Abstract
In this paper we investigate aspects of effectivity and computability on closed and compact subsets of locally compact spaces. We use the framework of the representation approach, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. This work is a generalization of the concepts introduced in [4] and [22] for the Euclidean case and in [3] for metric spaces. Whenever reasonable, we transfer a representation of the set of closed or compact subsets to locally compact spaces and discuss its properties and their relations to each other.
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References
Brattka, V.: Recursive quasi-metric spaces. Theoret. Comput. Sci. 305, 17–42 (2003)
Brattka, V., Hertling, P.: Continuity and computability of relations. Informatik Berichte 164, Fern University in Hagen, Hagen (1994)
Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theoret. Comput. Sci. 305, 43–76 (2003)
Brattka, V., Weihrauch, K.: Computability on subsets of Euclidean space I: Closed and compact subsets. Theoret. Comput. Sci. 219, 65–93 (1999)
Collins, P.: Continuity and computability on reachable sets. Theoret. Comput. Sci. 341, 162–195 (2005)
Engelking, R.: General Topology. Heldermann, Berlin (1989)
Grzegorczyk, A.: Computable functions. Fund. Math. 42, 168–202 (1955)
Grubba, T., Weihrauch, K.: A computable version of Dini’s theorem for topological spaces. In: Yolum, p., et al. (eds.) ISCIS 2005. LNCS, vol. 3733, pp. 927–936. Springer, Heidelberg (2005)
Grubba, T., Weihrauch, K.: On computable metrization. CCA 2006, Third International Conference on Computability and Complexity in Analysis. In: Cenzer, D., et al. (eds.) Informatik Berichte 333, pp. 176–191. Fern University in Hagen, Hagen (2006)
Hauck, J.: Berechenbare reelle Funktionen. Z. math. Logik Grundlagen Math. 19, 121–140 (1973)
Ko, K.I.: Complexity Theory of Real Functions. Birkhaeuser, Boston (1991)
Kreitz, C., Weihrauch, K.: A unified approach to constructive and recursive analysis. In: Richter, M., et al. (eds.) Computation and Proof Theory. Lecture Notes in Mathematics, vol. 1104, pp. 259–278. Springer, Berlin (1984)
Kusner, B.A.: Lectures on Constructive Mathematical Analysis vol. 60. AMS, Providence (1984)
Lacombe, D.: Extension de la notion de fonction recursive aux fonctions d’une ou plusieurs variables reelles I. Comptes Rendus Academie des Sciences Paris 240, 2478–2480 (1955)
Mazur, S.: Computable Analysis vol. 33. Razprawy Matematyczne, Warsaw (1963)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)
Schröder, M.: Effective metrization of regular spaces. In: Ko, K.I., et al. (eds.) Computability and Complexity in Analysis. Informatik Berichte, vol. 235, pp. 63–80. Fern University in Hagen, Hagen (1998)
Scott, D.: Outline of a mathematical theory of computation. Tech. Mono. PRG-2. Oxford University, Oxford (1970)
Stoltenberg-Hansen, V., Tucker, J.V.: Concrete models of computation for topological algebras. Theoret. Comput. Sci. 219, 347–378 (1999)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. 42, 230–265 (1936)
Weihrauch, K.: Computability on computable metric spaces. Theoret. Comput. Sci. 113, 191–210 (1993)
Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)
Weihrauch, K.: Multi-functions on multi-represented sets are closed under flowchart programming. CCA 2005, Second International Conference on Computability and Complexity in Analysis. In: Grubba, T., et al. (eds.) Informatik Berichte 326, pp. 267–300. Fern University in Hagen, Hagen (2005)
Yasugi, M., Mori, T., Tsujii, Y.: Effective properties of sets and functions in metric spaces with computability structure. Theoret. Comput. Sci. 219, 467–486 (1999)
Zhong, N., Weihrauch, K.: Computability theory of generalized functions. J. Asso. for Comp. Mach. 50(4), 469–505 (2003)
Zhou, Q.: Computable real-valued functions on recursive open and closed subsets of Euclidean space. Math. Logic Q. 42, 379–409 (1996)
Ziegler, M.: Computability on regular subsets of Euclidean space. Math. Logic Q. 48(Suppl. 1), 157–181 (2002)
Ziegler, M.: Computable operators on regular sets. Math. Logic Q. 50(4-5), 392–404 (2004)
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Xu, Y., Grubba, T. (2007). Computability on Subsets of Locally Compact Spaces. In: Cai, JY., Cooper, S.B., Zhu, H. (eds) Theory and Applications of Models of Computation. TAMC 2007. Lecture Notes in Computer Science, vol 4484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72504-6_9
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DOI: https://doi.org/10.1007/978-3-540-72504-6_9
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