Abstract
We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.”
This work was carried out in partial fulfillment of the requirements for the Ph.D. degree of the first author.
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References
Bowen, J.P.: Glossary of Z notation. Information and Software Technology 37, 333–334 (1995), Available at: http://staff.washington.edu/~jon/z/glossary.html
Minsky, M.L.: Matter, mind and models. Proc. International Federation of Information Processing Congress 1, 45–49 (1965), Available at: http://web.media.mit.edu/~minsky/papers/MatterMindModels.html
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998)
Bournez, O., Cucker, F., de Naurois, P.J., Marion, J.Y.: Computability over an arbitrary structure. sequential and parallel polynomial time. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 185–199. Springer, Heidelberg (2003)
Tucker, J.V., Zucker, J.I.: Abstract versus concrete computation on metric partial algebras. ACM Transactions on Computational Logic 5, 611–668 (2004)
Weihrauch, K.: Computable Analysis — An introduction. Springer, Berlin (2000)
Mycka, J., Costa, J.F.: Real recursive functions and their hierarchy. Journal of Complexity (2004) (in print)
Campagnolo, M.L., Moore, C., Costa, J.F.: Iteration, inequalities, and differentiability in analog computers. Journal of Complexity 16, 642–660 (2000)
Church, A.: An unsolvable problem of elementary number theory. American Journal of Mathematics 58, 345–363 (1936)
Kleene, S.C.: Lambda-definability and recursiveness. Duke Mathematical Journal 2, 340–353 (1936)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42, 230–265 (1936–1937)
Engeler, E.: Formal Languages: Automata and Structures. Lectures in Advanced Mathematics. Markham Publishing Company, Chicago (1968)
Hennie, F.: Introduction to Computability. Addison-Wesley, Reading (1977)
Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1966)
Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser, Boston (1998)
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Boker, U., Dershowitz, N. (2005). How to Compare the Power of Computational Models. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_7
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DOI: https://doi.org/10.1007/11494645_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26179-7
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