Skip to main content

A Tutorial on Computable Analysis

  • Chapter
New Computational Paradigms

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. O. Aberth. Computable Analysis. McGraw-Hill, New York, 1980.

    MATH  Google Scholar 

  2. K. Ambos-Spies, K. Weihrauch, and X. Zheng. Weakly computable real numbers. Journal of Complexity, 16(4):676-690, 2000.

    MATH  MathSciNet  Google Scholar 

  3. A. Avizienis. Signed-digit number representations for fast parallel arithmetic. IRE Transactions on Electronic Computers, 10:389-400, 1961.

    MathSciNet  Google Scholar 

  4. G. Baigger. Die Nichtkonstruktivität des Brouwerschen Fixpunktsatzes. Arch. Math. Logik Grundlag., 25:183-188, 1985.

    MATH  MathSciNet  Google Scholar 

  5. G. Barmpalias. The approximation structure of a computably approximable real. The Journal of Symbolic Logic, 68(3):885-922, 2003.

    MATH  MathSciNet  Google Scholar 

  6. G. Beer. Topologies on Closed and Closed Convex Sets, volume 268 of Mathematics and Its Applications. Kluwer Academic, Dordrecht, 1993.

    Google Scholar 

  7. I. Binder, M. Braverman, and M. Yampolsky. On computational complexity of Siegel Julia sets. Comm. Math. Phys., 264(2):317-334, 2006.

    MathSciNet  Google Scholar 

  8. E. Bishop and D. S. Bridges. Constructive Analysis, volume 279 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1985.

    Google Scholar 

  9. J. Blanck. Domain representations of topological spaces. Theoretical Computer Science, 247:229-255, 2000.

    MATH  MathSciNet  Google Scholar 

  10. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer, New York, 1998.

    Google Scholar 

  11. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers: NP -completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society, 21(1):1-46, 1989.

    MATH  MathSciNet  Google Scholar 

  12. J. M. Borwein and P. B. Borwein. Pi and the AGM. John Wiley & Sons, New York, 1987.

    MATH  Google Scholar 

  13. B. Branner. The Mandelbrot set. In R. L. Devaney and L. Keen, editors, Chaos and Fractals. The Mathematics Behind the Computer Graphics, volume 39 of Proceedings of Symposia in Applied Mathematics, pages 75-105, Providence, Rhode Island, 1989. American Mathematical Society, 1989.

    Google Scholar 

  14. V. Brattka. Computable invariance. Theoretical Computer Science, 210:3-20, 1999.

    MATH  MathSciNet  Google Scholar 

  15. V. Brattka. Computable versions of Baire’s category theorem. In J. Sgall, A. Pultr, and P. Kolman, editors, Mathematical Foundations of Computer Science 2001, volume 2136 of Lecture Notes in Computer Science, pages 224-235, Berlin, 2001. Springer. 26th International Symposium, MFCS 2001, Mariánské Lázn ě , Czech Republic, August 27-31, 2001.

    Google Scholar 

  16. V. Brattka. The inversion problem for computable linear operators. In H. Alt and M. Habib, editors, STACS 2003, volume 2607 of Lecture Notes in Computer Science, pages 391-402, Berlin, 2003. Springer. 20th Annual Symposium on Theoretical Aspects of Computer Science, Berlin, Germany, February 27-March 1, 2003.

    Google Scholar 

  17. V. Brattka. Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly, 51(1):19-44, 2005.

    MATH  MathSciNet  Google Scholar 

  18. V. Brattka. On the Borel complexity of Hahn-Banach extensions. In V. Brattka, L. Staiger, and K. Weihrauch, editors, Proceedings of the 6th Workshop on Computabil-ity and Complexity in Analysis, volume 120 of Electronic Notes in Theoretical Computer Science, pages 3-16, Amsterdam, 2005. Elsevier. 6th International Workshop, CCA 2004, Wittenberg, Germany, August 16-20, 2004.

