Lowness properties and approximations of the jump

• Autores: Figueira, S.; Nies, A.; Stephan, F.
• Año de publicación: 2008
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Ann. Pure Appl. Logic 2008;152(1-3):51-66
• Materia: ω-r.e.; K-triviality; Kolmogorov complexity; Lowness; Traceability
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.