Proof system representations of degrees of disjoint NP-pairs

  • Authors:

  • Liyu Zhang

  • Affiliations:

  • Department of Computer and Information Sciences, University of Texas at Brownsville, Brownsville, TX 78520, USA

  • Venue:
  • Information Processing Letters
  • Year:
  • 2011

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Abstract

Let D be a set of many-one degrees of disjoint NP-pairs. We define a proof system representation of D to be a set of propositional proof systems P such that each degree in D contains the canonical NP-pair of a corresponding proof system in P and the degree structure of D is reflected by the simulation order among the corresponding proof systems in P. We also define a nesting representation of D to be a set of NP-pairs S such that each degree in D contains a representative NP-pair in S and the degree structure of D is reflected by the inclusion relations among their representative NP-pairs in S. We show that proof system and nesting representations both exist for D if the lower span of each degree in D overlaps with D 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 Glaszer et al. (2009). We also show that in general D has a proof system representation if it has a nesting representation where all representative NP-pairs share the same set as their first components.