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Uncomputability and Physical Law

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The Incomputable

Part of the book series: Theory and Applications of Computability ((THEOAPPLCOM))

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Abstract

This paper investigates the role that uncomputability plays in the laws of physics. While uncomputability might seem like an abstract mathematical concept, many questions in physics involve uncomputability. In particular, the form of the energy spectrum of quantum systems capable of universal computation is uncomputable: to answer whether a Hamiltonian system is gapped or gapless in a particular sector requires one to solve the Halting problem. Finally, the problem of determining the most concise expression of physical laws requires one to determine the algorithmic complexity of those laws, and so, the answer to this problem is uncomputable.

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References

  1. A.M. Turing, Proc. Lond. Math. Soc. 242, 230–265, 243, 544–546 (1937)

    Google Scholar 

  2. K. Gödel, Monatsschr. Math. Phys. 38, 173–198 (1931)

    Article  Google Scholar 

  3. M. Cunningham, The History of Computation (AtlantiSoft, New York, 1997)

    Google Scholar 

  4. C. Shannon, A symbolic analysis of relay and switching circuits. M.S. thesis, MIT, 1937

    Google Scholar 

  5. S. Lloyd, Phys. Rev. Lett. 75, 346–349 (1995)

    Article  Google Scholar 

  6. D. Deutsch, A. Barenco, A. Ekert, Proc. R. Soc. Lond. A 8, 669–677 (1995)

    Article  Google Scholar 

  7. F. Barahona, J. Phys. A 15, 3241 (1982)

    Article  MathSciNet  Google Scholar 

  8. S. Lloyd, Science 261, 1569–1571 (1993)

    Article  Google Scholar 

  9. R.P. Feynman, Found. Phys. 16, 507–531 (1986)

    Article  MathSciNet  Google Scholar 

  10. S. Lloyd, Phys. Rev. Lett. 71, 943–946 (1993); J. Mod. Opt. 41, 2503–2520 (1994)

    Google Scholar 

  11. T. Cubitt et al. (in preparation)

    Google Scholar 

  12. R.J. Solomonoff, Inf. Control. 7, 224–254 (1964)

    Article  MathSciNet  Google Scholar 

  13. A.N. Kolmogorov, Probl. Inf. Transm. 1, 3–11 (1965)

    Google Scholar 

  14. G.J. Chaitin, J. Assoc. Comput. Mach. 13, 547–569 (1966)

    Article  Google Scholar 

  15. L. Blum, M. Shub, S. Smale, Bull. Am. Math. Soc. 21, 1–46 (1989)

    Article  Google Scholar 

  16. S. Lloyd, Science 273, 1073–1078 (1996)

    Article  MathSciNet  Google Scholar 

  17. W. Zhang, D. Cory, Phys. Rev. Lett. 80, 1324–1347 (1998)

    Article  Google Scholar 

  18. M.W. Johnson et al., Nature 473, 194–198 (2011)

    Article  Google Scholar 

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Acknowledgements

The author would like to thank Scott Aaronson for helpful discussions.

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Authors and Affiliations

  1. Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Seth Lloyd

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  1. Seth Lloyd
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Correspondence to Seth Lloyd .

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Editors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, United Kingdom

    S. Barry Cooper

  2. Dept. of Mathematical Logic & Applications, Sofia University, Sofia, Bulgaria

    Mariya I. Soskova

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Lloyd, S. (2017). Uncomputability and Physical Law. In: Cooper, S., Soskova, M. (eds) The Incomputable. Theory and Applications of Computability. Springer, Cham. https://doi.org/10.1007/978-3-319-43669-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-43669-2_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43667-8

  • Online ISBN: 978-3-319-43669-2

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