    Google Scholar 

  19. V. Brattka and R. Dillhage. On computable compact operators on Banach spaces. In D. Cenzer, R. Dillhage, T. Grubba, and K. Weihrauch, editors, Proceedings of the Third International Conference on Computability and Complexity in Analysis, volume 167 of Electronic Notes in Theoretical Computer Science, Amsterdam, 2007. Elsevier. CCA 2006, Gainesville, Florida, November 1-5, 2006.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. V. Brattka and G. Presser. Computability on subsets of metric spaces. Theoretical Computer Science, 305:43-76, 2003.

    MATH  MathSciNet  Google Scholar 

  22. V. Brattka and A. Yoshikawa. Towards computability of elliptic boundary value prob-lems in variational formulation. Journal of Complexity, 22(6):858-880, 2006.

    MATH  MathSciNet  Google Scholar 

  23. M. Braverman. Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are Poly-Time Computable. Master thesis, Department of Computer Science, University of Toronto, 2004.

    Google Scholar 

  24. M. Braverman. Hyperbolic Julia sets are poly-time computable. In V. Brattka, L. Staiger, and K. Weihrauch, editors, Proceedings of the 6th Workshop on Computability and Complexity in Analysis, volume 120 of Electronic Notes in Theoretical Computer Science, pages 17-30, Amsterdam, 2005. Elsevier. 6th International Workshop, CCA 2004, Wittenberg, Germany, August 16-20, 2004.

    Google Scholar 

  25. M. Braverman and S. Cook. Computing over the reals: Foundations for scientific com-puting. Notices of the AMS, 53(3):318-329, 2006.

    MATH  MathSciNet  Google Scholar 

  26. R. Brent. Fast multiple-precision evaluation of elementary functions. Journal of the Association for Computing Machinery, 23(2):242-251, 1976.

    MATH  MathSciNet  Google Scholar 

  27. D. Bridges and F. Richman. Varieties of Constructive Mathematics, volume 97 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1987.

    Google Scholar 

  28. L. Brouwer. Collected Works, Vol. 1, Philosophy and Foundations of Mathematics. North-Holland, Amsterdam, 1975. Heyting, A. (ed).

    Google Scholar 

  29. L. Brouwer. Collected Works, Vol. 2, Geometry, Analysis, Topology and Mechanics. North-Holland, Amsterdam, 1976. Freudenthal, H. (ed).

    Google Scholar 

  30. C. S. Calude, P. H. Hertling, B. Khoussainov, and Y. Wang. Recursively enumerable reals and Chaitin ω numbers. Theoretical Computer Science, 255:125-149, 2001.

    MATH  MathSciNet  Google Scholar 

  31. G. Ceĭtin. Algorithmic operators in constructive metric spaces. Tr. Mat. Inst. Steklov, 67:295-361, 1962. (in Russian, English trans. in AMS Trans. 64, 1967).

    Google Scholar 

  32. D. Cenzer and J. B. Remmel. Index sets for computable differential equations. Mathe-matical Logic Quarterly, 50(4,5):329-344, 2004.

    MATH  MathSciNet  Google Scholar 

  33. G. J. Chaitin. A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery, 22:329-340, 1975.

    MATH  MathSciNet  Google Scholar 

  34. A. Chou and K.-I. Ko. Computational complexity of two-dimensional regions. SIAM Journal on Computing, 24:923-947, 1995.

    MATH  MathSciNet  Google Scholar 

  35. A. W. Chou and K.-I. Ko. The computational complexity of distance functions of two-dimensional domains. Theoretical Computer Science, 337:360-369, 2005.

    MATH  MathSciNet  Google Scholar 

  36. R. G. Downey, D. R. Hirschfeldt, A. Nies, and F. Stephan. Trivial reals. In R. Downey, D. Decheng, T. S. Ping, Q. Y. Hui, and M. Yasugi, editors, Proceedings of the 7th and 8th Asian Logic Conferences, pages 63-102, Singapore, 2003. World Scientific. 7th Conference: Hsi-Tou, Taiwan, June 6-10, 1999; 8th Conference: Chongqing, China, August 29-September 2, 2002.

    Google Scholar 

  37. A. Edalat. Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic, 3(4):401-452, 1997.

    MATH  MathSciNet  Google Scholar 

  38. M. H. Escardó. PCF extended with real numbers. In K.-I. Ko and K. Weihrauch, editors, Computability and Complexity in Analysis, volume 190 of Informatik Berichte, pages 11-24. FernUniversität Hagen, Sept. 1995. CCA Workshop, Hagen, August 19-20, 1995.

