JUCS - Journal of Universal Computer Science 3(11): 1226-1240, doi: 10.3217/jucs-003-11-1226
Surjective Functions on Computably Growing Cantor Sets
expand article infoPeter Hertling
‡ Department of Computer Science, The University of Auckland, Auckland, New Zealand
Open Access
Every infinite binary sequence is Turing reducible to a random one. This is a corollary of a result of Peter Gacs stating that for every co-r.e. closed set with positive measure of infinite sequences there exists a computable mapping which maps a subset of the set onto the whole space of infinite sequences. Cristian Calude asked whether in this result one can replace the positive measure condition by a weaker condition not involving the measure. We show that this is indeed possible: it is sufficient to demand that the co-r.e. closed set contains a computably growing Cantor set. Furthermore, in the case of a set with positive measure we construct a surjective computable map which is more effective than the map constructed by Gacs. 1 Proceedings of the First Japan-New Zealand Workshop on Logic in Computer Science, special issue editors D.S. Bridges, C.S. Calude, M.J. Dinneen and B. Khoussainov.
Computable maps on infinite sequences, co-r.e. closed sets, Cantor sets, computability and measure