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### Abstract

We study the shrinking and separation properties (two notions well-known
in descriptive set theory) for NP and coNP and show that under reasonable
complexity-theoretic assumptions, both properties do not hold for NP and
the shrinking property does not hold for coNP. In particular we obtain the
following results.
1. NP and coNP do not have the shrinking property, unless PH is finite. In
general, Sigma_n and Pi_n do not have the shrinking property, unless PH is
finite. This solves an open question from [Selivanov 94].
2. The separation property does not hold for NP, unless UP \subseteq coNP.
3. The shrinking property does not hold for NP, unless there exist NP-hard
disjoint NP-pairs (existence of such pairs would contradict a conjecture
by Even, Selman, and Yacobi).
4. The shrinking property does not hold for NP, unless there exist
complete disjoint NP-pairs.
Moreover, we prove that the assumption NP \neq coNP is too weak to refute
the shrinking property for NP in a relativizable way. For this we
construct an oracle relative to which P = NP \cap coNP, NP \neq coNP, and
NP has the shrinking property. This solves an open question by Blass and
Gurevich who explicitly ask for such an oracle.