Computational complexity on computable metric spaces

Mathematical Logic Quarterly

Volumn 49, Issue 1, 2003, Pages 3-21

  Weirauch, Klaus ^a  

^a UNIVERSITY OF HAGEN   (Germany)

Author keywords

Admissible representation; Computable metric space; Computational complexity

Indexed keywords

EID: 0037280374     PISSN: 09425616     EISSN: None     Source Type: Journal    
DOI: 10.1002/malq.200310001     Document Type: Article

References (42)

1. H. Alt, Multiplication is the easiest nontrivial arithmetic function. Theor. Comp. Science 36, 333-339 (1985).
2. ∞-functions. In: Computability and Complexity in Analysis, CA Workshop Hagen, August 19-20, 1995 (Ker-I Ko and K. Weihrauch, eds.), Informatik Berichte 190, FernUniversität Hagen, September 1995, pp. 99-104.
3. L. Blum, L., F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation (Springer-Verlag, New York 1998).
4. V. Brattka and G. Presse, Computability on subsets of metric spaces. Theor. Comp. Science (accepted for publication).
5. V. Brattka and K. Weihrauch, Computability on subsets of Euclidean space I: Closed and compact subsets. Theor. Comp. Science 219, 65-93 (1999).
6. R. P. Brent, Fast multiple-precision evaluation of elementary functions. J. Assoc. Comp. Mach. 23, 242-251 (1976).
7. P. Bürgisser, M. Clausen, and M. Amin Shokrollahi, Algebraic Complexity Theory (Springer-Verlag, Berlin 1997).
8. S. A. Cook, Computability and complexity of higher type functions. In: Logic from Computer Science, Proceedings of the workshop held in Berkeley, California, November 13-17, 1989 (Y. N. Moschovakis, ed.), pp. 51-72. Mathematical Sciences Research Institute Publications, Vol. 21 (Springer-Verlag, New York 1992).
9. M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry: Algorithms and Applications, 2nd edition (Springer-Verlag, Berlin 2000).
10. J. Dieudonné, Foundations of Modern Analysis (Academic Press, New York 1960).
11. A. Edalat, Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbolic Logic 3, 401-452 (1997).
12. A. Grzegorczyk, Computable functionals. Fund. Math. 42, 168-202 (1955).
13. J. Hauck, Konstruktive Darstellungen reeller Zahlen und Folgen. Z. Math. Logik Grundlagen Math. 24, 365-374 (1978).
14. A. Hemmerling, On the time complexity of partial real functions. J. Complexity 16, 363-376 (2000).
15. H. J. Hoover, Feasible real functions and arithmetic circuits. SIAM J. Computing 19, 182-204 (1990).
16. H. Kamo and K. Kawamura, Computability of self-similar sets. Math. Logic Quarterly 45, 23-30 (1999).
17. H. Kamo, K. Kawamura, and I. Takeuti, Computational complexity of fractal sets. Real Analysis Exchange 26, 773-793 (2000/01).
18. B. M. Kapron, Feasibly continuous type-two functionals. Computational Complexity 8, 188-201 (1999).
19. Ker-I Ko, Complexity Theory of Real Functions (Birkhäuser, Boston 1991).
20. Ker-I Ko and K. Weihrauch, On the measure of two-dimensional regions with polynomial-time computable boundaries. In: Eleventh Annual IEEE Conference on Computational Complexity (St. Homer and Jin-Yi Cai, eds.), pp. 150-159 (IEEE Computer Society Press, Los Alamitos 1996).
21. C. Kreitz and K. Weihrauch, Theory of representations. Theor. Comp. Science 38, 35-53 (1985).
22. S. Labhalla, H. Lombardi, and E. Moutai, Espaces métriques rationnellement présentés et complexité, le mas de l'espace des fonctions réelles uniformément continues sur un intervalle compact. Theor. Comp. Science 250, 265-332 (2001).
23. K. Mehlhorn, Polynomial and abstract subrecursive classes. J. Comp. Syst. Science 12, 147-178 (1976).
24. N. Th. Müller, Subpolynomial complexity classes of real functions and real numbers. In: Proceedings 13th International Colloquium on Automata, Languages, and Programming (Laurent Kott, ed.), Lecture Notes in Computer Science, Vol. 226, pp. 284-293 (Springer-Verlag, Berlin 1986).
25. N. Th. Müller, Uniform computational complexity of Taylor series. In: Proceedings 14th International Colloquium on Automata, Languages, and Programming (Th. Ottmann, ed.), Lecture Notes in Computer Science, Vol. 267, pp. 435-444 (Springer-Verlag, Berlin 1987).
26. N. Th. Müller and B. Moiske, Solving initial value problems in polynomial time. In: Proceedings 22th JAIIO - Panel'93, Part 2, Buenos Aires 1993, pp. 283-293.
27. C. A. Neff and J. H. Reif, An efficient algorithm for the complex roots problem. J. Complexity 12, 81-115 (1996).
28. M. B. Pour-El and J. I. Richards, Computability in Analysis and Physics. Perspectives in Mathematical Logic (Springer-Verlag, Berlin 1989).
29. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, Vol. 44 (Cambridge University Press, Cambridge 1993).
30. A. Schönhage, A. F. W. Grotefeld, and E. Vetter, Fast Algorithms (B.I. Wissenschaftsverlag, Mannheim 1994).
31. A. Schönhage, Numerik analytischer Funktionen und Komplexität. Jahresbericht DMV 92, 1-20 (1990).
32. M. Schröder, Welche topologischen Räume erlauben Typ 2-Komplexitätstheorie? Informatik Berichte 178, FernUniversität Hagen, Hagen, May 1995.
33. M. Schröder, Fast online multiplication of real numbers. In: 14th Annual Symposium on Theoretical Aspects of Computer Science, Lübeck, Germany, February 27-March 1, 1997 (R. Reischuk and M. Morvan eds.), Lecture Notes in Computer Science, Vol. 1200, pp. 81 - 92 (Springer-Verlag, Berlin 1997).
34. M. Schröder, Online computations of differentiable functions. Theor. Comp. Science 219, 331-345 (1999).
35. M. Townsend, Complexity for type-2 relations. Notre Dame J. Formal Logic 31, 241-262 (1990).
36. J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski, Information-Based Complexity. Computer Science and Scientific Computing (Academic Press, New York 1988).
37. P. Walters, An Introduction to Ergodic Theory (Springer-Verlag, New York 1982).
38. K. Weihrauch, On the complexity of online computations of real functions. J. Complexity 7, 380-394 (1991).
39. K. Weihrauch, Computability on computable metric spaces. Theor. Comp. Science 113, 191-210 (1993).
40. K. Weihrauch, Computable Analysis (Springer-Verlag, Berlin 2000).
41. K. Weihrauch and Ch. Kreitz, Type 2 computational complexity of functions on Cantor's space. Theor. Comp. Science 82, 1-18 (1991).
42. E. Wiedmer, Computing with infinite objects. Theor. Comp. Science 10, 133-155 (1980).