Hypercomputation

Hypercomputation or super-Turing computation refers to models of computation that are more powerful than, or are incomparable with, Turing computability. This includes various hypothetical methods for the computation of non-Turing-computable functions, following super-recursive algorithms (see also supertask). The term "super-Turing computation" appeared in a 1995 Science paper by Hava Siegelmann. The term "hypercomputation" was introduced in 1999 by Jack Copeland and Diane Proudfoot.[1]

The terms are not quite synonymous: "super-Turing computation" usually implies that the proposed model is supposed to be physically realizable, while "hypercomputation" does not.

Contents

History

A model more powerful than Turing machines was introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals. This paper investigated mathematical systems in which an oracle was available, which could compute a single arbitrary (non-recursive) function from naturals to naturals. He used this device to prove that even in those more powerful systems, undecidability is still present. Turing's oracle machines are strictly mathematical abstractions, and are not physically realizable.[2]

Hypercomputation and the Church–Turing thesis

The Church–Turing thesis states that any function that is algorithmically computable can be computed by a Turing machine. Hypercomputers compute functions that a Turing machine cannot, hence, not computable in the Church-Turing sense.

An example of a problem a Turing machine cannot solve is the halting problem. A Turing machine cannot decide if an arbitrary program halts or runs forever. Some proposed hypercomputers can simulate the program for an infinite number of steps and tell the user whether or not the program halted.

Hypercomputer proposals

Analysis of capabilities

Many hypercomputation proposals amount to alternative ways to read an oracle or advice function embedded into an otherwise classical machine. Others allow access to some higher level of the arithmetic hierarchy. For example, supertasking Turing machines, under the usual assumptions, would be able to compute any predicate in the truth-table degree containing \Sigma^0_1 or \Pi^0_1. Limiting-recursion, by contrast, can compute any predicate or function in the corresponding Turing degree, which is known to be \Delta^0_2. Gold further showed that limiting partial recursion would allow the computation of precisely the \Sigma^0_2 predicates.

Model Computable predicates Notes Refs
supertasking tt(\Sigma^0_1, \Pi^0_1) dependent on outside observer [27]
limiting/trial-and-error  \Delta^0_2 [4]
iterated limiting (k times)  \Delta^0_{k%2B1} [6]
Blum-Shub-Smale machine incomparable with traditional computable real functions. [28]
Malament-Hogarth spacetime HYP Dependent on spacetime structure [29]
Analog recurrent neural network  \Delta^0_1[f] f is an advice function giving connection weights; size is bounded by runtime [30][31]
Infinite time Turing machine  \ge T(\Sigma^1_1) [32]
Classical fuzzy Turing machine  \Sigma^0_1 \cup \Pi^0_1 For any computable t-norm [33]
Increasing function oracle  \Delta^1_1 For the one-sequence model;  \Pi^1_1 are r.e. [26]

Taxonomy of "super-recursive" computation methodologies

Burgin has collected a list of what he calls "super-recursive algorithms" (from Burgin 2005: 132):

In the same book, he presents also a list of "algorithmic schemes":

Criticism

Martin Davis, in his writings on hypercomputation [38] [39] refers to this subject as "a myth" and offers counter-arguments to the physical realizability of hypercomputation. As for its theory, he argues against the claims that this is a new field founded in 1990s. This point of view relies on the history of computability theory (degrees of unsolvability, computability over functions, real numbers and ordinals), as also mentioned above.

Andrew Hodges wrote a critical commentary[40] on Copeland and Proudfoot's article[1].