    Google Scholar 

  39. H. Friedman. On the computational complexity of maximization and integration. Advances in Mathematics, 53:80-98, 1984.

    MATH  MathSciNet  Google Scholar 

  40. G. Gherardi. Effective Borel degrees of some topological functions. Mathematical Logic Quarterly, 52(6):625-642, 2006.

    MATH  MathSciNet  Google Scholar 

  41. A. Grzegorczyk. On the definitions of computable real continuous functions. Fundamenta Mathematicae, 44:61-71, 1957.

    MATH  MathSciNet  Google Scholar 

  42. J. Hauck. Berechenbare reelle Funktionen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 19:121-140, 1973.

    MATH  MathSciNet  Google Scholar 

  43. J. Hauck. Konstruktive Darstellungen reeller Zahlen und Folgen. Zeitschrift für Mathe-matische Logik und Grundlagen der Mathematik, 24:365-374, 1978.

    MATH  MathSciNet  Google Scholar 

  44. P. Hertling. A topological complexity hierarchy of functions with finite range. Tech-nical Report 223, Centre de recerca matematica, Institut d’estudis catalans, Barcelona, Barcelona, Oct. 1993. Workshop on Continuous Algorithms and Complexity, Barcelona, October, 1993.

    Google Scholar 

  45. P. Hertling. Topological complexity with continuous operations. Journal of Complexity, 12:315-338, 1996.

    MATH  MathSciNet  Google Scholar 

  46. P. Hertling. Unstetigkeitsgrade von Funktionen in der effektiven Analysis. PhD thesis, Fachbereich Informatik, FernUniversität Hagen, 1996.

    Google Scholar 

  47. P. Hertling. An effective Riemann Mapping Theorem. Theoretical Computer Science, 219:225-265, 1999.

    MATH  MathSciNet  Google Scholar 

  48. P. Hertling. A real number structure that is effectively categorical. Mathematical Logic Quarterly, 45(2):147-182, 1999.

    MATH  MathSciNet  Google Scholar 

  49. P. Hertling. A Banach-Mazur computable but not Markov computable function on the computable real numbers. Annals of Pure and Applied Logic, 132(2-3):227-246, 2005.

    MATH  MathSciNet  Google Scholar 

  50. P. Hertling. Is the Mandelbrot set computable? Mathematical Logic Quarterly, 51(1):5-18,2005.

    MATH  MathSciNet  Google Scholar 

  51. M. D. Hirsch. Applications of topology to lower bound estimates in computer science. In From Topology to Computation: Proceedings of the Smalefest, Berkeley, CA, 1990, pages 395-418, New York, 1993. Springer.

    Google Scholar 

  52. C.-K. Ho. Relatively recursive reals and real functions. Theoretical Computer Science, 210(1):99-120, 1999.

    MATH  MathSciNet  Google Scholar 

  53. K.-I. Ko. On the computational complexity of ordinary differential equations. Inform. Contr., 58:157-194, 1983.

    MATH  Google Scholar 

  54. K.-I. Ko. Approximation to measurable functions and its relation to probabilistic computation. Annals of Pure and Applied Logic, 30:173-200, 1986.

    MATH  MathSciNet  Google Scholar 

  55. K.-I. Ko. Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston, 1991.

    Google Scholar 

  56. K.-I. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323-352, 1982.

    MATH  MathSciNet  Google Scholar 

  57. G. Kreisel, D. Lacombe, and J. Shoenfield. Partial recursive functionals and effective operations. In A. Heyting, editor, Constructivity in Mathematics, Studies in Logic and the Foundations of Mathematics, pages 290-297, Amsterdam, 1959. North-Holland. Proc. Colloq., Amsterdam, Aug. 26-31, 1957.

    Google Scholar 

  58. C. Kreitz and K. Weihrauch. Theory of representations. Theoretical Computer Science, 38:35-53, 1985.

    MATH  MathSciNet  Google Scholar 

  59. B. A. Kušner. Lectures on Constructive Mathematical Analysis, volume 60 of Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island, 1984.

