Canonical Disjoint NP-Pairs of Propositional Proof Systems

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.