not any computable function f such that x is not in the Halting Problem iff f ( x ) belongs to set of Kolmogorov-random strings

Question

taking clue from this question set of Kolmogorov-random strings is co-re

the paper mentioned in the above link talks about the non existence of a computable function

how can I show that there is not any one-one computable function f such that x is not in the Halting Problem iff f ( x ) belongs to R_C

R_C = {x : C(x) ≥ |x|} is a set of Kolmogorov-random strings.

Answer 1

@xoff The paper What Can be Efficiently Reduced to the Kolmogorov-Random Strings? Talks about this question in the second page of the paper this is what the paper says

That is, the halting problem can be solved by a machine that, on input x , computes a set of queries to R _C and then uses the answers to those queries to decide whether x is in the halting problem. It is important to note that in this weak-truth-table reduction, there is no computable bound that can be placed on the time used by the oracle machine, for the part of the computation that takes place after the queries are answered.) This was improved significantly by Kummer.

I also went searched through the kum96 paper but was not able to make much sense out of it