17 research outputs found
Characterizing the strongly jump-traceable sets via randomness
We show that if a set is computable from every superlow 1-random set,
then is strongly jump-traceable. This theorem shows that the computably
enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets
computable from every superlow 1-random set.
We also prove the analogous result for superhighness: a c.e.\ set is strongly
jump-traceable if and only if it is computable from every superhigh 1-random
set.
Finally, we show that for each cost function with the limit condition
there is a 1-random set such that every c.e.\ set
obeys . To do so, we connect cost function strength and the strength of
randomness notions. This result gives a full correspondence between obedience
of cost functions and being computable from 1-random sets.Comment: 41 page
Calibrating the complexity of Delta 2 sets via their changes
The computational complexity of a Delta 2 set will be calibrated by the
amount of changes needed for any of its computable approximations. Firstly, we
study Martin-Loef random sets, where we quantify the changes of initial
segments. Secondly, we look at c.e. sets, where we quantify the overall amount
of changes by obedience to cost functions. Finally, we combine the two
settings. The discussions lead to three basic principles on how complexity and
changes relate
Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness
We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr
randomness (which we will call uniformly relative Schnorr randomness). An
immediate corollary is one direction of van Lambalgen's theorem for Schnorr
randomness. It has been claimed in the literature that this corollary (and the
analogous result for computable randomness) is a "straightforward modification
of the proof of van Lambalgen's Theorem." This is not so, and we point out why.
We also point out an error in Miyabe's proof of van Lambalgen's Theorem for
truth-table reducible randomness (which we will call uniformly relative
computable randomness). While we do not fix the error, we do prove a weaker
version of van Lambalgen's Theorem where each half is computably random
uniformly relative to the other