The Shrinking Property for NP and coNP

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, Σn and Πn do not have the shrinking property, unless PH is finite. This solves an open question from [Sel94a]. 2. The separation property does not hold for NP, unless UP ⊆ coNP. 3. The shrinking property does not hold for NP, unless there exist NP-hard disjoint NPpairs (existence of such pairs would contradict a conjecture by Even, Selman, and Yacobi [ESY84]). 4. The shrinking property does not hold for NP, unless there exist complete disjoint NP-pairs. Moreover, we prove that the assumption NP 6= 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 ∩ coNP, NP 6= coNP, and NP has the shrinking property. This solves an open question by Blass and Gurevich [BG84] who explicitly ask for such an oracle.