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Practical Intractability: A Critique of the Hypercomputation Movement

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Abstract

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable.

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Notes

  1. In places where clarification may be necessary, I will use the equivalent term “Turing-computable”. When referring to the non-Turing computable, I will simply use the standard term “non-computable”.

  2. Copeland (1998, pp. 130–131) provides an excellent list of papers concerned that present these speculations prior to the coining of the term “hypercomputation” by Copeland and Proudfoot (1999).

  3. Though one can never exclude the possibility that a groundbreaking new physical theory could provide for the physical implementation of the infinite; however, so far no such theory is even closely there.

  4. It was first called this by Church in his 1937 review (Church 1937) of Turing’s paper.

  5. I am only interested in the consideration of real machines here, though Gandy’s Thesis leaves open to interpretation the possible consideration of both notional and real machines (cf. Shagrir 2002, §4).

  6. In fact, given Turing’s description of partially random machines, it seems that the means by which the machine is to make its choices in order to appear it has a random element makes the restriction that p be a computable real number rather reasonable.

  7. “Modest” with respect to the bold physical CT, which states that any physical process is (Turing-) computable.

  8. There have been several recent improvements to the exponent on the theoretical runtime of matrix multiplication [by Stothers (2010) and Williams (2011)], which is conjectured to optimally be O(n 2) for the multiplication of two n × n matrices. But unlike Strassen’s algorithm (1969), these algorithms are not used for matrices of practical sizes (n < 1020, Nayebi 2012).

  9. In fact, the type of computation that usually involves actors and distributed process models does not relate to single algorithmic computations, but can be viewed as functional compositions of such algorithms, where despite the latency of their result (namely, the second property of finite delay), these interactions are oracles, no different from regular function calls. Latency is a rather natural property of abstract descriptions of concurrent systems; finite but unbounded delay gives rise to unbounded nondeterminism. Although the Actor model exhibits unbounded nondeterminism (due to the underlying property of fairness), it must be made clear that this does not mean that the Actor model can physically compute any functions outside the class of recursive functions.

References

  • Adleman, L. (1994). Molecular computation of solutions to combinatorial problems. Science, 266, 1021–1024.

    Google Scholar 

  • Agha, G. A. (1986). Actors: A model of concurrent computation. Doctoral dissertation. Cambridge: MIT Press.

    Google Scholar 

  • Barwise, J., Keisler, H. J., & Kunen, K. (Eds.). (1980). The Kleene symposium. Amsterdam: North-Holland.

    MATH  Google Scholar 

  • Blake, R. M. (1926). The paradox of temporal process. Journal of Philosophy, 23, 645–654.

    Article  Google Scholar 

  • Calude, C. S., & Păun, G. (2004). Bio-steps beyond Turing. Biosystems, 77, 175–194.

    Article  Google Scholar 

  • Church, A. (1936). An unsolvable problem of elementary number theory. American Journal of Mathematics, 58, 345–363.

    Article  MathSciNet  Google Scholar 

  • Church, A. (1937). Review of Turing 1936–1937. Journal of Symbolic Logic, 2, 42–43.

    Google Scholar 

  • Church, A. (1940). On the concept of a random sequence. Bulletin of the American Mathematical Society, 46, 130–135.

    Article  MathSciNet  Google Scholar 

  • Clinger, W. (1981). Foundations of actor semantics. Doctoral dissertation. Cambridge: MIT Press.

    Google Scholar 

  • Copeland, B. J. (1998). Turing’s o-machines, Searle, Penrose and the brain. Analysis, 58, 128–138.

    Article  MATH  MathSciNet  Google Scholar 

  • Copeland, B. J., & Proudfoot, D. (1999). Alan Turing’s forgotten ideas in computer science. Scientific American, 253, 98–103.

    Article  Google Scholar 

  • Copeland, B. J. (2002). Accelerating Turing machines. Minds and Machines, 12, 281–300.

    Article  MATH  Google Scholar 

  • Copeland, B. J. (2002). Hypercomputation. Minds and Machines, 12, 461–502.

    Article  MATH  Google Scholar 

  • Copeland, B. J., & Shagrir, O. (2007). Physical computation: How general are Gandy’s principles for mechanisms? Minds and Machines, 17, 217–231.

