We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is identical. Secondly, we show that this degree structure is not superficial: Assuming there exist P-inseparable disjoint pairs, there exist intermediate disjoint NP-pairs. That is, if (A, B) is a P-separable disjoint NP-pair and (C, D) is a P-inseparable disjoint NP-pair, then there exist P-inseparable, incomparable NP-pairs (E, F) and (G, H) whose degrees lie strictly between (A, B) and (C, D). Furthermore, between any two disjoint NP-pairs that are comparable and inequivalent, such a diamond exists.
a b s t r a c tWe 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, Σ P n and Π 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 ⊆ 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 NPpairs.Moreover, we prove that the assumption NP ̸ = 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 ̸ = 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.
a b s t r a c tWe study the question of the existence of non-mitotic sets in NP. We show under various hypotheses that• 1-tt-mitoticity and m-mitoticity differ on NP.• T-autoreducibility and T-mitoticity differ on NP (this contrasts the situation in the recursion theoretic setting, where Ladner showed that autoreducibility and mitoticity coincide).• 2-tt-autoreducibility does not imply weak 2-tt-mitoticity (from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2-tt and T, although the notions coincide for m-and 1-tt-reducibility).
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