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(Short) Survey of Real Hypercomputation

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Computation and Logic in the Real World (CiE 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4497))

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Abstract

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

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References

  1. 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, G.: A Transfinite Hierarchy of Reals. Mathematical Logic Quarterly 49(2), 163–172 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  4. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)

    Book  MATH  Google Scholar 

  5. 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. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2000)

    Book  MATH  Google Scholar 

  7. 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, W.W.: The word problem. Proc. Nat. Acad. Sci. U.S.A 44, 265–269 (1958)

    Article  MATH  Google Scholar 

  9. Boldi, P., Vigna, S.: Equality is a Jump. Theoretical Computer Science 219, 49–64 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  11. Brattka, V.: Recursive Characterization of Computable Real-Valued Functions and Relations. Theoretical Computer Science 162, 45–77 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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. Brattka, V.: Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51, 19–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brattka, V., Hertling, P.: Feasible real random access machines. Journal of Complexity 14(4), 490–526 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brattka, V., Hertling, P.: Topological Properties of Real Number Representations. Theoretical Computer Science 284, 241–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chadzelek, T., Hotz, G.: Analytic Machines. Theoretical Computer Science 219, 151–165 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Campagnolo, M.L., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18, 977–1000 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Copeland, J.: Hypercomputation. In: Minds and Machines, vol. 12, pp. 461–502. Kluwer, Dordrecht (2002)

    Google Scholar 

  20. Cucker, F.: The Arithmetical Hierarchy over the Reals. Journal of Logic and Computation 2(3), 375–395 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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. Escardó, M.H.: PCF extended with real numbers. Theoretical Computer Science 162, 79–115 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Etesi, G., Németi, I.: Non-Turing Computations Via Malament-Hogarth Space-Times. International Journal of Theoretical Physics 41(2), 341–370 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43 43, 236–238 (1957)

    Article  MATH  Google Scholar 

  25. Gathen, J. v. z., Gerhard, J.: Modern Computer Algebra, 2nd edn., Cambridge (2003)

    Google Scholar 

  26. Gassner, C.: The Addititve Halting Problem is Not Decidable by means of the Rationals as an Oracle (pre-print)

    Google Scholar 

  27. Geroch, R., Hartle, J.B.: Computability and Physical Theories. Foundations of Physics 16(6), 533–550 (1986)

    Article  MathSciNet  Google Scholar 

  28. Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. Journal of Complexity 19, 644–664 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grzegorczyk, A.: On the Definitions of Computable Real Continuous Functions. Fundamenta Mathematicae 44, 61–77 (1957)

    MathSciNet  MATH  Google Scholar 

  30. Hamkins, J.D., Lewis, A.: Infinite Time Turing machines. Journal of Symbolic Logic 65(2), 567–604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hinman, P.G.: Recursion-Theoretic Hierarchies. In: Perspectives in Mathematical Logic, Springer, Heidelberg (1978)

    Google Scholar 

  32. Ho, C.-K.: Relatively recursive reals and real functions. Theoretical Computer Science 210, 99–120 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  Google Scholar 

  34. Hotz, G., Vierke, G., Schieffer, B.: Analytic Machines, Electronic Colloquium on Computational Complexity vol. 025 (1995)

    Google Scholar 

  35. Kawamura, A.: Type-2 Computability and Moore’s Recursive Functions. Electronic Notes in Theoretical Computer Science 120, 83–95 (2005)

    Article  MathSciNet  Google Scholar 

  36. Kechris, A.S.: Classical Descriptive Set Theory. In: Graduate Texts in Mathematics, Springer, Heidelberg (1995)

    Google Scholar 

  37. Kieu, T.D.: A reformulation of Hilbert’s tenth problem through quantum mechanics. Proc. Royal Soc. A 460, 1535–1545 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Koiran, P.: Computing over the Reals with Addition and Order. Theoretical Computer Science 133, 35–48 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ko, K.: Complexity Theory of Real Functions, Birkhäuser (1991)

