Abstract
In January 2005 I showed that universality in computation cannot be achieved. Specifically, I exhibited a number of distinct counterexamples, each of which sufficing to demonstrate that no finite and fixed computer can be universal in the sense of being able to simulate successfully any computation that can be executed on any other computer. The number and diversity of the counterexamples attest to the general nature of the nonuniversality result. This not only put to rest the “Church–Turing Thesis”, by proving it to be a false conjecture, but also was seen to apply, in addition to the Turing Machine , to all computational models, past, present, and future, conventional and unconventional. While ten years have now passed since nonuniversality in computation was established, the result remains largely misunderstood. There appear to be at least two main reasons for this state of affairs. As often happens to new ideas, the nonuniversality result was confronted with a stubborn entrenchment in a preconceived, deeply held, and quasi-religious belief in the existence of a universal computer. This was exacerbated by a failure to read the literature that demonstrates why such a belief is unfounded. Behavior of this sort, sadly not uncommon in science, explains the enduring mirage of the universal computer. The purpose of this chapter is to rectify the most notorious misconceptions associated with nonuniversality in computation. These misconceptions were expressed to the author in personal communications, orally, by email, and in referee reports. Each misconception is quoted verbatim and a detailed response to it is provided. The chapter concludes by inviting the reader to take a computational challenge.
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References
Abramsky, S., et al.: Handbook of Logic in Computer Science. Clarendon Press, Oxford (1992)
Akl, S.G.: Parallel Computation: Models and Methods. Prentice Hall, Upper Saddle River (1997)
Akl, S.G.: Superlinear performance in real-time parallel computation. J. Supercomput. 29, 89–111 (2004)
Akl, S.G.: The myth of universal computation. In: Trobec, R., Zinterhof, P., Vajteršic, M., Uhl, A. (eds.) Parallel Numerics, pp. 211–236. University of Salzburg, Salzburg and Jozef Stefan Institute, Ljubljana (2005)
Akl, S.G.: Three counterexamples to dispel the myth of the universal computer. Parallel Proc. Lett. 16, 381–403 (2006)
Akl, S.G.: Conventional or unconventional: is any computer universal? In: Adamatzky, A., Teuscher, C. (eds.) From Utopian to Genuine Unconventional Computers, pp. 101–136. Luniver Press, Frome (2006)
Akl, S.G.: Gödel’s incompleteness theorem and nonuniversality in computing. In: Nagy, M., Nagy, N. (eds.) Proceedings of the Workshop on Unconventional Computational Problems, Sixth International Conference on Unconventional Computation, pp. 1–23. Kingston (2007)
Akl, S.G.: Even accelerating machines are not universal. Int. J. Unconv. Comp. 3, 105–121 (2007)
Akl, S.G.: Unconventional computational problems with consequences to universality. Int. J. Unconv. Comp. 4, 89–98 (2008)
Akl, S.G.: Evolving computational systems. In: Rajasekaran, S., Reif, J.H. (eds.) Parallel Computing: Models, Algorithms, and Applications, pp. 1–22. Taylor and Francis, Boca Raton (2008)
Akl, S.G.: Ubiquity and simultaneity: the science and philosophy of space and time in unconventional computation. Keynote address, Conference on the Science and Philosophy of Unconventional Computing, The University of Cambridge, Cambridge (2009)
Akl, S.G.: Time travel: a new hypercomputational paradigm. Int. J. Unconv. Comp. 6, 329–351 (2010)
Akl, S.G.: What is computation? Int. J. Parallel, Emerg. Distrib. Syst. 29, 337–345 (2014)
Akl, S.G.: Nonuniversality explained. Int. J. Parallel Emerg. Distrib. Syst. (To appear in)
Akl, S.G., Nagy, M.: Introduction to parallel computation. In: Trobec, R., Vajteršic, M., Zinterhof, P. (eds.) Parallel Computing: Numerics, Applications, and Trends, pp. 43–80. Springer, London (2009)
Akl, S.G.: Non-Universality in Computation: The Myth of the Universal Computer. School of Computing, Queen’s University. http://research.cs.queensu.ca/Parallel/projects.html
Akl, S.G.: A computational challenge. School of Computing, Queen’s University. http://www.cs.queensu.ca/home/akl/CHALLENGE/A_Computational_Challenge.htm
Akl, S.G.: Universality in computation: Some quotes of interest. Technical Report No. 2006-511, School of Computing, Queen’s University. http://www.cs.queensu.ca/home/akl/techreports/quotes.pdf
Akl, S.G., Nagy: M.: The future of parallel computation. In: Trobec, R., Vajteršic, M., Zinterhof, P. (eds.) Parallel Computing: Numerics, Applications, and Trends, pp. 471-510. Springer, London (2009)
Akl, S.G., Salay, N.: On computable numbers, nonuniversality, and the genuine power of parallelism. Int. J. Unconv. Comput. 11, 283–297 (2015)
Akl, S.G., Yao, W.: Parallel computation and measurement uncertainty in nonlinear dynamical systems. J. Math. Model. Alg. 4, 5–15 (2005)
Davies, E.B.: Building infinite machines. Br. J. Phil. Sci. 52, 671–682 (2001)
