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

Authors Titus Dose, Christian Glaßer

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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
  • propositional proof system
  • disjoint NP-pair
  • oracle

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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.

Cite As Get BibTex

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

Author Details

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

Funding

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

References

  1. T. Baker, J. Gill, and R. Solovay. Relativizations of the P=NP problem. SIAM Journal on Computing, 4:431-442, 1975. Google Scholar
  2. O. Beyersdorff. Representable disjoint NP-pairs. In Proceedings 24th International Conference on Foundations of Software Technology and Theoretical Computer Science, volume 3328 of Lecture Notes in Computer Science, pages 122-134. Springer, 2004. Google Scholar
  3. O. Beyersdorff. Disjoint NP-pairs from propositional proof systems. In Proceedings of Third International Conference on Theory and Applications of Models of Computation, volume 3959 of Lecture Notes in Computer Science, pages 236-247. Springer, 2006. Google Scholar
  4. O. Beyersdorff. Classes of representable disjoint NP-pairs. Theoretical Computer Science, 377(1-3):93-109, 2007. Google Scholar
  5. O. Beyersdorff. The deduction theorem for strong propositional proof systems. Theory of Computing Systems, 47(1):162-178, 2010. Google Scholar
  6. S. Cook and R. Reckhow. The relative efficiency of propositional proof systems. Journal of Symbolic Logic, 44:36-50, 1979. Google Scholar
  7. T. Dose. P-optimal proof systems for each set in coNP and no complete problems in NP∩coNP relative to an oracle. CoRR, abs/1910.08571, 2019. URL: http://arxiv.org/abs/1910.08571.
  8. T. Dose. P ≠ NP and all sets in NP∪coNP have P-optimal proof systems relative to an oracle. CoRR, abs/1909.02839, 2019. URL: http://arxiv.org/abs/1909.02839.
  9. T. Dose. An oracle separating conjectures about incompleteness in the finite domain. Theoret. Comput. Sci., 2020. URL: https://doi.org/10.1016/j.tcs.2020.01.003.
  10. T. Dose and C. Glaßer. NP-completeness, proof systems, and disjoint NP-pairs. Technical Report 19-050, Electronic Colloquium on Computational Complexity (ECCC), 2019. Google Scholar
  11. S. Even, A. L. Selman, and J. Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61:159-173, 1984. Google Scholar
  12. S. Even and Y. Yacobi. Cryptocomplexity and NP-completeness. In Proceedings 7th International Colloquium on Automata, Languages and Programming, volume 85 of Lecture Notes in Computer Science, pages 195-207. Springer, 1980. Google Scholar
  13. C. Glaßer, J. M. Hitchcock, A. Pavan, and S. Travers. Unions of disjoint NP-complete sets. ACM Trans. Comput. Theory, 6(1):3:1-3:10, 2014. Google Scholar
  14. C. Glaßer, A. Pavan, A. L. Selman, and S. Sengupta. Properties of NP-complete sets. SIAM Journal on Computing, 36(2):516-542, 2006. Google Scholar
  15. C. Glaßer, A. L. Selman, and S. Sengupta. Reductions between disjoint NP-pairs. Information and Computation, 200:247-267, 2005. Google Scholar
  16. C. Glaßer, A. L. Selman, S. Sengupta, and L. Zhang. Disjoint NP-pairs. SIAM Journal on Computing, 33(6):1369-1416, 2004. Google Scholar
  17. C. Glaßer, A. L. Selman, S. Travers, and K. W. Wagner. The complexity of unions of disjoint sets. Journal of Computer and System Sciences, 74(7):1173-1187, 2008. Google Scholar
  18. C. Glaßer, A. L. Selman, and L. Zhang. Canonical disjoint NP-pairs of propositional proof systems. Theoretical Computer Science, 370:60-73, 2007. Google Scholar
  19. C. Glaßer, A. L. Selman, and L. Zhang. The informational content of canonical disjoint NP-pairs. International Journal of Foundations of Computer Science, 20(3):501-522, 2009. Google Scholar
  20. J. Grollmann and A. L. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309-335, 1988. Google Scholar
  21. E. Hemaspaandra, L. A. Hemaspaandra, and H. Hempel. All superlinear inverse schemes are conp-hard. Theoretical Computer Science, 345(2-3):345-358, 2005. Google Scholar
  22. S. Homer and A. L. Selman. Oracles for structural properties: The isomorphism problem and public-key cryptography. Journal of Computer and System Sciences, 44(2):287-301, 1992. Google Scholar
  23. Erfan Khaniki. New relations and separations of conjectures about incompleteness in the finite domain. CoRR, abs/1904.01362, 2019. URL: http://arxiv.org/abs/1904.01362.
  24. J. Köbler, J. Messner, and J. Torán. Optimal proof systems imply complete sets for promise classes. Information and Computation, 184(1):71-92, 2003. Google Scholar
  25. J. Krajíček and P. Pudlák. Propositional proof systems, the consistency of first order theories and the complexity of computations. Journal of Symbolic Logic, 54:1063-1079, 1989. Google Scholar
  26. L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25(1):1-7, 1979. Google Scholar
  27. J. Myhill. Creative sets. Mathematical Logic Quarterly, 1(2):97-108, 1955. Google Scholar
  28. M. Ogiwara and L. Hemachandra. A complexity theory of feasible closure properties. Journal of Computer and System Sciences, 46:295-325, 1993. Google Scholar
  29. C. M. Papadimitriou. Computational complexity. Addison-Wesley, Reading, Massachusetts, 1994. Google Scholar
  30. P. Pudlák. On the lengths of proofs of consistency. In Collegium Logicum, pages 65-86. Springer Vienna, 1996. Google Scholar
  31. P. Pudlák. On reducibility and symmetry of disjoint NP pairs. Theoretical Computer Science, 295:323-339, 2003. Google Scholar
  32. P. Pudlák. Incompleteness in the finite domain. The Bulletin of Symbolic Logic, 23(4):405-441, 2017. Google Scholar
  33. C. Rackoff. Relativized questions involving probabilistic algorithms. Journal of the ACM, 29:261-268, 1982. Google Scholar
  34. A. A. Razborov. On provably disjoint NP-pairs. Electronic Colloquium on Computational Complexity (ECCC), 1(6), 1994. Google Scholar
  35. H. Rogers Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967. Google Scholar
  36. Z. Sadowski. On an optimal propositional proof system and the structure of easy subsets of TAUT. Theoretical Computer Science, 288(1):181-193, 2002. Google Scholar
  37. A. L. Selman. Natural self-reducible sets. SIAM Journal on Computing, 17(5):989-996, 1988. Google Scholar
  38. E. Tardos. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica, 8(1):141-142, 1988. Google Scholar
  39. S. Travers. Structural Properties of NP-Hard Sets and Uniform Characterisations of Complexity Classes. PhD thesis, Julius-Maximilians-Universität Würzburg, 2007. Google Scholar
  40. O. V. Verbitskii. Optimal algorithms for coNP-sets and the EXP =? NEXP problem. Mathematical notes of the Academy of Sciences of the USSR, 50(2):796-801, August 1991. Google Scholar
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