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Algorithmic Randomness, Quantum Physics, and Incompleteness

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Machines, Computations, and Universality (MCU 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3354))

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Abstract

Is randomness in quantum mechanics “algorithmically random”? Is there any relation between Heisenberg’s uncertainty relation and Gödel’s incompleteness? Can quantum randomness be used to trespass the Turing’s barrier? Can complexity shed more light on incompleteness? In this paper we use variants of “algorithmic complexity” to discuss the above questions.

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References

  1. Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P (August 6, 2002), http://www.cse.iitk.ac.in/primality.pdf

  2. Allender, E., Buhrman, H., Koucký, M.: What can be efficiently reduced to the Kolmogorov-random strings? Electronic Colloquium on Computational Complexity, Report 44, 19 (2004)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Berkeland, D.J., Raymondson, D.A., Tassin, V.M.: Tests for non-randomness in quantum jumps, Los Alamos preprint archive (April 2, 2004), http://arxiv.org/abs/physics/0304013

  5. Calude, C.S.: Information and Randomness. An Algorithmic Perspective, 2nd edn. Revised and Extended. Springer, Berlin (2002)

    MATH  Google Scholar 

  6. Calude, C.S.: Chaitin Ω numbers, Solovay machines and incompleteness. Theoret. Comput. Sci. 284, 269–277 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Calude, C.S., Jürgensen, H.: Is Complexity a Source of Incompleteness? CDMTCS Research Report, 241(15) (2004) Los Alamos preprint archive 12 pp (August 11, 2004), http://arxiv.org/abs/math.LO/0408144

  8. Calude, C., Hertling, P., Khoussainov, B.: Do the zeros of Riemann’s zeta–function form a random sequence? Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 62, 199–207 (1997)

    MATH  Google Scholar 

  9. Calude, C., Jürgensen, H., Zimand, M.: Is independence an exception? Appl. Math. Comput. 66, 63–76 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Calude, C.S., Pavlov, B.: Coins, quantum measurements, and Turing’s barrier. Quantum Information Processing 1(1-2), 107–127 (2002)

    Article  MathSciNet  Google Scholar 

  11. Calude, C.S., Stay, M.A.: From Heinsenberg to Gödel via Chaitin. International Journal of Theoretical Physics, accepted. E-print as CDMTCS Research Report 235, 2004, 15 pp. and Los Alamos preprint archive (February 26, 2004), http://arXiv:quant-ph/0402197

    Google Scholar 

  12. Calude, C., Zimand, M.: A relation between correctness and randomness in the computation of probabilistic algorithms. Internat. J. Comput. Math. 16, 47–53 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22, 329–340 (1975)

    MATH  MathSciNet  Google Scholar 

  14. Chaitin, G.J.: Information–Theoretic Incompleteness. World Scientific, Singapore (1992)

    Google Scholar 

  15. Chaitin, G.J.: Leibniz, Information, Math and Physics, http://www.cs.auckland.ac.nz/CDMTCS/chaitin/kirchberg.html

  16. Chaitin, G.J.: META MATH! The Quest for Omega. Pantheon Books, New York (2005) (to appear)

    Google Scholar 

  17. Chaitin, G.J., Schwartz, J.T.: A note on Monte-Carlo primality tests and algorithmic information theory. Comm. Pure Appl. Math. 31, 521–527 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chang, R., Chor, B., Goldreich, O., Hartmanis, J., Hastad, J., Ranjan, D., Rohatgi, P.: The random oracle hypothesis is false. J. Comput. System Sci. 49(1), 24–39 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. http://www.claymath.org/millennium/Riemann_Hypothesis/

  20. Davis, M.: The Universal Computer: The Road from Leibniz to Turing, Norton, New York (2000)

    Google Scholar 

  21. Davis, M.: The myth of hypercomputation. In: Teuscher, C. (ed.) Alan Turing: Life and Legacy of a Great Thinker, pp. 195–211. Springer, Heidelberg (2003)

    Google Scholar 

  22. Delahaye, J.-P.: L’Intelligence and le Calcul, BELIN. Pour la Science, Paris (2002)

    Google Scholar 

  23. Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2005) (to appear)

    Google Scholar 

  24. Eastlake 3rd, D.E., Crocker, S., Schiller, J.: Randomness Recommendations for Security. RFC 1750, 30 (December 1994)

    Google Scholar 

  25. Etesi, G., Németi, I.: Non-Turing computations via Malament-Hogarth space-times. International Journal of Theoretical Physics 41, 341–370 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21, 467–488 (1982)

    Article  MathSciNet  Google Scholar 

  27. http://www.fourmilab.ch/hotbits/

  28. Golenko, D.I.: Generation of uniformly distributed random variables on electronic computers. In: Shreider, Y.A. (ed.) The Monte Carlo Method: The Method Statistical Trials, pp. 257–305. Pergamon Press, Oxford (1966) (translated from Russian by G. J. Tee)

    Google Scholar 

  29. Heisenberg, W.: Über den Anschaulichen Inhalt der Quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43, 172–198 (1927) English translation In Wheeler, J.A., Zurek, H. (eds.): Quantum Theory and Measurement, pp. 62–84. Princeton Univ. Press, Princeton (1983)

    Google Scholar 

  30. http://www.idquantique.com/

  31. http://www.idquantique.com/img/QuantisBoth.jpg

  32. Jennewein, T., Achleitner, U., Weihs, G., Weinfurter, H., Zeilinger, A.: A fast and compact quantum random number generator. Rev. Sci. Instr. 71, 1675–1680 (2000)

    Article  Google Scholar 

  33. Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326–352 (1927)

    Article  Google Scholar 

  34. Kieu, T.D.: Computing the non-computable. Contemporary Physics 44(1), 51–71 (2003)

    Article  Google Scholar 

  35. Kurtz, S.A.: On the random oracle hypothesis. Information and Control 57(1), 40–47 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  36. Leibniz, G.W.: Discours de métaphysique, Gallimard, Paris (1995)

    Google Scholar 

  37. http://www.mathstat.dal.ca/~joerg/pic/g-letter.jpg

  38. Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)

    Google Scholar 

  39. Milburn, G.: The Feynman Processor. An Introduction to Quantum Computation, Allen & Unwin, St. Leonards (1998)

    Google Scholar 

  40. von Neumann, J.: Various techniques used in connection with random digits. National Bureau of Standards Applied Mathematics Series 12, 36–38 (1951)

    Google Scholar 

  41. Oliver, D.: Email to C. Calude, (August 20, 2004)

    Google Scholar 

  42. http://www.randomnumbers.info

  43. Miller, G.L.: Riemann’s hypothesis and tests of primality. J. Comput. System Sci. 13, 300–317 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  44. Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Dordrecht (1993)

    MATH  Google Scholar 

  45. Peres, Y.: Iterating von Neumann’s procedure for extracting random bits. Ann. Stat. 20, 590–597 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  46. Rabin, M.O.: Probabilistic algorithms. In: Traub, J.F. (ed.) Algorithms and Complexity, New Directions and Recent Results, pp. 21–39. Academic Press, New York (1976)

    Google Scholar 

  47. A Million Random Digits with 100,000 Normal Deviates, The RAND Corporation, The Free Press, Glencoe, IL (1955), online edition http://www.rand.org/publications/classics/randomdigits/

  48. du Sautoy, M.: The Music of the Primes. HarperCollins, New York (2003)

    MATH  Google Scholar 

  49. Sinha, S., Ditto, W.L.: Dynamics based computation. Physical Letters Review 81(10), 2156–2159 (1998)

    Article  Google Scholar 

  50. Sinha, S., Ditto, W.L.: Computing with distributed chaos. Physical Review E 60(1), 363–377 (1999)

    Article  Google Scholar 

  51. Solovay, R.M.: A version of Ω for which ZFC can not predict a single bit. In: Calude, C.S., Păun, G. (eds.) Finite Versus Infinite. Contributions to an Eternal Dilemma, pp. 323–334. Springer, London (2000)

    Google Scholar 

  52. Svozil, K.: The quantum coin toss-testing microphysical undecidability. Physics Letters A143, 433–437

    Google Scholar 

  53. Svozil, K.: Randomness & Undecidability in Physics. World Scientific, Singapore (1993)

    MATH  Google Scholar 

  54. Tadaki, K.: Upper bound by Kolmogorov complexity for the probability in computable POVM measurement, Los Alamos preprint archive (December 11, 2002), http://www.arXiv:quantph/0212071

  55. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55, 601–644 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)

    Google Scholar 

  57. Yurtsever, U.: Quantum mechanics and algorithmic randomness. Complexity 6(1), 27–31 (2002)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

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  1. Cristian S. Calude
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Editors and Affiliations

  1. IUT de Metz, LITA, Université Paul Verlaine - Metz, EA 3097, UFR MIM, Île du Saulcy, 57045, Metz, Cédex, France

    Maurice Margenstern

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Calude, C.S. (2005). Algorithmic Randomness, Quantum Physics, and Incompleteness. In: Margenstern, M. (eds) Machines, Computations, and Universality. MCU 2004. Lecture Notes in Computer Science, vol 3354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31834-7_1

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  • DOI: https://doi.org/10.1007/978-3-540-31834-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25261-0

  • Online ISBN: 978-3-540-31834-7

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