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The Informational Content of Canonical Disjoint NPPairs
, 2007
"... We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: Where does the implication ..."
Abstract

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We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: Where does the implication [can(f) ≤ pp m can(g) ⇒ f ≤s g] hold, and where does it fail? Q2: Where can we find proof systems of different strengths, but equivalent canonical pairs? Q3: What do (non)equivalent canonical pairs tell about the corresponding proof systems? Q4: Is every NPpair (A, B), where A is NPcomplete, strongly manyone equivalent to the canonical pair of some proof system? In short, we show that Q1 and Q2 can be answered with ‘everywhere’, which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the proof systems are not “very similar”, while equivalent, Pinseparable canonical pairs tell that they are not “very different”. We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NPcomplete sets are NPcomplete. This demonstrates a new connection between proof systems, disjoint NPpairs, and unions of disjoint NPcomplete sets. 1
Unions of Disjoint NPComplete Sets
"... Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1 ..."
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Abstract. We study the following question: if A and B are disjoint NPcomplete sets, then is A ∪ B NPcomplete? We provide necessary and sufficient conditions under which the union of disjoint NPcomplete sets remain complete. 1