Abstract
In this paper we extend computability theory to the spaces of continuous, upper and lower semi-continuous real functions. We apply the framework of TTE, Type-2 Theory of Effectivity, where not only computable objects but also computable functions on the spaces can be considered. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on different intiuitive concepts of “effectivity” and prove their equivalence. Computability of several operations on the function spaces is investigated, among others limits, mappings to open sets, images of compact sets and preimages of open sets, maximum and minimum values. The positive results usually show computability in all arguments, negative results usually express non-continuity. Several of the problems have computable but not extensional solutions. Since computable functions map computable elements to computable elements, many previously known results on computability are obtained as simple corollaries.
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References
A. Brown & C. Pearcy Introduction to Analysis. Springer-Verlag, Berlin, Heidelberg, 1995.
R. Engelking General Topology, Heldermann Verlag, Berlin, 1989.
X. Ge & A. Nerode Effective content of the calculus of variations I: semicontinuity and the chattering lemma. Annals of Pure and Applied Logic, 78 (1996), 127–146.
Ker-I Ko Complexity Theory of Real Functions, Birkhäuser, Berlin, 1991.
Ch. Kreitz & K. Weihrauch Theory of representations, Theoret. Comput. Sci. 38 (1985), 35–53.
Ch. Kreitz & K. Weihrauch Compactness in constructive analysis, Annals of Pure and Applied Logic 36 (1987), 29–38.
M. Pour-El & J. Richards Computability in Analysis and Physics. Springer-Verlag, Berlin, Heidelberg, 1989.
A. Reiser & K. Weihrauch Natural numberings and generalized computability, Elektronische Informationsverarbeitung und Kybernetik 16 (1980)1–3, 11–20.
H. Jr. Rogers Theory of Recursive Functions and Effective Computability McGraw-Hill, Inc. New York, 1967.
J. Ian Richards & Q. Zhou Computability of closed and open sets in Euclidean space, Preprint, 1992.
H. TongSome characterizations of normal and perfectly normal space, Duke Math. J. 19 (1952), 289–292.
K. Weihrauch Type-2 recursion theory. Theoret. Comput. Sci. 38 (1985), 17–33.
K. Weihrauch Computability. EATCS Monographs on Theoretical Computer Secience Vol. 9, Springer-Verlag, Berlin, Heidelberg, 1987.
K. Weihrauch Effektive Analysis. Lecture Notes for Corresponding Course, Fern-Universität Hagen, 1994.
K. Weihrauch A foundation for computable analysis. in Douglas S. Bridges, Cristian S. Calude, Jeremy Gibbons, Steves Reeves and Ian H. Witten (eds.), Combinatorics, Complexity, Logic, Proceedings of DMTCS'96, pp 66–89, Springer-Verlag, Sigapore, 1996.
K. Weihrauch & Ch. Kreitz Representations of the real numbers and of the open subsets of the real numbers, Annals of Pure and Applied Logic 35 (1987), 247–260.
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© 1997 Springer-Verlag Berlin Heidelberg
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Weihrauch, K., Zheng, X. (1997). Computability on continuous, lower semi-continuous and upper semi-continuous real functions. 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/BFb0045083
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DOI: https://doi.org/10.1007/BFb0045083
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