Proof system representations of degrees of disjoint NP-pairs

Let DD be a set of many-one degrees of disjoint NP-pairs. We define a proof system representation   of DD to be a set of propositional proof systems PP such that each degree in DD contains the canonical NP-pair of a corresponding proof system in PP and the degree structure of DD is reflected by the simulation order among the corresponding proof systems in PP. We also define a nesting representation   of DD to be a set of NP-pairs SS such that each degree in DD contains a representative NP-pair in SS and the degree structure of DD is reflected by the inclusion relations among their representative NP-pairs in SS. We show that proof system and nesting representations both exist for DD if the lower span of each degree in DD overlaps with DD on a finite set only. In particular, a linear chain of many-one degrees of NP-pairs has both a proof system representation and a nesting representation. This extends a result by Glaßer et al. (2009). We also show that in general DD has a proof system representation if it has a nesting representation where all representative NP-pairs share the same set as their first components.

Research highlights► We define proof systems and nesting representations of degrees of NP-pairs. ► Both representations exist for certain nontrivial collections of degrees of NP-pairs. ► A special type of nesting representations yield proof system representations.