We show that if a set A is computable from every superlow 1-random set,
then A 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 c with the limit condition
there is a 1-random Δ20​ set Y such that every c.e.\ set A≤T​Y
obeys c. 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 Δ20​ 1-random sets.Comment: 41 page