11 research outputs found
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
Lowness and Î nullsets
We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Lof randomness
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
K-Triviality and Computable Measures
Algorithmic randomness is primarily concerned with quantifying the degree of randomness of infinite binary strings, and is usually carried out in the setting of Cantor space. One characterization of randomness involves prefixes ``being as hard as possible to describe . Also of interest are the infinite binary strings whose prefixes are as easy as possible to describe i.e., the -trivial strings. We will study these strings in the setting of computable metric spaces, and investigate several definitions which attempt to correctly generalize -triviality. We describe some of the difficulties inherent in a natural-seeming approach, and offer partial results where new definitions relate to a more established definition of -triviality under the right conditions