    Article  Google Scholar 

  • da Costa, N. C. A., & Doria, F. A. (1991). Classical physics and Penrose’s thesis. Foundations of Physics Letters, 4, 363–374.

    Article  MathSciNet  Google Scholar 

  • Davies, E. B. (2001). Building infinite machines. British Journal for the Philosophy of Science, 52, 671–682.

    Article  MATH  MathSciNet  Google Scholar 

  • Davis, M. (2004). The myth of hypercomputation”. In Teuscher et al. (2004), pp. 195–211.

  • Davis, M. (2006a). The Church–Turing thesis: Consensus and opposition. Lecture Notes in Computer Science, 3988, 125–132.

    Article  Google Scholar 

  • Davis, M. (2006b). Why there is no such discipline as hypercomputation. Applied Mathematics and Computation, 178, 4–7.

    Article  MATH  MathSciNet  Google Scholar 

  • de Leeuw, K., Moore, E. F., Shannon C. E., & Shapiro N. (1956). Computability by probabilistic machines. In McCarthy and Shannon (1956), pp. 183–212.

  • Deutsch, D. (1985). Quantum theory, the Church–Turing principle, and the universal quantum computer. Proceedings of the Royal Society, Series A, 400, 97–117.

    Article  MATH  MathSciNet  Google Scholar 

  • Eberbach, E., & Wegner, P. (2003). Beyond Turing machines. Bulletin of the European Association for Theoretical Computer Science, 81, 279–304.

    MATH  MathSciNet  Google Scholar 

  • Eberbach, E., Goldin, D. Q., & Wegner, P. (2004). Turing’s ideas and models of computation. In Teuscher et al. (2004), pp. 159–194.

  • Echeverría, J., Ibarra, A., & Mormann, T. (Eds.). (1992). The space of mathematics: Philosophical, epistemological, and historical explorations. Berlin: Walter de Gruyter.

    MATH  Google Scholar 

  • Feferman, S. (1992). Turing’s ‘oracle’: From absolute to relative computability—And back. In Echeverría et al. (1992), pp. 314–348.

  • Gandy, R. O. (1980). Church’s thesis and principles for mechanisms”. In Barwise et al. (1980), pp. 123–148.

  • Gandy, R. O. (1993). On the impossibility of using analogue machines to calculate non-computable functions (unpublished).

  • Gold, E. M. (1965). Limiting recursion. Journal of Symbolic Logic 30, 28–46.

    Article  MATH  MathSciNet  Google Scholar 

  • Gurevich, Y. (2011). What is an algorithm? Technical Report MSR-TR-2011-116. http://research.microsoft.com/pubs/155608/209-3.pdf.

  • Goldin, D, Q. & Wegner, P. (2003). Computation beyond Turing machines. Communications of the ACM, 46, 100–102.

    Article  Google Scholar 

  • Hagar, A., & Korolev, A. (2007). Quantum hypercomputation—Hype or computation?. Philosophy of Science, 74, 347–363.

    Article  MathSciNet  Google Scholar 

  • Hewitt, C., Bishop, P., & Steiger, R. (1973). A universal modular Actor formalism for artificial intelligence. In Proceedings of the 3rd international joint conference on Artificial intelligence (pp. 235–245).

  • Hopcroft, J. E., Ullman, J. D. (1979). Introduction to automata theory, languages, and computation. Reading, MA: Addison-Wesley.

  • Ince, D. C. (Ed). (1992). Collected works of A. M. Turing: mechanical intelligence. Amsterdam: Elsevier Science.

  • Israel, D. (2002). Reflections on Gödel’s and Gandy’s reflections on Turing’s thesis. Minds and Machines, 12, 181–201.

    Article  MATH  Google Scholar 

  • Kelly, K. T. (1996). The logic of reliable inquiry. New York: Oxford University Press.

    MATH  Google Scholar 

  • Kieu, T. D. (2002). Quantum hypercomputation. Minds and Machines, 12, 541–561.

    Article  MATH  Google Scholar 

  • Kieu, T. D. (2003). Quantum algorithm for Hilbert’s tenth problem. International Journal of Theoretical Physics, 42, 1461–1478.

    Article  MATH  MathSciNet  Google Scholar 

  • Kieu, T. D. (2003). Quantum adiabatic algorithm for Hilbert’s tenth problem: I. The algorithm. http://arxiv.org/abs/quant-ph/0310052.

  • Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: North-Holland.

    MATH  Google Scholar 

  • Kleene, S. C. (1967). Mathematical logic. New York: Wiley.

    MATH  Google Scholar 

  • Kreisel, G. (1974). A notion of mechanistic theory. Synthese, 29, 11–26.

    Article  MATH  Google Scholar 

  • Kreisel, G. (1982). Review of Pour-El and Richards. Journal of Symbolic Logic, 47, 900–902.

    Article  Google Scholar 

  • Losseva, E. (2003). Building reliable systems from unreliable components and the DNA computing paradigm. Bulletin of the European Association for Theoretical Computer Science, 81, 305–317.

    MATH  MathSciNet  Google Scholar 

  • Löwe, B., Sorbi, A., & Cooper, S. B. (Eds.). (2007). New computational paradigms: Changing conceptions of what is computable. Heidelberg: Springer.

    Google Scholar 

  • McCarthy, J., & Shannon, C. E. (Eds.). (1956). Automata studies. Princeton, NJ: Princeton University Press.

    MATH  Google Scholar 

  • Mok, W. (2009). How Twitter is scaling. https://waimingmok.wordpress.com/2009/06/27/how-twitter-is-scaling/.

  • Nayebi, A. (2012). Fast matrix multiplication techniques based on the Adleman–Lipton model. International Journal of Computer Engineering Research, 3, 10–19. Also available at: http://arxiv.org/abs/0912.0750, December 2011.

  • Networked cyber-physical systems at SRI. http://ncps.csl.sri.com/.

  • Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Odersky, M., Spoon, L., & Venners, B. (2008). Scala: A scalable language. http://www.artima.com/scalazine/articles/scalable-language.html.

  • Ord, T. (2002). Hypercomputation: Computing more than the Turing machine. Honours thesis. The University of Melbourne. http://arxiv.org/abs/math/0209332.

  • OSL (2012). Open systems laboratory: Research. http://osl.cs.uiuc.edu/research.php.

  • Păun, G. (2000). Computing with membranes. Journal of Computer and System Sciences, 61, 108–143.

    Article  MATH  MathSciNet  Google Scholar 

  • Păun, G. (2001). P systems with active membranes: Attacking NP-complete problems. Journal of Automata, Languages and Combinatorics, 6, 75–90.

    MATH  MathSciNet  Google Scholar 

  • Păun, G. (2003). Membrane computing. In Fundamentals of computation theory (Vol. 2751, pp. 177–220). Lecture Notes in Computer Science.

  • Piccinini, G. (2011). The physical Church–Turing thesis: Modest or bold? British Journal for the Philosophy of Science, 62, 733–769.

    Article  MATH  MathSciNet  Google Scholar 

  • Post, E. L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50, 284–316.

    Article  MATH  MathSciNet  Google Scholar 

  • Pour-El, M. B., & Richards, J. I. (1979). A computable ordinary differential equation which possesses no computable solution. Annals of Mathematical Logic, 17, 61–90.

    Article  MATH  MathSciNet  Google Scholar 

  • Pour-El, M. B., & Richards, J. I. (1989). Computability in analysis and physics. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Putnam, H. (1965). Trial and error predicates and the solution to a problem of Mostowski. Journal of Symbolic Logic, 30, 49–57.

    Article  MATH  MathSciNet  Google Scholar 

  • Russell, B. A. W. (1936). The limits of empiricism. Proceedings of the Aristotelian Society, 36, 131–150.

    Google Scholar 

  • Shagrir, O. (2002). Effective computation by humans and machines. Minds and Machines, 12, 221–240.

    Article  MATH  Google Scholar 

  • Shagrir, O., & Pitowsky, I. (2003). Physical hypercomputation and the Church–Turing thesis. Minds and Machines, 13, 87–101.

    Article  MATH  Google Scholar 

  • Shagrir, O. (2004). Super-tasks, accelerating Turing machines and uncomputability. Theoretical Computer Science, 317, 105–114.

    Article  MATH  MathSciNet  Google Scholar 

  • Shagrir, O. (2012). Supertasks do not increase computational power. Natural Computing, 11, 51–58.

    Article  MATH  MathSciNet  Google Scholar 

  • Shor, P. W. (1997). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26, 1484–1509.

