The shrinking property for NP and coNP

作者：Glasser Christian; Reitwiessner Christian^*; Selivanov Victor

来源：Theoretical Computer Science, 2011, 412(8-10): 853-864.

DOI：10.1016/j.tcs.2010.11.035

• Computational complexity
• Polynomial hierarchy
• NP-pairs
• P-separability
• Multivalued functions

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(P)(n), and Pi(P)(n) do not have the shrinking property unless PH is finite. This solves an open question posed by Selivanov (1994) [33].
2. The separation property does not hold for NP unless UP subset of 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 of Even et al. (1984) [6]).
4. The shrinking property does not hold for NP unless there exist complete disjoint NP-pairs.
Moreover, we prove that the assumption NP not equal 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 boolean AND coNP, NP not equal coNP, and NP has the shrinking property. This solves an open question posed by Blass and Gurevich (1984) [3] who explicitly ask for such an oracle.

• 出版日期2011-3-4