Are Gandy Machines Really Local?

  • Vincenzo Fano
  • Pierluigi Graziani
  • Roberto Macrelli
  • Gino Tarozzi
Chapter
Part of the Synthese Library book series (SYLI, volume 375)

Abstract

This paper discusses the empirical question concerning the physical realization (or implementation) of a computation. We give a precise definition of the realization of a Turing-computable algorithm into a physical situation. This definition is not based, as usual, on an interpretation function of physical states, but on an implementation function from machine states to physical states (as suggested by Piccinini G, Computation in physical systems. The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/fall2012/entries/computation-physicalsystems. Accessed 5 Dec 2013, 2012). We show that our definition avoids difficulties posed by Putnam’s theorem (Putnam H, Representation and reality. MIT Press, Cambridge, 1988) and Kripke’s objections (Stabler EP Jr, Kripke on functionalism and automata. Synthese 70(1):1–22, 1987; Scheutz M, What is not to implement a computation: a critical analysis of Chalmers’ notion of implementation. http://hrilab.tufts.edu/publications/scheutzcogsci12chalmers.pdf. Accessed 5 Dec 2013, 2001). Using our notion of representation, we analyse Gandy machines, intended in a physical sense, as a case study and show an inaccuracy in Gandy’s analysis with respect to the locality notion. This shows the epistemological relevance of our realization concept. We also discuss Gandy machines in quantum context. In fact, it is well known that in quantum mechanics, locality is seriously questioned, therefore it is worthwhile to analyse briefly, whether quantum machines are Gandy machines.

Keywords

Machine State Physical Theory Turing Machine Transitive Closure Rigid Designator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We thank Claudio Calosi, Maurizio Colucci, Paola Gentili, Marco Giunti, Rossella Lupacchini, Wilfried Sieg and Guglielmo Tamburrini for their comments and suggestions. We thank the reviewers who have commented our paper and the scholars attending the IACAP 2014 Conference for their questions.

