Complicated complementations

Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. Using Kolmogorov complexity for oracle construction, we obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself: In particular we construct relativized worlds where: 1. NP/spl cap/CoNP/spl isin/P/poly. 2. NP has a set that is both simple and NP/spl cap/CoNP-immune. 3. CoNP has a set that is both simple and NP/spl cap/CoNP-immune. 4. /spl Pi//sub 2//sup p/ has a set that is both simple and /spl Pi//sub 2//sup p//spl cap//spl Sigma//sup 2p/-immune.