Information-based complexity
Complexity theory of real functions
Complexity theory of real functions
Selected papers of the workshop on Topology and completion in semantics
Computability on random variables
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computability on the probability measures on the Borel sets of the unit interval
Theoretical Computer Science - Special issue on computability and complexity in analysis
Computable analysis: an introduction
Computable analysis: an introduction
Theoretical Computer Science
Dynamical Systems, Measures and Fractals Via Domain Theory
Proceedings of the First Imperial College Department of Computing Workshop on Theory and Formal Methods
Three concepts of decidability for general subsets of uncountable spaces
Theoretical Computer Science - Real numbers and computers
A computable version of the Daniell-Stone theorem on integration and linear functionals
Theoretical Computer Science
Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
Representing probability measures using probabilistic processes
Journal of Complexity
Admissible Representations of Probability Measures
Electronic Notes in Theoretical Computer Science (ENTCS)
Electronic Notes in Theoretical Computer Science (ENTCS)
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We define and compare several probabilistically weakened notions of computability for mappings from represented spaces (that are equipped with a measure or outer measure) into effective metric spaces. We thereby generalize definitions by Ko [Ko, K.-I., ''Complexity Theory of Real Functions,'' Birkhauser, Boston, 1991] and Parker [Parker, M.W., Undecidability inR^n: Riddled basins, the KAM tori, and the stability of the solar system, Philosophy of Science 70 (2003), pp. 359-382; Parker, M.W., Three concepts of decidability for general subsets of uncountable spaces, Theoretical Computer Science 351 (2006), pp. 2-13], and furthermore introduce the new notion of computability in the mean. Some results employ a notion of computable measure that originates in definitions by Weihrauch [Weihrauch, K., Computability on the probability measures on the Borel sets of the unit interval., Theoretical Computer Science 219 (1999), pp. 421-437] and Schroder [Schroder, M., Admissible representations of probability measures, Electronic Notes in Theoretical Computer Science 167 (2007), pp. 61-78]. In the spirit of the well-known Representation Theorem, we establish dependencies between the weakened computability notions and classical properties of mappings. We finally present some positive results on the computability of vector-valued integration on metric spaces, and discuss certain measurability issues arising in connection with our definitions.