Hierarchies of function classes defined by the first-value operator

Armin Hemmerling

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 42, Issue: 2, page 253-270
  • ISSN: 0988-3754

Abstract

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The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within Δ 2 0 . The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class  Δ 2 B . Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

How to cite

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Hemmerling, Armin. "Hierarchies of function classes defined by the first-value operator." RAIRO - Theoretical Informatics and Applications 42.2 (2007): 253-270. <http://eudml.org/doc/92870>.

@article{Hemmerling2007,
abstract = { The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within $\Delta^\{0\}_2$. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class $\Delta^\{B\}_2$. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis. },
author = {Hemmerling, Armin},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Hierarchies of functions; degree of discontinuity; computable analysis; effective descriptive set theory; Hausdorff hierarchy; Ershov hierarchy; hierarchies of functions},
language = {eng},
month = {12},
number = {2},
pages = {253-270},
publisher = {EDP Sciences},
title = {Hierarchies of function classes defined by the first-value operator},
url = {http://eudml.org/doc/92870},
volume = {42},
year = {2007},
}

TY - JOUR
AU - Hemmerling, Armin
TI - Hierarchies of function classes defined by the first-value operator
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/12//
PB - EDP Sciences
VL - 42
IS - 2
SP - 253
EP - 270
AB - The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value at any point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within $\Delta^{0}_2$. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class $\Delta^{B}_2$. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.
LA - eng
KW - Hierarchies of functions; degree of discontinuity; computable analysis; effective descriptive set theory; Hausdorff hierarchy; Ershov hierarchy; hierarchies of functions
UR - http://eudml.org/doc/92870
ER -

References

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