Abstract
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,” Birkhäuser, Boston, 1991] and Parker [Parker, M.W., Undecidability inRn: 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 Schröder [Schröder, 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.
Highlights
1.1 MotivationThe considerations in this article are inspired by real-world situations like the following: An agent has the task to perform a measurement ξ of a physical magnitude
We do not model the details of this process, so we can make no assumptions about what particular δ-name of ξ will be extracted from the measurement
We introduce severalrepresentations of Borel measures and prove a result on computable measures on computable metric spaces
Summary
The considerations in this article are inspired by real-world situations like the following: An agent (i.e. a person, a machine or a combination of such) has the task to perform a measurement ξ of a physical magnitude. A 2−kapproximation to the value f (ξ) shall be computed, where k ∈ N is a given precision parameter and f : X → Y is a given function that maps the state space X of the magnitude into a metric space (Y, d). When it comes to computations, the abilities of the agent shall be modeled by a Turing machine; so the results of the measurement must be available in machine readable form, i.e. encoded as a string over some finite alphabet Σ. All facts we use can be found in any introductory textbook; we occasionally refer to [Kallenberg 2002, Kechris 1995]
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