Demuth randomness and computational complexity

Abstract

a b s t r a c t Demuth tests generalize Martin-Löf tests (G m ) m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the G m . If we only allow Demuth tests such that G m ⊇ G m+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and ∆ 0 2 , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable.We also prove a basis theorem for non-empty Π 0 1 classes P. It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function.

Characterizing Lowness for Demuth Randomness

“…In [29] it is shown that a weakly Demuth random set is never superhigh. On the other hand, they build a high ∆ 0 2 set that is weakly Demuth random (while a Demuth random is always generalized low 1 ).…”

“…For instance, the strongly jump-traceable c.e. sets were characterized (in one direction by Kučera and Nies [29], in the other by Greenberg and Turetsky [22]) as the c.e. sets computable from a Demuth random set.…”

Counting the changes of random Δ20 sets

“…set Turing below a Demuth random set must be strongly jump-traceable. In [3] further evidence for this direction is given by showing that a weakly Demuth random set Z is not superhigh, namely, Z ≥ tt ∅ . (However, it can be high.)…”

“…Evidence for the direction from right to left in the thesis above is given by the fact that a Demuth random set bounds only generalized low 1 sets, and the result of [3] that a c.e. set Turing below a Demuth random set must be strongly jump-traceable.…”

Strong jump-traceability II: K-triviality

“…set is strongly jump-traceable if and only if it is computable from every superlow random sets, if and only if it is computable from every superhigh random set. Kučera and Nies [15] showed that every c.e. set which is computable from a Demuth random set is strongly jump-traceable, relating such random sets with the "benign cost functions" which by work of Greenberg and Nies [11] characterise c.e., strong jump-traceability.…”