References

  1. Beggs, E. J., & Tucker, J. V. (2007). Can Newtonian systems, bounded in space, time, mass and energy, compute all functions? Theoretical Computer Science, 371(1), 4–19.CrossRefGoogle Scholar
  2. Calosi, C., & Graziani, P. (2014). Mereology and the sciences. Parts and wholes in the contemporary scientific context (Synthese library, Vol. 371). Heidelberg/New York/Dordrecht/London: Springer.Google Scholar
  3. Carnap, R. (1945). The two concepts of probability. Philosophy and Phenomenological Research, 5(4), 513–532.CrossRefGoogle Scholar
  4. Carnap, R. (1950). Logical foundations of probability. Chicago: Chicago University of Chicago Press.Google Scholar
  5. Chalmers, D. J. (1996). Does a rock implement every finite-state automaton? Synthese, 108, 309–333.CrossRefGoogle Scholar
  6. Chalmers, D. J. (2011). A computational foundation for the study of cognition. Journal of Cognitive Science, 12, 325–359.CrossRefGoogle Scholar
  7. Copeland, B. J. (2015). The Church-Turing thesis. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Stanford: Metaphysics Research Lab. http://plato.stanford.edu/archives/sum2015/entries/church-turing. Accessed 24 Sep 2015.
  8. Copeland, B. J., & Shagrir, O. (2007). Physical computation: How general are Gandy’s principles for mechanism. Mind and Machines, 17(2), 217–231.CrossRefGoogle Scholar
  9. Cotogno, P. (2003). Hypercomputation and the physical Church-Turing Thesis. British Journal for the Philosophy of Science, 54(2), 181–223.CrossRefGoogle Scholar
  10. Cushing, J. T., & McMullin, E. (1989). Philosophical consequences of quantum theory. Notre Dame: University of Notre Dame Press.Google Scholar
  11. Deutsch, D. (1985). Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society A, 400, 97–117.CrossRefGoogle Scholar
  12. Gandy, R. (1980). Church’s thesis and principles for mechanisms. In J. Barwise, H. J. Keisler, & K. Kunen (Eds.), The Kleene symposium (pp. 123–148). Dordrecht: North Holland.CrossRefGoogle Scholar
  13. Gandy, R. (1993). On the impossibility of using analogue machines to calculate non-computable functions. Unpublished.Google Scholar
  14. Giunti, M. (1997). Computation, dynamics and cognition. Oxford: Oxford University Press.Google Scholar
  15. Herken, R. (1995). The Universal Turing machine: A half-century survey. Wien/New York: Springer.CrossRefGoogle Scholar
  16. Horsman, C., Stepney, S., Wagner, R., & Kendon, V. (2014). When does a physical system compute? Proceeding of the Royal Society A, 470(2169), 20140182.Google Scholar
  17. Jarrett, J. P. (1984). On the physical significance of the locality conditions in Bell argument. Noûs, 18, 569–589.CrossRefGoogle Scholar
  18. Kieu, T. D. (2002). Quantum hypercomputation. Minds and Machines, 12(4), 461–502.CrossRefGoogle Scholar
  19. Kleene, S. C. (1952). Introduction to metamathematics. Princeton: Van Nostrand.Google Scholar
  20. Kripke, S. A. (2013). The Church-Turing ‘Thesis’ as a special corollary of Gödel’s completeness theorem. In B. J. Copeland, C. J. Posy, & O. Shagrir (Eds.), Computability: Turing, Gödel, Church, and Beyond (pp. 77–104). Cambridge: MIT Press.Google Scholar
  21. Maudlin, T. (1994). Quantum non-locality and relativity. Oxford: Blackwell.Google Scholar
  22. Mendelson, E. (1990). Second thoughts about Church’s thesis and mathematical proof’s. The Journal of Philosophy, 87(5), 225–233.CrossRefGoogle Scholar
  23. Olszewski, A., Wolenski, J., & Janusz, R. (2007). Church’s thesis after 70 years. Frankfurt: Ontos Verlag.Google Scholar
  24. Piccinini, G. (2012). Computation in physical systems. The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/fall2012/entries/computation-physicalsystems. Accessed 5 Dec 2013.
  25. Putnam, H. (1988). Representation and reality. Cambridge: MIT Press.Google Scholar
  26. Rovelli, C. (2004). Quantum gravity. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  27. Scheutz, M. (1999). When physical systems realize functions…. Minds and Machines, 9(2), 161–196.CrossRefGoogle Scholar
  28. Scheutz, M. (2001). What is not to implement a computation: A critical analysis of Chalmers’ notion of implementation. http://hrilab.tufts.edu/publications/scheutzcogsci12chalmers.pdf. Accessed 5 Dec 2013.
  29. Shagrir, O. (1997). Two dogmas of computationalism. Minds and Machines, 7(3), 321–344.CrossRefGoogle Scholar
  30. Shagrir, O. (2002). Effective computation by humans and machines. Minds and Machines, 12(2), 221–240.CrossRefGoogle Scholar
  31. Shagrir, O., & Pitowsky, I. (2003). Physical hypercomputation and the Church-Turing Thesis. Minds and Machines, 13(1), 87–101.CrossRefGoogle Scholar
  32. Sieg, W. (2002). Calculations by man & machine: Mathematical presentation. In Synthese series, proceedings of the Cracow international congress of logic, methodology and philosophy of science (pp. 245–260). Dordrecht: Kluwer Academic Publishers.Google Scholar
  33. Sieg, W. (2008). Church without dogma: Axioms for computability. In S. B. Cooper, B. Löwe, & A. Sorbi (Eds.), New computational paradigms: Changing conceptions of what is computable (pp. 139–152). Berlin: Springer.CrossRefGoogle Scholar
  34. Sieg, W., & Byrnes, J. (1999). An abstract model for parallel computations. Gandy’s thesis. The Monist, 82(1), 150–164.CrossRefGoogle Scholar
  35. Simons, P. (1987). Parts. Oxford: Clarendon.Google Scholar
  36. Soare, R. I. (1996). Computability and recursion. The Bulletin of Symbolic Logic, 2(3), 284–321.CrossRefGoogle Scholar
  37. Stabler, E. P., Jr. (1987). Kripke on functionalism and automata. Synthese, 70(1), 1–22.CrossRefGoogle Scholar
  38. Syropoulus, A. (2008). Hypercomputation: Computing beyond Church-Turing barrier. New York: Springer.CrossRefGoogle Scholar
  39. Tamburrini, G. (2002). I matematici e le macchine intelligenti. Milano: Bruno Mondadori.Google Scholar
  40. Turing, A. M. (1936). On computable numbers with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42(2), 230–265. A correction in 43(2) (1937), (pp. 544–546). Oxford: Oxford University Press.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vincenzo Fano
    • 1
  • Pierluigi Graziani
    • 2
  • Roberto Macrelli
    • 1
  • Gino Tarozzi
    • 1
  1. 1.Department of Basic Sciences and FoundationsUniversity of UrbinoUrbinoItaly
  2. 2.Department of Philosophical, Pedagogical and Economic-Quantitative SciencesUniversity of Chieti-PescaraChietiItaly

Personalised recommendations