Computability on computable metric spaces
Theoretical Computer Science
Computable analysis: an introduction
Computable analysis: an introduction
A computable version of the Daniell-Stone theorem on integration and linear functionals
Theoretical Computer Science
Absolutely non-effective predicates and functions in computable analysis
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
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In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that are absolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.