Lowness properties and approximations of the jump
We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informa...
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todo:paper_01680072_v152_n1-3_p51_Figueira2023-10-03T15:05:40Z Lowness properties and approximations of the jump Figueira, S. Nies, A. Stephan, F. ω-r.e. K-triviality Kolmogorov complexity Lowness Traceability We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA (e), and the number of values enumerated is at most h (e). A′ is well-approximable if can be effectively approximated with less than h (x) changes at input x, for each order function h. We prove that there is a strongly jump-traceable set which is not computable, and that if A′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and strong jump-traceability in terms of Kolmogorov complexity. We also investigate other properties of these lowness properties. © 2007 Elsevier B.V. All rights reserved. Fil:Figueira, S. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_01680072_v152_n1-3_p51_Figueira |
| institution |
Universidad de Buenos Aires |
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I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
ω-r.e. K-triviality Kolmogorov complexity Lowness Traceability |
| spellingShingle |
ω-r.e. K-triviality Kolmogorov complexity Lowness Traceability Figueira, S. Nies, A. Stephan, F. Lowness properties and approximations of the jump |
| topic_facet |
ω-r.e. K-triviality Kolmogorov complexity Lowness Traceability |
| description |
We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA (e), and the number of values enumerated is at most h (e). A′ is well-approximable if can be effectively approximated with less than h (x) changes at input x, for each order function h. We prove that there is a strongly jump-traceable set which is not computable, and that if A′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and strong jump-traceability in terms of Kolmogorov complexity. We also investigate other properties of these lowness properties. © 2007 Elsevier B.V. All rights reserved. |
| format |
JOUR |
| author |
Figueira, S. Nies, A. Stephan, F. |
| author_facet |
Figueira, S. Nies, A. Stephan, F. |
| author_sort |
Figueira, S. |
| title |
Lowness properties and approximations of the jump |
| title_short |
Lowness properties and approximations of the jump |
| title_full |
Lowness properties and approximations of the jump |
| title_fullStr |
Lowness properties and approximations of the jump |
| title_full_unstemmed |
Lowness properties and approximations of the jump |
| title_sort |
lowness properties and approximations of the jump |
| url |
http://hdl.handle.net/20.500.12110/paper_01680072_v152_n1-3_p51_Figueira |
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AT figueiras lownesspropertiesandapproximationsofthejump AT niesa lownesspropertiesandapproximationsofthejump AT stephanf lownesspropertiesandapproximationsofthejump |
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