TR08-029 Authors: Christian Glaßer, Christian Reitwießner, Victor Selivanov

Publication: 18th March 2008 16:26

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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.