See also

References

  1. ^ a b Copeland and Proudfoot, Alan Turing's forgotten ideas in computer science. Scientific American, April 1999
  2. ^ "Let us suppose that we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine" (Undecidable p. 167, a reprint of Turing's paper Systems of Logic Based On Ordinals)
  3. ^ These models have been independently developed by many different authors, including Hermann Weyl (1927). Philosophie der Mathematik und Naturwissenschaft. ; the model is discussed in Shagrir, O. (June 2004). "Super-tasks, accelerating Turing machines and uncomputability". Theor. Comput. Sci. 317, 1-3 317: 105–114. doi:10.1016/j.tcs.2003.12.007. http://edelstein.huji.ac.il/staff/shagrir/papers/Supertasks_Accelerating_Turing_Machines_and_Uncomputability.pdf.  and in Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040. 
  4. ^ a b c E. M. Gold (1965). "Limiting Recursion". Journal of Symbolic Logic 30 (1): 28–48. doi:10.2307/2270580. JSTOR 2270580. , E. Mark Gold (1967). "Language identification in the limit". Information and Control 10 (5): 447–474. doi:10.1016/S0019-9958(67)91165-5. 
  5. ^ a b c Hilary Putnam (1965). "Trial and Error Predicates and the Solution to a Problem of Mostowksi". Journal of Symbolic Logic 30 (1): 49–57. doi:10.2307/2270581. JSTOR 2270581. 
  6. ^ a b L. K. Schubert (July 1974). "Iterated Limiting Recursion and the Program Minimization Problem". Journal of the ACM 21 (3): 436–445. doi:10.1145/321832.321841. http://portal.acm.org/citation.cfm?id=321832.321841. 
  7. ^ Arnold Schönhage, "On the power of random access machines", in Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520-529, 1979. Source of citation: Scott Aaronson, "NP-complete Problems and Physical Reality"[1] p. 12
  8. ^ Edith Spaan, Leen Torenvliet and Peter van Emde Boas (1989). "Nondeterminism, Fairness and a Fundamental Analogy". EATCS bulletin 37: 186–193. 
  9. ^ Hajnal Andréka, István Németi and Gergely Székely, Closed Timelike Curves in Relativistic Computation, 2011.[2]
  10. ^ Todd A. Brun, Computers with closed timelike curves can solve hard problems, Found.Phys.Lett. 16 (2003) 245-253.[3]
  11. ^ S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent [4]
  12. ^ Hogarth, M., 1992, ‘Does General Relativity Allow an Observer to View an Eternity in a Finite Time?’, Foundations of Physics Letters, 5, 173–181.
  13. ^ István Neméti; Hajnal Andréka (2006). "Can General Relativistic Computers Break the Turing Barrier?". Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006. Proceedings. Lecture Notes in Computer Science. 3988. Springer. doi:10.1007/11780342. 
  14. ^ Etesi, G., and Nemeti, I., 2002 'Non-Turing computations via Malament-Hogarth space-times', Int.J.Theor.Phys. 41 (2002) 341–370, Non-Turing Computations via Malament-Hogarth Space-Times:.
  15. ^ Earman, J. and Norton, J., 1993, ‘Forever is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes’, Philosophy of Science, 5, 22–42.
  16. ^ Verifying Properties of Neural Networks p.6
  17. ^ Joel David Hamkins and Andy Lewis, Infinite time Turing machines, Journal of Symbolic Logic, 65(2):567-604, 2000.[5]
  18. ^ a b Jan van Leeuwen; Jiří Wiedermann (September 2000). "On Algorithms and Interaction". MFCS '00: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science. Springer-Verlag. 
  19. ^ Jürgen Schmidhuber (2000). "Algorithmic Theories of Everything". Sections in: Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. International Journal of Foundations of Computer Science ():587-612 (). Section 6 in: the Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions. in J. Kivinen and R. H. Sloan, editors, Proceedings of the 15th Annual Conference on Computational Learning Theory (COLT ), Sydney, Australia, Lecture Notes in Artificial Intelligence, pages 216--228. Springer, . 13 (4): 1–5. arXiv:quant-ph/0011122. 
