Lowness properties and approximations of the jump

Abstract 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 T e of possible values for the jump J A ( 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.

[1]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[2]  Rodney G. Downey,et al.  Presentations of computably enumerable reals , 2002, Theor. Comput. Sci..

[3]  Ludomir Newelski,et al.  Logic Colloquium 2005 (Lecture Notes in Logic) , 2007 .

[4]  Wolfgang Merkle,et al.  Kolmogorov Complexity and the Recursion Theorem , 2006, STACS.

[5]  Peter A. Cholak,et al.  STRONG JUMP-TRACEABILITY I : THE COMPUTABLY ENUMERABLE CASE , 2008 .

[6]  Guohua Wu,et al.  Randomness and universal machines , 2006, J. Complex..

[7]  Paul M. B. Vitányi,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1993, Graduate Texts in Computer Science.

[8]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[9]  ComplexityValentine KabanetsDecember.  Randomness and Complexity , 1997 .

[10]  Jeanleah Mohrherr.  A Refinement of Low n and High n for the R.E. Degrees , 1986, Math. Log. Q..

[11]  Qiu Yu Hui,et al.  Proceedings of the 7th and 8th Asian Logic Conferences , 2003 .

[12]  Gregory J. Chaitin.  Information-Theoretic Characterizations of Recursive Infinite Strings , 1976, Theor. Comput. Sci..

[13]  N. Cutland.  Computability: An Introduction to Recursive Function Theory , 1980 .

[14]  Robert I. Soare,et al.  An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees , 1984 .

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

[16]  A. Nies.  Lowness properties and randomness , 2005 .

[17]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[18]  André Nies,et al.  Reals which Compute Little , 2002 .

[19]  André Nies,et al.  Relativizing Chaitin's Halting Probability , 2005, J. Math. Log..

[20]  Bakhadyr Khoussainov,et al.  Recursively enumerable reals and Chaitin Ω numbers , 1998 .

[21]  Sebastiaan Terwijn,et al.  Computational randomness and lowness* , 2001, Journal of Symbolic Logic.