Computability on Regular Subsets of Euclidean Space

For the computability of subsets of real numbers, several reasonable notions have been suggested in the literature. We compare these notions in a systematic way by relating them to pairs of 'basic' ones. They turn out to coincide for full-dimensional convex sets; but on the more general class of regular sets, they reveal rather interesting 'weaker/stronger' relations. This is in contrast to single real numbers and vectors where all 'reasonable' notions coincide. Mathematics Subject Classification: 03F60,51M04,54H05,65D18.

[1]  Elham Kashefi,et al.  The convex hull in a new model of computation , 2001, CCCG.

[2]  Vasco Brattka,et al.  Computability on subsets of metric spaces , 2003, Theor. Comput. Sci..

[3]  Klaus Weihrauch,et al.  Computability on Subsets of Euclidean Space I: Closed and Compact Subsets , 1999, Theor. Comput. Sci..

[4]  Thomas Deil,et al.  Darstellungen und Berechenbarkeit reeller Zahlen , 1983 .

[5]  Boris A. Kushner,et al.  Markov's Constructive Analysis: A Participant's View , 1999, Theor. Comput. Sci..

[6]  Martin Kummer,et al.  Computability of Convex Sets (Extended Abstract) , 1995, STACS.

[7]  Martin Ziegler,et al.  Turing computability of (non-)linear optimization , 2001, CCCG.

[8]  Matthias Schröder,et al.  Admissible Representations of Limit Spaces , 2000, CCA.

[9]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[10]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[11]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[12]  Peter Hertling,et al.  Topological properties of real number representations , 2002, Theor. Comput. Sci..

[13]  Peter Hertling,et al.  The Effective Riemann Mapping Theorem , 1999, Theor. Comput. Sci..

[14]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[15]  Anil Nerode,et al.  On Extreme Points of Convex Compact Turing Located Set , 1994, LFCS.

[16]  Peter Hertling.  A Comparison of Certain Representations of Regularly Closed Sets , 2002, Electron. Notes Theor. Comput. Sci..

[17]  Christoph Kreitz,et al.  Representations of the real numbers and of the open subsets of the set of real numbers , 1987, Ann. Pure Appl. Log..

[18]  Ernst Specker,et al.  Nicht konstruktiv beweisbare Sätze der Analysis , 1949, Journal of Symbolic Logic.

[19]  A. Grzegorczyk.  On the definitions of computable real continuous functions , 1957 .

[20]  Marian Boykan Pour-El,et al.  Computability in analysis and physics , 1989, Perspectives in Mathematical Logic.