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(Short) Survey of Real Hypercomputation

  • Martin Ziegler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

Abstract

We survey and compare models of computation on real numbers exceeding the Church–Turing Hypothesis.

Keywords

Turing Machine Baire Space Analytic Machine Random Access Machine Recursive Analysis 
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.

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References

  1. Calude.
    Adamyan, V.A., Calude, C.S., Pavlov, B.S.: Transcending the limits of Turing computability. In: Hida, T., Saito, K., Si, S. (eds.) Proc. Meijo Winter School, pp. 119–137. World Scientific, Singapore (2003)Google Scholar
  2. Barmpalias.
    Barmpalias, G.: A Transfinite Hierarchy of Reals. Mathematical Logic Quarterly 49(2), 163–172 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. ACT.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  4. BCSS.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  5. Beggs.
    Beggs, E.J., Tucker, J.V.: Can Newtonian systems, bounded in space, time, mass and energy compute all functions? Theoretical Computer Science (to appear, 2007)Google Scholar
  6. deBerg.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  7. Bournez.
    Bournez, O., Campagnolo, M.: A Survey On Continuous Time Computations. In: submitted as a chapter of the book New Computational Paradigms, Springer, Heidelberg (2007)Google Scholar
  8. Boone.
    Boone, W.W.: The word problem. Proc. Nat. Acad. Sci. U.S.A 44, 265–269 (1958)CrossRefzbMATHGoogle Scholar
  9. Boldi.
    Boldi, P., Vigna, S.: Equality is a Jump. Theoretical Computer Science 219, 49–64 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Basu.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  11. Recursive.
    Brattka, V.: Recursive Characterization of Computable Real-Valued Functions and Relations. Theoretical Computer Science 162, 45–77 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Emperor.
    Brattka, V.: The Emperor’s New Recursiveness: the Epigraph of the Exponential Function in Two Models of Computability. In: Ito, M., Imaoka, T. (eds.) Words, Languages & Combinatorics, vol. III, pp. 63–72. World Scientific Publishing, Singapore (2000)Google Scholar
  13. EffBorel.
    Brattka, V.: Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51, 19–44 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. feasible.
    Brattka, V., Hertling, P.: Feasible real random access machines. Journal of Complexity 14(4), 490–526 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Naive.
    Brattka, V., Hertling, P.: Topological Properties of Real Number Representations. Theoretical Computer Science 284, 241–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. BSS.
    Blum, L., Shub, M., Smale, S.: On a Theory of Computation and Complexity over the Real Numbers: \(\mathcal{NP}\)-Completeness, Recursive Functions, and Universal Machines. Bulletin of the American Mathematical Society (AMS Bulletin) 21, 1–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hotz2.
    Chadzelek, T., Hotz, G.: Analytic Machines. Theoretical Computer Science 219, 151–165 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Campagnolo.
    Campagnolo, M.L., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18, 977–1000 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. CopelandSurvey.
    Copeland, J.: Hypercomputation. In: Minds and Machines, vol. 12, pp. 461–502. Kluwer, Dordrecht (2002)Google Scholar
  20. Cucker.
    Cucker, F.: The Arithmetical Hierarchy over the Reals. Journal of Logic and Computation 2(3), 375–395 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. vdDries.
    van den Dries, L.: Remarks on Tarski’s problem concerning (R, +, x, exp). In: Longi, G., Longo, G., Marcja, A. (eds.) Logic Colloquium ’82, North-Holland, Amsterdam (1984)Google Scholar
  22. realPCF.
    Escardó, M.H.: PCF extended with real numbers. Theoretical Computer Science 162, 79–115 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Etesi.
    Etesi, G., Németi, I.: Non-Turing Computations Via Malament-Hogarth Space-Times. International Journal of Theoretical Physics 41(2), 341–370 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Friedberg.
    Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43 43, 236–238 (1957)CrossRefzbMATHGoogle Scholar
  25. MCA2.
    Gathen, J. v. z., Gerhard, J.: Modern Computer Algebra, 2nd edn., Cambridge (2003)Google Scholar
  26. Gassner.
