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. Informally, a set A is strongly jumptraceable 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.

• Type: Journal
• Year: 2008
• Where: APAL
• Authors: Santiago Figueira, André Nies, Frank Stephan
• Keywords: APAL 2008, Combinatorial Lowness Notions, Order Function, Strong Jump-traceability