an:06915722
Zbl 1415.03047
Downey, Rod; Stephenson, Jonathan
Avoiding effective packing dimension 1 below array noncomputable c.e. degrees
EN
J. Symb. Log. 83, No. 2, 717-739 (2018).
00411460
2018
j
03D32 03D28 68Q30
Kolmogorov complexity; effective packing dimension; array noncomputability
The present article is concerned with the notion of effective packing dimension when applied to Turing lower cones. \textit{L. Bienvenu} et al. [Theory Comput. Syst. 45, No. 4, 740--755 (2009; Zbl 1183.68281)] and \textit{L. Fortnow} et al. [Lect. Notes Comput. Sci. 4051, 335--345 (2006; Zbl 1223.68060)] have independently shown that when taking the supremum over the effective packing dimensions of all sets in the Turing lower cone of a set \(X\), then that number must either be 0 or 1. This behaviour is notable as it is in stark contrast to that observed by \textit{J. S. Miller} [Adv. Math. 226, No. 1, 373--384 (2011; Zbl 1214.03030)] for the closely related notion of effective Hausdorff dimension.
An obvious example for when the supremum takes the value 0 is if \(X\) is computable: then all sets in \(X\)'s lower cone are also computable, and all of them have effective packing dimension 0. For the value 1, there are two possible cases: The supremum might be an attained maximum over the lower cone; or it might be an unattained supremum. Clearly, lower cones of the first type exist: if \(X\) is Martin-L??f random, \(X\) has effective packing dimension 1. \textit{C. J. Conidis} [J. Symb. Log. 77, No. 2, 447--474 (2012; Zbl 1251.03047)] showed that lower cones of the second type exist as well.
A natural question was then to ask which lower cones exactly are of the second type. The authors precisely answer this question for lower cones that are below the halting problem. A result of \textit{M. Kummer} [SIAM J. Comput. 25, No. 6, 1123--1143 (1996; Zbl 0859.03015)] suggest that array noncomputability may have a role to play in this particular case. And indeed, within the c.e. sets, the present article characterizes the array noncomputable sets as exactly those below which a lower cone of the second type can be found.
The result is proved using pruned clumpy trees, building on the notion of clumpy trees introduced by \textit{R. Downey} and \textit{N. Greenberg} [Inf. Process. Lett. 108, No. 5, 298--303 (2008; Zbl 1191.68304)].
Rupert H??lzl (Neubiberg)
Zbl 1183.68281; Zbl 1223.68060; Zbl 1214.03030; Zbl 1251.03047; Zbl 0859.03015; Zbl 1191.68304