    Gassner, C.: The Addititve Halting Problem is Not Decidable by means of the Rationals as an Oracle (pre-print)Google Scholar
  27. Geroch.
    Geroch, R., Hartle, J.B.: Computability and Physical Theories. Foundations of Physics 16(6), 533–550 (1986)MathSciNetCrossRefGoogle Scholar
  28. Graca.
    Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. Journal of Complexity 19, 644–664 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Grzegorczyk.
    Grzegorczyk, A.: On the Definitions of Computable Real Continuous Functions. Fundamenta Mathematicae 44, 61–77 (1957)MathSciNetzbMATHGoogle Scholar
  30. Hamkins.
    Hamkins, J.D., Lewis, A.: Infinite Time Turing machines. Journal of Symbolic Logic 65(2), 567–604 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Hinman.
    Hinman, P.G.: Recursion-Theoretic Hierarchies. In: Perspectives in Mathematical Logic, Springer, Heidelberg (1978)Google Scholar
  32. Ho.
    Ho, C.-K.: Relatively recursive reals and real functions. Theoretical Computer Science 210, 99–120 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Hogarth.
    Hogarth, M.L.: Does General Relativity Allow an Observer to View an Eternity in a Finite Time? Foundations of Physics Letters 5(2), 173–181 (1992)MathSciNetCrossRefGoogle Scholar
  34. Hotz1.
    Hotz, G., Vierke, G., Schieffer, B.: Analytic Machines, Electronic Colloquium on Computational Complexity vol. 025 (1995)Google Scholar
  35. Kawamura.
    Kawamura, A.: Type-2 Computability and Moore’s Recursive Functions. Electronic Notes in Theoretical Computer Science 120, 83–95 (2005)MathSciNetCrossRefGoogle Scholar
  36. Kechris.
    Kechris, A.S.: Classical Descriptive Set Theory. In: Graduate Texts in Mathematics, Springer, Heidelberg (1995)Google Scholar
  37. KieuRoy.
    Kieu, T.D.: A reformulation of Hilbert’s tenth problem through quantum mechanics. Proc. Royal Soc. A 460, 1535–1545 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  38. Koiran.
    Koiran, P.: Computing over the Reals with Addition and Order. Theoretical Computer Science 133, 35–48 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Ko.
    Ko, K.: Complexity Theory of Real Functions, Birkhäuser (1991)Google Scholar
  40. Koepf.
    Koepf, W.: Besprechungen zu Büchern der Computeralgebra: Klaus Weihrauch Computable Analysis. in Computeralgebra Rundbrief 29, 29 (2001), http://fachgruppe-computeralgebra.de/CAR/CAR29/node19.html
  41. Kreitz.
    Kreitz, C., Weihrauch, K.: Theory of representations. Theoretical Computer Science 38, 35–53 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Lacombe.
    Lacombe, D.: Les ensembles récursivement ouverts ou fermés, et leurs applications à l’analyse récursive, pp. 1040–1043 in Compt. Rend. Acad. des Sci. Paris vol. 245 (1957); sequel pp. 28–31 in Compt. Rend. Acad. des Sci. Paris, vol. 246 (1958)Google Scholar
  43. Lambov.
    Lambov, B.: RealLib: an Efficient Implementation of Exact Real Arithmetic, to appear in Mathematical Structures in Computer ScienceGoogle Scholar
  44. Macintyre.
    Macintyre, A.: Schanuel’s conjecture and free exponential rings. Ann. Pure Appl. Logic 51, 241–246 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Marker.
    Marker, D.: Model Theory and Exponentiation. Notices of the AMS, 753–759 (1996)Google Scholar
  46. Macintyre, A., Wilkie, A.J.: On the decidability of the real exponential field, in Kreiseliana. About and Around Georg Kreisel, A.K. Peters pp. 441–467 (1996)Google Scholar
  47. Matiyasevich.
    Matiyasevich, Y.V.: Enumerable Sets are Diophantine. Soviet Math.Dokl 11, 354–357 (1970)zbMATHGoogle Scholar
  48. MeerSinus.
    Meer, K.: Real Number Models under Various Sets of Operations. Journal of Complexity 9, 366–372 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  49. RealPost.
    Meer, K., Ziegler, M.: An Explicit Solution to Post’s Problem over the Reals. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 456–467. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  50. CiE06.
    Meer, K., Ziegler, M.: Uncomputability below the Real Halting Problem. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 368–377. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  51. Moore.
    Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162, 23–44 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  52. Moschovakis.
    Moschovakis, Y.N.: Descriptive Set Theory. In: Studies in Logic, North-Holland, Amsterdam (1980)Google Scholar
  53. iRRAM.
    Müller, N.: The iRRAM: Exact Arithmetic in C++. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  54. Novikov.
    Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44, 1–143 (1959)Google Scholar
  55. Ord02.
    Ord, T.: Hypercomputation: computing more than the Turing machine, Honour’s Thesis, University of Melbourne (2002)Google Scholar
  56. Orponen.
    Orponen, P.: A survey of continuous-time computation theory. In: Du, D.-Z., Ko, K.-I. (eds.) Advances in Algorithms, Languages, and Complexity, pp. 209–224. Kluwer, Dordrecht (1997)CrossRefGoogle Scholar
  57. PER.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)CrossRefzbMATHGoogle Scholar
  58. Rogers.
    Rogers, J.H.: Theory of Recursive Functions and Effective Computability. Series in Higher Mathematics. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  59. Rossello.
    Cucker, F., Rosselló, F.: Recursiveness over the Complex Numbers is Time-Bounded. In: Shyamasundar, R.K. (ed.) Foundations of Software Technology and Theoretical Computer Science, vol. 761, pp. 260–267. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  60. Sacks.
    Sacks, G.: Higher Recursion Theory. Springer, Heidelberg (1990)CrossRefzbMATHGoogle Scholar
  61. RAM.
    Schönhage, A.: On the Power of Random Access Machines. In: Maurer, H.A. (ed.) Automata, Languages, and Programming. LNCS, vol. 71, pp. 520–529. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  62. Scott.
    Scott, D.S.: Outline of a Mathematical Theory of Computation. In: Technical Monograph PRG-2, Oxford University, Oxford (1970)Google Scholar
  63. Schoening2.
    Schöning, U., Pruim, R.: Gems of Theoretical Computer Science. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  64. SmithCM.
    Smith, W.D.: Church’s Thesis meets the N-body Problem. J. Applied Mathematics and Computation 178, 154–183 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  65. Soare.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)CrossRefzbMATHGoogle Scholar
  66. Specker.
    Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. Journal of Symbolic Logic 14(3), 145–158 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  67. Tucker.
    Tucker, J.V.: Computability and the algebra of fields. J. Symbolic Logic 45, 103–120 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  68. Turing.
    Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc 42(2), 230–265 (1936)MathSciNetzbMATHGoogle Scholar
  69. Turing2.
    Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. A correction. Proc. London Math. Soc. 43(2), 544–546 (1937)MathSciNetzbMATHGoogle Scholar
  70. TuringDiss.
    Turing, A.M.: Systems of Logic Based on Ordinals. Proc. London Math. Soc. 45, 161–228 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  71. Zucker.
    Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many-sorted algebras. In: Abramsky, S., Gabbay, D.M., Maybaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 5, pp. 317–523. Oxford Science Publications, Oxford (2000)Google Scholar
  72. WeihrauchIntro.
    Weihrauch, K.: A Simple Introduction to Computable Analysis, Monographs of the Electronic Colloquium on Computational Complexity (1995)Google Scholar
  73. Weihrauch.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  74. WiedermannUMC.
    Wiedermann, J., van Leeuwen, J.: Relativistic Computers and Non-uniform Complexity Theory. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002. LNCS, vol. 2509, pp. 287–299. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  75. Zheng.
    Zheng, X.: On the hierarchy of Δ 2 real numbers. Theoretical Informatics and Application (to appear)Google Scholar
  76. Xizhong.
    Zheng, X., Weihrauch, K.: The Arithmetical Hierarchy of Real Numbers. Mathematical Logic Quarterly 47, 51–65 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  77. CiE05.
    Ziegler, M.: Computability and Continuity on the Real Arithmetic Hierarchy. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 562–571. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  78. IJTP2.
    Ziegler, M.: Computational Power of Infinite Quantum Parallelism. International Journal of Theoretical Physics 44, 2059–2071 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  79. CCA06.
    Ziegler, M.: Revising Type-2 Computation and Degrees of Discontinuity. In: Proc. 3rd International Conference on Computability and Complexity in Analysis, pp. 347–366Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Martin Ziegler
    • 1
  1. 1.University of PaderbornGermany

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