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P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle

Author Titus Dose

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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
  • proof systems
  • disjoint NP-pair
  • oracle
  • UP

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Abstract

Pudlák [P. Pudlák, 2017] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: 
- DisjNP: The class of all disjoint NP-pairs has no many-one complete elements. 
- SAT: NP contains no many-one complete sets that have P-optimal proof systems. 
- UP: UP has no many-one complete problems. 
- NP cap coNP: NP cap coNP has no many-one complete problems. 
 As one answer to this question, we construct an oracle relative to which DisjNP, neg SAT, UP, and NP cap coNP hold, i.e., there is no relativizable proof for the implication DisjNP wedge UP wedge NP cap coNP ==> SAT. In particular, regarding the conjectures by Pudlák this extends a result by Khaniki [Khaniki, 2019]. Since Khaniki [Khaniki, 2019] constructs an oracle showing that the implication SAT ==> DisjNP has no relativizable proof, we obtain that the conjectures DisjNP and SAT are independent in relativized worlds, i.e., none of the implications DisjNP ==> SAT and SAT ==> DisjNP can be proven relativizably.

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Titus Dose. P-Optimal Proof Systems for Each NP-Set but no Complete Disjoint NP-Pairs Relative to an Oracle. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.47

Author Details

Titus Dose
  • Julius-Maximilians-Universität Würzburg, Germany

References

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