    Article  MATH  MathSciNet  Google Scholar 

  • Sieg, W. (2002a). Calculations by man and machines: Conceptual analysis. In Sieg et al. (2002), pp. 396–415.

  • Sieg, W. (2002b). Calculations by man and machine: Mathematical presentation. In Proceedings of the Cracow international congress of logic, methodology and philosophy of science. Synthese series (pp. 245–260). Dordrecht: Kluwer.

  • Sieg, W. (2007). Church without dogma—Axioms for computability. In Löwe et al. (2007), pp. 18–44.

  • Sieg, W., Sommer, R., & Talcott, C. (Eds.). (2002). Reflections on the foundations of mathematics. Association for Symbolic Logic.

  • Sieg, W., & Byrnes, J. (1999). An abstract model for parallel computations: Gandy’s thesis. The Monist, 82, 150–164.

    Article  Google Scholar 

  • Soare, R. I. (1996). Computability and recursion. Bulletin of Symbolic Logic, 2, 284–321.

    Article  MATH  MathSciNet  Google Scholar 

  • Soare, R. I. (2009). Turing oracle machines, online computing, and three displacements in computability theory. Annals of Pure and Applied Logic, 160, 368–399.

    Article  MATH  MathSciNet  Google Scholar 

  • Stannett, M. (2006). The case for hypercomputation. Applied Mathematics and Computation, 178, 8–24.

    Article  MATH  MathSciNet  Google Scholar 

  • Stehr, M.-O., Talcott, C., Rushby, J., Lincoln, P., Kim, M., Cheung, S., et al. (2011). Fractionated software for networked cyber-physical systems: Research directions and long-term vision. Lecture Notes in Computer Science, Formal Modeling: Actors, Open Systems, Biological Systems, 7000, 110–143.

    Article  Google Scholar 

  • Stothers, A. (2010). On the complexity of matrix multiplication. Ph.D. thesis. University of Edinburgh. http://www.maths.ed.ac.uk/pg/thesis/stothers.pdf.

  • Strassen, V. (1969). Gaussian elimination is not optimal. Numerische Mathematik, 13, 354–356.

    Article  MATH  MathSciNet  Google Scholar 

  • Teuscher, C. (Ed.). (2004). Alan Turing: Life and legacy of a great thinker. Berlin: Springer.

    Google Scholar 

  • Turing, A. M. (1936–1937). On computable numbers, with an application to the Entscheidungsproblem. (1938) Proceedings of the London Mathematical Society, 42, 230–265 (On computable numbers, with an application to the Entscheidungsproblem. A correction. Proceedings of the London Mathematical Society, 43, 544–546.

  • Turing, A. M. (1939). Systems of logic based on ordinals. Proceedings of the London Mathematical Society, 45, 161–228.

    Article  MathSciNet  Google Scholar 

  • Turing, A. M. (1948). Intelligent machinery. In Ince (1992), pp. 107–127. Amsterdam: Elsevier Science.

  • Wang, H. (1974). From mathematics to philosophy. London: Routledge and Kegan Paul.

    MATH  Google Scholar 

  • Wegner, P. (1997). Why interaction is more powerful than algorithms. Communications of the ACM, 40, 81–91.

    Article  Google Scholar 

  • Weyl, H. (1928). Philosophie der Mathematik und Natureissenschaft. München; und Berlin: R. Oldenbourg (German). Philosophy of mathematics and natural science. Princeton: Princeton University Press, 1949 (English version).

  • Williams, V. V. (2011). Breaking the Coppersmith–Winograd barrier. http://www.cs.berkeley.edu/~virgi/matrixmult.pdf.

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Acknowledgments

I am grateful to Professor Solomon Feferman for his invaluable guidance throughout the process of this research. I also express my gratitude to Steven Ericsson-Zenith for discussions on interactive computing and the historical premises of Turing’s work, and to Carolyn L. Talcott for discussions on the Actor model and cyber-physical systems. I further thank Martin Davis, Wilfried Sieg, and the two anonymous referees for their helpful comments on a draft of this article.

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    Aran Nayebi

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Nayebi, A. Practical Intractability: A Critique of the Hypercomputation Movement. Minds & Machines 24, 275–305 (2014). https://doi.org/10.1007/s11023-013-9317-3

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