    Google Scholar 

  60. A. Ku čera and T. A. Slaman. Randomness and recursive enumerability. SIAM J. Comput., 31(1):199-211, 2001.

    MathSciNet  Google Scholar 

  61. S. Labhalla, H. Lombardi, and E. Moutai. Espaces métriques rationnellement présentés et complexité, le cas de l’espace des fonctions réelles uniformément continues sur un intervalle compact. Theoretical Computer Science, 250:265-332, 2001.

    MATH  MathSciNet  Google Scholar 

  62. D. Lacombe. Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles III. Comptes Rendus Académie des Sciences Paris, 241:151-153,1955. Théorie des fonctions.

    Google Scholar 

  63. S. Mazur. Computable Analysis, volume 33. Razprawy Matematyczne, Warsaw, 1963.

    Google Scholar 

  64. G. Metakides and A. Nerode. The introduction of non-recursive methods into mathemat-ics. In A. Troelstra and D. v. Dalen, editors, The L.E.J. Brouwer Centenary Symposium, volume 110 of Studies in Logic and the foundations of mathematics, pages 319-335, Amsterdam, 1982. North-Holland. Proceedings of the conference held in Noordwijker-hout, June 8-13, 1981.

    Google Scholar 

  65. G. Metakides, A. Nerode, and R. Shore. Recursive limits on the Hahn-Banach theorem. In M. Rosenblatt, editor, Errett Bishop: Reflections on Him and His Research, volume 39 of Contemporary Mathematics, pages 85-91, Providence, Rhode Island, 1985. Ameri-can Mathematical Society. Proceedings of the memorial meeting for Errett Bishop, University of California, San Diego, September 24, 1983.

    Google Scholar 

  66. J. Miller and A. Nies. Randomness and computability: Open questions. Bull. Symb.Logic, 12(3):390-410, 2006.

    MATH  MathSciNet  Google Scholar 

  67. J. S. Miller. Pi-0-1 Classes in Computable Analysis and Topology. PhD thesis, Cornell University, Ithaca, New York, 2002.

    Google Scholar 

  68. J. S. Miller. Degrees of unsolvability of continuous functions. The Journal of SymbolicLogic, 69(2):555-584, 2004.

    MATH  Google Scholar 

  69. R. E. Moore. Interval Analysis. Prentice Hall, Englewood Cliffs, New Jersey, 1966.

    MATH  Google Scholar 

  70. Y. N. Moschovakis. Recursive metric spaces. Fundamenta Mathematicae, 55:215-238, 1964.

    MATH  MathSciNet  Google Scholar 

  71. Y. N. Moschovakis. Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1980.

    Google Scholar 

  72. A. Mostowski. On computable sequences. Fundamenta Mathematicae, 44:37-51, 1957.

    MATH  MathSciNet  Google Scholar 

  73. J. M. Muller. Elementary Functions. Birkhäuser, Boston, 2nd edition, 2006.

    Google Scholar 

  74. N. T. Müller. Subpolynomial complexity classes of real functions and real numbers. In L. Kott, editor, Proceedings of the 13th International Colloquium on Automata, Languages, and Programming, volume 226 of Lecture Notes in Computer Science, pages 284-293, Berlin, 1986. Springer.

    Google Scholar 

  75. N. T. Müller. Polynomial time computation of Taylor series. In Proceedings of the 22th JAIIO - Panel’93, Part 2, pages 259-281, 1993. Buenos Aires, 1993.

    Google Scholar 

  76. J. Myhill. A recursive function defined on a compact interval and having a continuous derivative that is not recursive. Michigan Math. J., 18:97-98, 1971.

    MATH  MathSciNet  Google Scholar 

  77. C. A. Neff and J. H. Reif. An efficient algorithm for the complex roots problem. Journal of Complexity, 12:81-115, 1996.

    MATH  MathSciNet  Google Scholar 

  78. V. Orevkov. A constructive mappping of the square onto itself displacing every construc-tive point (Russian). Doklady Akademii Nauk, 152:55-58, 1963. Translated in: Soviet Math. - Dokl., 4 (1963) 1253-1256.

    MathSciNet  Google Scholar 

  79. M. B. Pour-El and J. I. Richards. The wave equation with computable inital data such that its unique solution is not computable. Advances in Math., 39:215-239, 1981.