    Google Scholar 

  40. 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, C., Weihrauch, K.: Theory of representations. Theoretical Computer Science 38, 35–53 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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, B.: RealLib: an Efficient Implementation of Exact Real Arithmetic, to appear in Mathematical Structures in Computer Science

    Google Scholar 

  44. Macintyre, A.: Schanuel’s conjecture and free exponential rings. Ann. Pure Appl. Logic 51, 241–246 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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, Y.V.: Enumerable Sets are Diophantine. Soviet Math.Dokl 11, 354–357 (1970)

    MATH  Google Scholar 

  48. Meer, K.: Real Number Models under Various Sets of Operations. Journal of Complexity 9, 366–372 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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)

    Chapter  Google Scholar 

  50. 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)

    Chapter  Google Scholar 

  51. Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162, 23–44 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. Moschovakis, Y.N.: Descriptive Set Theory. In: Studies in Logic, North-Holland, Amsterdam (1980)

    Google Scholar 

  53. 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)

    Chapter  Google Scholar 

  54. 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. Ord, T.: Hypercomputation: computing more than the Turing machine, Honour’s Thesis, University of Melbourne (2002)

    Google Scholar 

  56. 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)

    Chapter  Google Scholar 

  57. Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)

    Book  MATH  Google Scholar 

  58. Rogers, J.H.: Theory of Recursive Functions and Effective Computability. Series in Higher Mathematics. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  59. 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)

    Chapter  Google Scholar 

  60. Sacks, G.: Higher Recursion Theory. Springer, Heidelberg (1990)

    Book  MATH  Google Scholar 

  61. 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)

    Chapter  Google Scholar 

  62. Scott, D.S.: Outline of a Mathematical Theory of Computation. In: Technical Monograph PRG-2, Oxford University, Oxford (1970)

    Google Scholar 

  63. Schöning, U., Pruim, R.: Gems of Theoretical Computer Science. Springer, Heidelberg (1998)

    Book  MATH  Google Scholar 

  64. Smith, W.D.: Church’s Thesis meets the N-body Problem. J. Applied Mathematics and Computation 178, 154–183 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)

    Book  MATH  Google Scholar 

  66. Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. Journal of Symbolic Logic 14(3), 145–158 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  67. Tucker, J.V.: Computability and the algebra of fields. J. Symbolic Logic 45, 103–120 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  68. Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc 42(2), 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  69. Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. A correction. Proc. London Math. Soc. 43(2), 544–546 (1937)

    MathSciNet  MATH  Google Scholar 

  70. Turing, A.M.: Systems of Logic Based on Ordinals. Proc. London Math. Soc. 45, 161–228 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  71. 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. Weihrauch, K.: A Simple Introduction to Computable Analysis, Monographs of the Electronic Colloquium on Computational Complexity (1995)

    Google Scholar 

  73. Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)

    Book  MATH  Google Scholar 

  74. 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)

    Chapter  Google Scholar 

  75. Zheng, X.: On the hierarchy of Δ 2 real numbers. Theoretical Informatics and Application (to appear)

    Google Scholar 

  76. Zheng, X., Weihrauch, K.: The Arithmetical Hierarchy of Real Numbers. Mathematical Logic Quarterly 47, 51–65 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  77. 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)

    Chapter  Google Scholar 

  78. Ziegler, M.: Computational Power of Infinite Quantum Parallelism. International Journal of Theoretical Physics 44, 2059–2071 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  79. Ziegler, M.: Revising Type-2 Computation and Degrees of Discontinuity. In: Proc. 3rd International Conference on Computability and Complexity in Analysis, pp. 347–366

    Google Scholar 

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Ziegler, M. (2007). (Short) Survey of Real Hypercomputation. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_86

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  • DOI: https://doi.org/10.1007/978-3-540-73001-9_86

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