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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Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P (August 6, 2002), http://www.cse.iitk.ac.in/primality.pdf
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)
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)
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
Calude, C.S.: Information and Randomness. An Algorithmic Perspective, 2nd edn. Revised and Extended. Springer, Berlin (2002)
Calude, C.S.: Chaitin Ω numbers, Solovay machines and incompleteness. Theoret. Comput. Sci. 284, 269–277 (2002)
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
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)
Calude, C., Jürgensen, H., Zimand, M.: Is independence an exception? Appl. Math. Comput. 66, 63–76 (1994)
Calude, C.S., Pavlov, B.: Coins, quantum measurements, and Turing’s barrier. Quantum Information Processing 1(1-2), 107–127 (2002)
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
Calude, C., Zimand, M.: A relation between correctness and randomness in the computation of probabilistic algorithms. Internat. J. Comput. Math. 16, 47–53 (1984)
Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22, 329–340 (1975)
Chaitin, G.J.: Information–Theoretic Incompleteness. World Scientific, Singapore (1992)
Chaitin, G.J.: Leibniz, Information, Math and Physics, http://www.cs.auckland.ac.nz/CDMTCS/chaitin/kirchberg.html
Chaitin, G.J.: META MATH! The Quest for Omega. Pantheon Books, New York (2005) (to appear)
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)
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)
Davis, M.: The Universal Computer: The Road from Leibniz to Turing, Norton, New York (2000)
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)
Delahaye, J.-P.: L’Intelligence and le Calcul, BELIN. Pour la Science, Paris (2002)
Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2005) (to appear)
Eastlake 3rd, D.E., Crocker, S., Schiller, J.: Randomness Recommendations for Security. RFC 1750, 30 (December 1994)
Etesi, G., Németi, I.: Non-Turing computations via Malament-Hogarth space-times. International Journal of Theoretical Physics 41, 341–370 (2002)
Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21, 467–488 (1982)
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)
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)
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)
Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326–352 (1927)
Kieu, T.D.: Computing the non-computable. Contemporary Physics 44(1), 51–71 (2003)
Kurtz, S.A.: On the random oracle hypothesis. Information and Control 57(1), 40–47 (1983)
Leibniz, G.W.: Discours de métaphysique, Gallimard, Paris (1995)
Matiyasevich, Y.V.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)
Milburn, G.: The Feynman Processor. An Introduction to Quantum Computation, Allen & Unwin, St. Leonards (1998)
von Neumann, J.: Various techniques used in connection with random digits. National Bureau of Standards Applied Mathematics Series 12, 36–38 (1951)
Oliver, D.: Email to C. Calude, (August 20, 2004)
Miller, G.L.: Riemann’s hypothesis and tests of primality. J. Comput. System Sci. 13, 300–317 (1976)
Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Dordrecht (1993)
Peres, Y.: Iterating von Neumann’s procedure for extracting random bits. Ann. Stat. 20, 590–597 (1992)
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)
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/
du Sautoy, M.: The Music of the Primes. HarperCollins, New York (2003)
Sinha, S., Ditto, W.L.: Dynamics based computation. Physical Letters Review 81(10), 2156–2159 (1998)
Sinha, S., Ditto, W.L.: Computing with distributed chaos. Physical Review E 60(1), 363–377 (1999)
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)
Svozil, K.: The quantum coin toss-testing microphysical undecidability. Physics Letters A143, 433–437
Svozil, K.: Randomness & Undecidability in Physics. World Scientific, Singapore (1993)
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
Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55, 601–644 (1983)
Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)
Yurtsever, U.: Quantum mechanics and algorithmic randomness. Complexity 6(1), 27–31 (2002)
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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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