Abstract
In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for non-computability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is sufficiently discontinuous and sufficiently effective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf Pour-El and Richards (cf. [18]). We apply our theorem to prove that several set-valued operators are not computably invariant.
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References
V. Brattka, Grade der Nichtstetigkeit in der Analysis, Diplomarbeit, Fern Univer-sität Hagen (1993)
V. Brattka, Recursive characterization of computable real-valued functions and relations, Theoretical Computer Science 162 (1996) 45–77
V. Brattka, Computable selection in analysis, in: K.-I Ko & K. Weihrauch, eds., Proc. of the Workshop on Computability and Complexity in Analysis, Informatik Berichte, Fern Universität Hagen 190 (1995) 125–138
V. Brattka, Order-free recursion on the real numbers, Mathematical Logic Quarterly 43 (1997)
A. Chou, K.-I Ko, Computational complexity of two-dimensional regions, SIAM J. on Comput. 24 (1995) 923–947
R. Engelking, General Topology, Heldermann, Berlin (1989)
A. Grzegorczyk, On the definition of computable real continuous functions, Fund. Math. 44 (1957) 61–71
P. Hertling, Topologische Komplexitätsgerade von Funktionen mit endlichem Bild, Informatik-Berichte, Fern Universität Hagen 152 (1993)
K.-I Ko, Complexity Theory of Real Functions, Birkhäuser, Boston (1991)
K.-I Ko, A polynomial-time computable curve whose interior has a nonrecursive measure, Theoretical Computer Science 145 (1995) 241–270
K.-I Ko & K. Weihrauch, On the measure of two-dimensional regions with polynomial-time computable boundaries, in: Porc. of the 11 th IEEE Conference on Computational Complexity, to appear (1996)
K.-I Ko, On the complexity of fractal dimensions and Julia sets, Theoretical Computer Science, submitted (1996)
K. Kuratowski, Topology, Vol. I&II, Academic Press, New York (1966, 1968)
Ch. Kreitz & K. Weihrauch, Compactness in constructive analysis revisited, Annals of Pure and Applied Logic 36 (1987) 29–38
D. Lacombe, Extension de la notion de fonction recursive aux fonctions d'une ou plusieurs variables relles I-III, Comptes Rendus 240/241 (1955) 2478–2480/13-14,151–153,1250–1252
J.R. Myhill, A recursive function, defined on a compact interval and having a continuous derivative that is not recursive, Michigan Math. J. 18 (1971) 97–98
U. Mylatz, Vergleich unstetiger Funktionen in der Analysis, Diplomarbeit, FernUniversität Hagen (1992)
M.B. Pour-El & J. Richards, Computability in Analysis and Physics, Springer, Berlin (1989)
Th. von Stein, Vergleich nicht konstruktiv lösbarer Probleme in der Analysis, Diplomarbeit, Fern Universität Hagen (1989)
K. Weihrauch, Computability, Springer, Berlin (1987)
K. Weihrauch, The degrees of discontinuity of some Translators between representations of the real numbers, Informatik-Berichte, FernUniversität Hagen 129 (1992)
K. Weihrauch, The TTE-interpretation of three hierarchies of omniscience principles, Informatik-Berichte, FernUniversität Hagen 130 (1992)
K. Weihrauch, Computability on computable metric spaces, Theoretical Computer Science 113 (1993) 191–210
K. Weihrauch, A Simple Introduction to Computable Analysis, Informatik Berichte 171, FernUniversität Hagen (1995)
K. Weihrauch & Ch. Kreitz, Representations of the real numbers and of the open subsets of the set of real numbers, Annals of Pure and Applied Logic 35 (1987) 247–260
Q. Zhou, Computable real-valued functions on recursive open and closed subsets of Euclidean space, Mathematical Logic Quarterly 42 (1996) 379–409
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© 1997 Springer-Verlag Berlin Heidelberg
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Brattka, V. (1997). Computable invariance. In: Jiang, T., Lee, D.T. (eds) Computing and Combinatorics. COCOON 1997. Lecture Notes in Computer Science, vol 1276. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0045081
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DOI: https://doi.org/10.1007/BFb0045081
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