    MATH  MathSciNet  Google Scholar 

  80. M. B. Pour-El and J. I. Richards. Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin, 1989.

    MATH  Google Scholar 

  81. 81. R. Rettinger. A fast algorithm for Julia sets of hyperbolic rational functions. In V. Brattka, L. Staiger, and K. Weihrauch, editors, Proceedings of the 6th Workshop on Com-putability and Complexity in Analysis, volume 120 of Electronic Notes in Theoretical Computer Science, pages 145-157, Amsterdam, 2005. Elsevier. 6th International Workshop, CCA 2004, Wittenberg, Germany, August 16-20, 2004.

    Google Scholar 

  82. R. Rettinger and K. Weihrauch. The computational complexity of some Julia sets. In M. X. Goemans, editor, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages 177-185, New York, 2003. ACM Press. San Diego, California, June 9-11, 2003.

    Google Scholar 

  83. H. Rice. Recursive real numbers. Proc. Amer. Math. Soc., 5:784-791, 1954.

    MATH  MathSciNet  Google Scholar 

  84. H. Rogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  85. N. Šanin. Constructive Real Numbers and Constructive Function Spaces, volume 21 of Translations of Mathematical Monographs. American Mathematical Society, Provi-dence, 1968.

    Google Scholar 

  86. M. Schröder. Topological spaces allowing type 2 complexity theory. In K.-I. Ko and K. Weihrauch, editors, Computability and Complexity in Analysis, volume 190 of Informatik Berichte, pages 41-53. FernUniversität Hagen, 1995. CCA Workshop, Hagen, August 19-20, 1995.

    Google Scholar 

  87. M. Schröder. Fast online multiplication of real numbers. In R. Reischuk and M. Morvan, editors, STACS 97, volume 1200 of Lecture Notes in Computer Science, pages 81-92, Berlin, 1997. Springer. 14th Annual Symposium on Theoretical Aspects of Computer Science, Lübeck, Germany, February 27-March 1, 1997.

    Google Scholar 

  88. M. Schröder. Online computations of differentiable functions. Theoretical Computer Science, 219:331-345, 1999.

    MATH  MathSciNet  Google Scholar 

  89. M. Schröder. Admissible representations of limit spaces. In J. Blanck, V. Brattka, and P. Hertling, editors, Computability and Complexity in Analysis, volume 2064 of Lecture Notes in Computer Science, pages 273-295, Berlin, 2001. Springer. 4th International Workshop, CCA 2000, Swansea, UK, September 2000.

    Google Scholar 

  90. M. Schröder. Effectivity in spaces with admissible multirepresentations. Mathematical Logic Quarterly, 48(Suppl. 1):78-90, 2002.

    MATH  MathSciNet  Google Scholar 

  91. M. Schröder. Extended admissibility. Theoretical Computer Science, 284(2):519-538, 2002.

    MATH  MathSciNet  Google Scholar 

  92. M. Schröder. Spaces allowing type-2 complexity theory revisited. Mathematical Logic Quarterly, 50(4,5):443-459, 2004.

    MATH  Google Scholar 

  93. A. Slisenko. Examples of a nondiscontinuous but not continuous constructive operator in a metric space. Trudy Mat. Inst. Steklov, 72:524-532, 1964.(in Russian, English trans. in AMS Trans. 100, 1972).

    MATH  MathSciNet  Google Scholar 

  94. S. Smale. On the topology of algorithms, I. Journal of Complexity, 3:81-89, 1987.

    MATH  MathSciNet  Google Scholar 

  95. R. Soare. Cohesive sets and recursively enumerable Dedekind cuts. Pacific J. Math., 31:215-231, 1969.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  97. E. Specker. The fundamental theorem of algebra in recursive analysis. In B. Dejon and P. Henrici, editors, Constructive Aspects of the Fundamental Theorem of Algebra, pages 321-329, London, 1969. Wiley-Interscience.

    Google Scholar 

  98. V. Stoltenberg-Hansen, I. Lindström, and E. Griffor. Mathematical Theory of Domains, volume 22 of Cambrige Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1994.

