Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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

[2]  S. M. Srivastava,et al.  SELECTION THEOREMS AND INVARIANCE OF BOREL POINTCLASSES , 1986 .

[3]  Christoph Kreitz,et al.  Theory of Representations , 1985, Theor. Comput. Sci..

[4]  Douglas Cenzer,et al.  Index Sets in Computable Analysis , 1999, Theor. Comput. Sci..

[5]  K. Weihrauch.  The Degrees of Discontinuity of some Translators Between Representations of the Real Numbers , 1992 .

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

[7]  Douglas A. Cenzer,et al.  Effectively closed sets and graphs of computable real functions , 2002, Theor. Comput. Sci..

[8]  Matthias Schröder,et al.  Extended admissibility , 2002, Theor. Comput. Sci..

[9]  Peter Hertling,et al.  Unstetigkeitsgrade von Funktionen in der effektiven Analysis , 1996 .

[10]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[11]  Chun-Kuen Ho,et al.  Relatively Recursive Reals and Real Functions , 1994, Theor. Comput. Sci..

[12]  Vasco Brattka.  Computable Invariance , 1996, Theor. Comput. Sci..

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

[14]  Klaus Weihrauch,et al.  The Arithmetical Hierarchy of Real Numbers , 1999, MFCS.

[15]  Vasco Brattka,et al.  Recursive and computable operations over topological structures , 1999 .

[16]  K. Weihrauch.  The TTE-Interpretation of Three Hierarchies of Omniscience Principles , 1992 .

[17]  Vasco Brattka,et al.  Computability of Banach space principles , 2001 .

[18]  Vasco Brattka,et al.  Computability over Topological Structures , 2003 .

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

[20]  A. Kechris.  Classical descriptive set theory , 1987 .

[21]  P. Odifreddi.  Classical recursion theory , 1989 .

[22]  Armin Hemmerling,et al.  Approximate decidability in euclidean spaces , 2003, Math. Log. Q..

[23]  K. Weirauch.  Computational complexity on computable metric spaces , 2003 .