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NP-Completeness, Proof Systems, and Disjoint NP-Pairs

Authors Titus Dose, Christian Glaßer

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Titus Dose
  • Julius-Maximilians-Universität Würzburg, Germany
Christian Glaßer
  • Julius-Maximilians-Universität Würzburg, Germany

Funding

  • Dose, Titus: supported by the German Academic Scholarship Foundation.

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Titus Dose and Christian Glaßer. NP-Completeness, Proof Systems, and Disjoint NP-Pairs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.9

Abstract

The article investigates the relation between three well-known hypotheses.  
- H_{union}: the union of disjoint ≤^p_m-complete sets for NP is ≤^p_m-complete 
- H_{opps}: there exist optimal propositional proof systems 
- H_{cpair}: there exist ≤^{pp}_m-complete disjoint NP-pairs  The following results are obtained:  
- The hypotheses are pairwise independent under relativizable proofs, except for the known implication H_{opps} ⇒ H_{cpair}. 
- An answer to Pudlák’s question for an oracle relative to which ¬H_{cpair}, ¬H_{opps}, and UP has ≤^p_m-complete sets. 
- The converse of Köbler, Messner, and Torán’s implication NEE ∩ TALLY ⊆ coNEE ⇒ H_{opps} fails relative to an oracle, where NEE =^{df} NTIME(2^O(2ⁿ)). 
- New characterizations of H_{union} and two variants in terms of coNP-completeness and p-producibility of the set of hard formulas of propositional proof systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Proof complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • NP-complete
  • propositional proof system
  • disjoint NP-pair
  • oracle

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