    Google Scholar 

  99. J. F. Traub, G. Wasilkowski, and H. Wo zniakowski. Information-Based Complexity. Computer Science and Scientific Computing. Academic Press, New York, 1988.

    MATH  Google Scholar 

  100. A. Troelstra and D. v. Dalen. Constructivism in Mathematics, Volume 1, volume 121 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1988.

    Google Scholar 

  101. A . Troelstra and D. v. Dalen. Constructivism in Mathematics, Volume 2, volume 123 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1988.

    Google Scholar 

  102. A. M. Turing. On computable numbers, with an application to the “Entscheidungsproblem”. Proceedings of the London Mathematical Society, 42(2):230-265, 1936.

    MATH  Google Scholar 

  103. A. M. Turing. On computable numbers, with an application to the “Entscheidungsproblem”. A correction. Proceedings of the London Mathematical Society, 43(2):544-546, 1937.

    MATH  Google Scholar 

  104. V. Vassiliev. Cohomology of braid groups and the complexity of algorithms. Funktsional. Anal. i Prilozhen., 22(3):15 - 24, 1989. Englische Übers. in Functional. Anal. Appl., 22:182-190, 1989.

    Google Scholar 

  105. M. Washihara. Computability and Fréchet spaces. Mathematica Japonica, 42(1):1-13, 1995.

    MATH  MathSciNet  Google Scholar 

  106. K. Weihrauch. Computability, volume 9 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1987.

    Google Scholar 

  107. K. Weihrauch. On the complexity of online computations of real functions. Journal of Complexity, 7:380-394, 1991.

    MATH  MathSciNet  Google Scholar 

  108. K. Weihrauch. The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen, Sept. 1992.

    Google Scholar 

  109. K. Weihrauch. Computable Analysis. Springer, Berlin, 2000.

    MATH  Google Scholar 

  110. K. Weihrauch. Computational complexity on computable metric spaces. Mathematical Logic Quarterly, 49(1):3-21, 2003.

    MATH  MathSciNet  Google Scholar 

  111. K. Weihrauch and N. Zhong. Is wave propagation computable or can wave computers beat the Turing machine? Proceedings of the London Mathematical Society, 85(2):312-332,2002.

    MATH  MathSciNet  Google Scholar 

  112. H. Weyl. Randbemerkungen zu Hauptproblemen der Mathematik. Math. Zeitschrift, 20:131-150, 1924.

    MathSciNet  Google Scholar 

  113. H. Wo zniakowski. Why does information-based complexity use the real number model? Theoretical Computer Science, 219:451-465, 1999.

    MathSciNet  Google Scholar 

  114. M. Yasugi, T. Mori, and Y. Tsujii. Effective properties of sets and functions in metric spaces with computability structure. Theoretical Computer Science, 219:467-486, 1999.

    MATH  MathSciNet  Google Scholar 

  115. X. Zheng. Recursive approximability of real numbers. Mathematical Logic Quarterly, 48(Suppl. 1):131-156, 2002.

    MATH  Google Scholar 

  116. X. Zheng and R. Rettinger. Weak computability and representation of reals. Mathematical Logic Quarterly, 50(4,5):431-442, 2004.

    MATH  MathSciNet  Google Scholar 

  117. X. Zheng and K. Weihrauch. The arithmetical hierarchy of real numbers. Mathematical Logic Quarterly, 47(1):51-65, 2001.

    MATH  MathSciNet  Google Scholar 

  118. N. Zhong. Recursively enumerable subsets of Rq in two computing models: BlumShub-Smale machine and Turing machine. Theoretical Computer Science, 197:79-94, 1998.

    MATH  MathSciNet  Google Scholar 

  119. N. Zhong and K. Weihrauch. Computability theory of generalized functions. Journal of the Association for Computing Machinery, 50(4):469-505, 2003.

    MathSciNet  Google Scholar 

  120. M. Ziegler and V. Brattka. Computability in linear algebra. Theoretical Computer Science, 326(1-3):187-211, 2004.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Brattka, V., Hertling, P., Weihrauch, K. (2008). A Tutorial on Computable Analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) New Computational Paradigms. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68546-5_18

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-68546-5_18

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-36033-1

  • Online ISBN: 978-0-387-68546-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics