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Oracle-dependent properties of the lattice of NP sets. (English) Zbl 0543.03024
The paper considers questions about the lattice of NP sets, together with the sublattice P, under the assumption $$P\neq NP$$. Two properties of NP sets are considered; whether there are any simple sets in NP and whether every infinite NP set contains an infinite subset in P. An NP-simple set is a coinfinite set in NP whose complement contains no infinite NP set. Oracles are constructed relative to which $$P\neq NP$$ and for which the statement that NP-simple sets exist is true, respectively false. Similarly, it is shown that any argument which solves the problem of whether every infinite NP set contains an infinite P subset does not relativize. In particular an oracle B is constructed relative to which $$P^ B\neq NP^ B$$ and every infinite set in $$NP^ B$$ contains an infinite $$P^ B$$ subset. The considerations are analogous to the study of the lattices of recursively enumerable and recursive sets. The constructions are somewhat more sophisticated than those previously used in this context.

##### MSC:
 03D15 Complexity of computation (including implicit computational complexity)
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##### References:
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