Davis, M.: The Universal Computer. W.W. Norton, New York (2000)
Denning, P.J., Dennis, J.B., Qualitz, J.E.: Machines, Languages, and Computation. Prentice-Hall, Englewood Cliffs (1978)
Deutsch, D.: The Fabric of Reality. Penguin Books, London (1997)
Durand-Lose, J.: Abstract geometrical computation for black hole computation. Research Report No. 2004-15, Laboratoire de l’Informatique du Parallélisme, École Normale Supérieure de Lyon, Lyon (2004)
Einstein, A.: Letter to Erwin Schrödinger. In: Gilder, L., The Age of Entanglement, p. 170. Vintage Books, New York (2009)
Fortnow, L.: The enduring legacy of the Turing machine. http://ubiquity.acm.org/article.cfm?id=1921573
Fraser, R., Akl, S.G.: Accelerating machines: a review. Int. J. Parallel, Emerg. Distrib. Syst. 23, 81–104 (2008)
Gleick, J.: The Information: A History, a Theory, a Flood. HarperCollins, London (2011)
Gould, S.J.: Evolution as fact and theory. Discover 2, 34–37 (1981)
Harel, D.: Algorithmics: The Spirit of Computing. Addison-Wesley, Boston (Reading (1992))
Hillis, D.: The Pattern on the Stone. Basic Books, New York (1998)
Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974)
Hopcroft, J.E., Ullman, J.D.: Formal Languages and their Relations to Automata. Addison-Wesley, Boston, Reading (1969)
Hypercomputation. http://en.wikipedia.org/wiki/Hypercomputation
James, W.: Pragmatism, A New Name for Some Old Ways of Thinking. Longman Green and Co., New York (1907)
Kelly, K.: God is the machine. Wired 10 (2002)
Kleene, S.C.: Introduction to Metamathematics. North Holland, Amsterdam (1952)
Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation. Prentice Hall, Englewood Cliffs (1981)
Lloyd, S., Ng, Y.J.: Black Hole Comput. Sci. Am. 291, 53–61 (2004)
Lloyd, S.: Programming the Universe. Knopf, New York (2006)
Mandrioli, D., Ghezzi, C.: Theoretical Foundations of Computer Science. Wiley, New York (1987)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)
Nagy, M., Akl, S.G.: On the importance of parallelism for quantum computation and the concept of a universal computer. In: Calude, C.S., Dinneen, M.J., Paun, G., Pérez-Jiménez, M. de J., Rozenberg, G. (eds.) Unconventional Computation, pp. 176-190. Springer, Heildelberg (2005)
Nagy, M., Akl, S.G.: Quantum measurements and universal computation. Int. J. Unconv. Comput. 2, 73–88 (2006)
Nagy, M., Akl, S.G.: Quantum computing: Beyond the limits of conventional computation. Int. J. Parallel Emerg. Distrib. Syst. 22, 123–135 (2007)
Nagy, M., Akl, S.G.: Parallelism in quantum information processing defeats the Universal Computer. Par. Proc. Lett. 17, 233–262 (2007)
Nagy, N., Akl, S.G.: Computations with uncertain time constraints: effects on parallelism and universality. In: Calude, C.S., Kari, J., Petre, I., Rozenberg, G. (eds.) Unconventional Computation, pp. 152–163. Springer, Heidelberg (2011)
Nagy, N., Akl, S.G.: Computing with uncertainty and its implications to universality. Int. J. Parallel Emerg. Distrib. Syst. 27, 169–192 (2012)
Prusinkiewicz, P., Lindenmayer, A.: The Algorithmic Beauty of Plants. Springer, New York (1990)
Rucker, R.: The Lifebox, the Seashell, and the Soul. Thunder’s Mouth Press, New York (2005)
Savage, J.E.: Models of Computation. Addison-Wesley, Boston, Reading (1998)
Seife, C.: Decoding the Universe. Viking Penguin, New York (2006)
Siegfried, T.: The Bit and the Pendulum. Wiley, New York (2000)
Sipser, M.: Introduction to the Theory of Computation. PWS Publishing Company, Boston (1997)
Stepney, S.: Journeys in non-classical computation. In: Hoare, T., Milner, R. (eds.) Grand Challenges in Computing Research, pp. 29–32. BCS, Swindon (2004)
Stepney, S.: The neglected pillar of material computation. Phys. D 237, 1157–1164 (2004)
Tipler, F.J.: The Physics of Immortality: Modern Cosmology. God and the Resurrection of the Dead. Macmillan, London (1995)
Toffoli, T.: Physics and Computation. Int. J. Theor. Phys. 21, 165–175 (1982)
Turing, A.M.: Systems of logic based on ordinals. Proc. Lond. Math. Soc. 2(45), 161–228 (1939)
Vedral, V.: Decoding Reality. Oxford University Press, Oxford (2010)
Wegner, P., Goldin, D.: Computation beyond Turing Machines. Commun. ACM 46, 100–102 (1997)
Wheeler, J.A.: Information, physics, quanta: The search for links. In: Proceedings of the Third International Symposium on Foundations of Quantum Mechanics in Light of New Technology, pp. 354-368. Tokyo (1989)
Wheeler, J.A.: Information, physics, quantum: the search for links. In: Zurek, W. (ed.) Complexity, Entropy, and the Physics of Information. Addison-Wesley, Redwood City (1990)
Wheeler, J.A.: At Home in the Universe. American Institute of Physics Press, Woodbury (1994)
Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign (2002)
Zuse, K.: Calculating space. MIT Technical Translation AZT-70-164-GEMIT, Massachusetts Institute of Technology (Project MAC). Cambridge (1970)
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Akl, S.G. (2017). Nonuniversality in Computation: Fifteen Misconceptions Rectified. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_1
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