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Physical Computation and First-Order Logic

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Machines, Computations, and Universality (MCU 2018)

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Abstract

We introduce a computational formalism that is deployable within an arbitrary logical system. This formalism is intended to capture computation on an arbitrary system, both physical and unphysical, including quantum computers, Blum-Shub-Smale machines, and infinite time Turing machines. We demonstrate that for finite problems, the computational power of any device describable via a finite first-order theory is equivalent to that of a Turing machine. Whereas for infinite problems, their computational power is equivalent to that of a type-2 machine.

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Notes

  1. 1.

    N.B. despite the similar name, theory machines are not related to the concept of a “logic theory machine” found in [19].

  2. 2.

    This is the only second-order sentence in the theory of \(\mathcal {N}\), and as we shall see in Example 4.2, it is unnecessary for describing a Turing machine computation.

  3. 3.

    Unlike BSS machines, if such a device requires only finite precision to be implemented correctly then the second order least upper bound axiom in RA is not required.

References

  1. Apelian, C., Surace, S.: Real and Complex Analysis. CRC Press, Boca Raton (2009)

    Book  Google Scholar 

  2. Baumeler, Ä., Wolf, S.: Computational tameness of classical non-causal models. Proc. R. Soc. A 474(2209) (2018). https://doi.org/10.1098/rspa.2017.0698

    Article  MathSciNet  Google Scholar 

  3. Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic. Graph. Darst, vol. 53. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  4. Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: \(NP\)-completeness, recursive functions and universal machines. Bull. Am. Math. Soc. 21, 1–46 (1989). https://doi.org/10.1090/S0273-0979-1989-15750-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Bournez, O., Dershowitz, N., Néron, P.: Axiomatizing analog algorithms. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds.) CiE 2016. LNCS, vol. 9709, pp. 215–224. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40189-8_22

    Chapter  Google Scholar 

  6. Barry Cooper, S.: Computability Theory. CRC Press, Boca Raton (2003)

    MATH  Google Scholar 

  7. Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. A 400(1818), 97–117 (1985). https://doi.org/10.1098/rspa.1985.0070

    Article  MathSciNet  MATH  Google Scholar 

  8. Trillas, E., Eciolaza, L.: Fuzzy Logic. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14203-6

    Book  Google Scholar 

  9. Epstein, R.L.: Classical Mathematical Logic (2011)

    Google Scholar 

  10. Grigorieff, S., Valarcher, P.: Evolving MultiAlgebras unify all usual sequential computation models. Leibniz Int. Proc. Inform. (2010). https://doi.org/10.4230/LIPIcs.STACS.2010.2473

  11. Gurevich, Y.: Sequential abstract-state machines capture sequential algorithms. ACM Trans. Comput. Log. 77–111 (2000). https://doi.org/10.1145/343369.343384

    Article  MathSciNet  Google Scholar 

  12. Hamkins, J.D., Lewis, A.: Infinite time Turing machines. J. Symb. Logic 65, 567–604 (2000). https://doi.org/10.2307/2586556

    Article  MathSciNet  MATH  Google Scholar 

  13. Henkin, L.: The completeness of the first-order functional calculus. J. Symb. Logic 14(3) (1949). https://doi.org/10.2307/2267044

  14. Horsman, C., Stepney, S., Wagner, R.C., Kendon, V.: When does a physical system compute? Proc. R. Soc. A 470(2169) (2014). https://doi.org/10.1098/rspa.2014.0182

    Article  Google Scholar 

  15. Kaye, R.: Models of Peano arithmetic (1991)

    Google Scholar 

  16. Kitaev, A.Y., Shen, A., Vyalyi, M.: Classical and Quantum Computation. American Mathematical Society, Boston (2002)

    Book  Google Scholar 

  17. Koepke, P., Koerwien, M.: Ordinal computations. Math. Struct. Comput. Sci. 16, 867–884 (2006). https://doi.org/10.1017/S0960129506005615

    Article  MathSciNet  MATH  Google Scholar 

  18. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Course of Theoretical Physics. Pergamon, Oxford (1987)

    MATH  Google Scholar 

  19. Newell, A., Simon, H.: The logic theory machine-a complex information processing system. IRE Trans. Info. Theor. 2, 61–79 (1956)

    Article  Google Scholar 

  20. Sierpiński, W.: Cardinal and ordinal numbers. Państwowe Wydawn. Naukowe, vol. 34 (1958)

    Google Scholar 

  21. Smullyan, R.M.: Theory of Formal Systems. Princeton University Press, Princeton (1961)

    MATH  Google Scholar 

  22. Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 2, 230–265 (1937)

    Article  MathSciNet  Google Scholar 

  23. Vakil, N.: Real Analysis Through Modern Infinitesimals. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  24. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)

    Book  Google Scholar 

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Authors and Affiliations

  1. The University of Leeds, Leeds, West Yorkshire, LS2 9JT, UK

    Richard Whyman

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  1. Richard Whyman
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Correspondence to Richard Whyman .

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Editors and Affiliations

  1. Université d'Orléans, Orleans, France

    Jérôme Durand-Lose

  2. Université Paris Est, Creteil, France

    Sergey Verlan

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Whyman, R. (2018). Physical Computation and First-Order Logic. In: Durand-Lose, J., Verlan, S. (eds) Machines, Computations, and Universality. MCU 2018. Lecture Notes in Computer Science(), vol 10881. Springer, Cham. https://doi.org/10.1007/978-3-319-92402-1_9

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  • DOI: https://doi.org/10.1007/978-3-319-92402-1_9

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