  20. ^ a b J. Schmidhuber (2002). "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit". International Journal of Foundations of Computer Science 13 (4): 587–612. doi:10.1142/S0129054102001291. http://www.idsia.ch/~juergen/kolmogorov.html. 
  21. ^ There have been some claims to this effect; see Tien Kieu (2003). "Quantum Algorithm for the Hilbert's Tenth Problem". Int. J. Theor. Phys. 42 (7): 1461–1478. arXiv:quant-ph/0110136. doi:10.1023/A:1025780028846. . & the ensuing literature. Errors have been pointed out in Kieu's approach by Warren D. Smith in Three counterexamples refuting Kieu’s plan for “quantum adiabatic hypercomputation”; and some uncomputable quantum mechanical tasks
  22. ^ Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411-1473, 1997. [6]
  23. ^ Santos, Eugene S. (1970). "Fuzzy Algorithms". Information and Control 17 (4): 326–339. doi:10.1016/S0019-9958(70)80032-8. 
  24. ^ Biacino, L.; Gerla, G. (2002). "Fuzzy logic, continuity and effectiveness". Archive for Mathematical Logic 41 (7): 643–667. doi:10.1007/s001530100128. ISSN 0933-5846. 
  25. ^ a b Wiedermann, Jiří (2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theor. Comput. Sci. 317 (1–3): 61–69. doi:10.1016/j.tcs.2003.12.004. http://portal.acm.org/citation.cfm?id=1011188. 
  26. ^ a b Dmytro Taranovsky (July 17, 2005). "Finitism and Hypercomputation". http://web.mit.edu/dmytro/www/FinitismPaper.htm. Retrieved Apr 26, 2011. 
  27. ^ Petrus H. Potgieter (July 2006). "Zeno machines and hypercomputation". Theoretical Computer Science 358 (1): 23–33. doi:10.1016/j.tcs.2005.11.040. 
  28. ^ Lenore Blum, Felipe Cucker, Michael Shub, and Stephen Smale. Complexity and Real Computation. ISBN 0387982817. 
  29. ^ P. D. Welch (10-Sept-2006). The extent of computation in Malament-Hogarth spacetimes. arXiv:gr-qc/0609035. 
  30. ^ Hava Siegelmann (April 1995). "Computation Beyond the Turing Limit". Science 268 (5210): 545–548. doi:10.1126/science.268.5210.545. PMID 17756722. 
  31. ^ Hava Siegelmann; Eduardo Sontag (1994). "Analog Computation via Neural Networks". Theoretical Computer Science 131 (2): 331–360. doi:10.1016/0304-3975(94)90178-3. 
  32. ^ Joel David Hamkins; Andy Lewis (2000). "Infinite Time Turing machines". Journal of Symbolic Logic 65 (2): 567=604. http://jdh.hamkins.org/Publications/2000e. 
  33. ^ Jiří Wiedermann (June 4, 2004). "Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines". Theoretical Computer Science (Elsevier Science Publishers Ltd. Essex, UK) 317 (1–3). 
  34. ^ Hintikka, Ja; Mutanen, A. (1998). "An Alternative Concept of Computability". Language, Truth, and Logic in Mathematics. Dordrecht. pp. 174–188. 
  35. ^ Darko Roglic (24–Jul–2007). "The universal evolutionary computer based on super-recursive algorithms of evolvability". arXiv:0708.2686 [cs.NE]. 
  36. ^ Eugene Eberbach (2002). "On expressiveness of evolutionary computation: is EC algorithmic?". Computational Intelligence, WCCI 1: 564–569. doi:10.1109/CEC.2002.1006988. http://www.computer.org/portal/web/csdl/doi/10.1109/CEC.2002.1006988. 
  37. ^ Borodyanskii, Yu M; Burgin, M. S. (1994). "Operations and compositions in transrecursive operators". Cybernetics and Systems Analysis 30 (4): 473–478. doi:10.1007/BF02366556. http://www.springerlink.com/content/a70r23722wqu43t7/. 
  38. ^ Davis, Martin, Why there is no such discipline as hypercomputation, Applied Mathematics and Computation, Volume 178, Issue 1, 1 July 2006, Pages 4–7, Special Issue on Hypercomputation
  39. ^ Davis, Martin (2004). "The Myth of Hypercomputation". Alan Turing: Life and Legacy of a Great Thinker. Springer. 
  40. ^ Andrew Hodges (retrieved 23 September 2011). "The Professors and the Brainstorms". The Alan Turing Home Page. http://www.turing.org.uk/philosophy/sciam.html. 

Further reading

External links