsproblem, in Proceedings of the London Mathematical Society, Series 2, vol. 42, pp. 230–265. http://www.turingarchive.org/viewer/?id=466&title=01bb, and at https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf. (Errata (1937): Vol. 43, pp. 544–546. http://www.abelard.org/turpap2/tp2-ie.asp

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Acadinnet Education Services IndiaBangaloreIndia

Personalised recommendations