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An Oracle with no $\mathrm{UP}$-Complete Sets, but $\mathrm{NP}=\mathrm{PSPACE}$ (2404.19104v1)

Published 29 Apr 2024 in cs.CC

Abstract: We construct an oracle relative to which $\mathrm{NP} = \mathrm{PSPACE}$, but $\mathrm{UP}$ has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [HH88] and one by Ogiwara and Hemachandra [OH93]. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudl\'ak [Pud17], e.g., the implications $\mathsf{UP} \Longrightarrow \mathsf{CON}{\mathsf{N}}$ and $\mathsf{SAT} \Longrightarrow \mathsf{TFNP}$ are false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that $\mathrm{TFNP}$-complete problems exist, while at the same time $\mathrm{SAT}$ has no p-optimal proof systems.

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References (54)
  1. The relative complexity of NP search problems. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’95, page 303–314, New York, NY, USA, 1995. Association for Computing Machinery.
  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.
  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.
  4. O. Beyersdorff. Classes of representable disjoint NP-pairs. Theoretical Computer Science, 377(1-3):93–109, 2007.
  5. O. Beyersdorff. The deduction theorem for strong propositional proof systems. Theory of Computing Systems, 47(1):162–178, 2010.
  6. Nondeterministic functions and the existence of optimal proof systems. Theoretical Computer Science, 410(38-40):3839–3855, 2009.
  7. On the cryptographic hardness of finding a nash equilibrium. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1480–1498, 2015.
  8. O. Beyersdorff and Z. Sadowski. Do there exist complete sets for promise classes? Mathematical Logic Quarterly, 57(6):535–550, 2011.
  9. S. A. Cook. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC ’71, page 151–158, New York, NY, USA, 1971. Association for Computing Machinery.
  10. S. Cook and R. Reckhow. The relative efficiency of propositional proof systems. Journal of Symbolic Logic, 44:36–50, 1979.
  11. T. Dose and C. Glaßer. NP-completeness, proof systems, and disjoint NP-pairs. In C. Paul and M. Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science, STACS 2020, March 10-13, 2020, Montpellier, France, volume 154 of LIPIcs, pages 9:1–9:18. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2020.
  12. T. Dose. Balance Problems for Integer Circuits and Separations of Relativized Conjectures on Incompleteness in Promise Classes. PhD thesis, Fakultät für Mathematik und Informatik, Universität Würzburg, 2020.
  13. T. Dose. Further oracles separating conjectures about incompleteness in the finite domain. Theoretical Computer Science, 847:76–94, 2020.
  14. T. Dose. An oracle separating conjectures about incompleteness in the finite domain. Theoretical Computer Science, 809:466–481, 2020.
  15. J. Edmonds. Optimum branchings. Journal of Research of the national Bureau of Standards B, 71(4):233–240, 1966.
  16. Oracle with P = NP ∩\cap∩ coNP, but No Many-One Completeness in UP, DisjNP, and DisjCoNP. In S. Szeider, R. Ganian, and A. Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 45:1–45:15, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
  17. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2):159–173, 1984.
  18. On inverting onto functions. In Proceedings 11th Conference on Computational Complexity, pages 213–223. IEEE Computer Society Press, 1996.
  19. Inverting onto functions. Information and Computation, 186(1):90–103, 2003.
  20. Further Collapses in TFNP. In S. Lovett, editor, 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1–33:15, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
  21. TFNP: An update. In D. Fotakis, A. Pagourtzis, and V. Th. Paschos, editors, Algorithms and Complexity, pages 3–9, Cham, 2017. Springer International Publishing.
  22. Revisiting the cryptographic hardness of finding a nash equilibrium. In M. Robshaw and J. Katz, editors, Advances in Cryptology – CRYPTO 2016, pages 579–604, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg.
  23. The shrinking property for NP and coNP. Theoretical Computer Science, 412(8-10):853–864, 2011.
  24. J. Grollmann and A. L. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309–335, 1988.
  25. Disjoint NP-pairs. SIAM Journal on Computing, 33(6):1369–1416, 2004.
  26. Canonical disjoint NP-pairs of propositional proof systems. Theoretical Computer Science, 370:60–73, 2007.
  27. The informational content of canonical disjoint NP-pairs. International Journal of Foundations of Computer Science, 20(3):501–522, 2009.
  28. J. Hartmanis and L. A. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.
  29. E. Jeřábek. Integer factoring and modular square roots. arXiv e-prints, page arXiv:1207.5220, July 2012.
  30. Snargs for bounded depth computations and ppad hardness from sub-exponential lwe. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, page 708–721, New York, NY, USA, 2021. Association for Computing Machinery.
  31. How easy is local search? Journal of Computer and System Sciences, 37(1):79–100, 1988.
  32. R. Kannan. Sipser [Sip82] cites an unpublished work by Kannan for asking if there is a set complete for NP ∩\cap∩ coNP, 1979.
  33. E. Khaniki. New relations and separations of conjectures about incompleteness in the finite domain. The Journal of Symbolic Logic, 87(3):912–937, 2022.
  34. Optimal proof systems imply complete sets for promise classes. Information and Computation, 184(1):71–92, 2003.
  35. 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.
  36. J. Krajíček. Bounded Arithmetic, Propositional Logic and Complexity Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1995.
  37. L. A. Levin. Universal search problems. Problemy Peredachi Informatsii, 9(3):115–116, 1973.
  38. J. Messner. On the Simulation Order of Proof Systems. PhD thesis, Universität Ulm, 2000.
  39. N. Megiddo and C. H. Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317–324, 1991.
  40. M. Ogiwara and L.A. Hemachandra. A complexity theory for feasible closure properties. In [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pages 16–29, 1991.
  41. C. H. Papadimitriou. On inefficient proofs of existence and complexity classes. In J. Neŝetril and M. Fiedler, editors, Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity, volume 51 of Annals of Discrete Mathematics, pages 245–250. Elsevier, 1992.
  42. C. H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498–532, 1994.
  43. P. Pudlák. The lengths of proofs. In S. R. Buss, editor, Handbook of Proof Theory, pages 547–637. Elsevier, Amsterdam, 1998.
  44. P. Pudlák. Logical foundations of mathematics and computational complexity: A gentle introduction. Springer, 2013.
  45. P. Pudlák. On some problems in proof complexity. In O. Beyersdorff, E. A. Hirsch, J. Krajíček, and R. Santhanam, editors, Optimal algorithms and proofs (Dagstuhl Seminar 14421), volume 4, pages 63–63, Dagstuhl, Germany, 2014. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
  46. P. Pudlák. Incompleteness in the finite domain. The Bulletin of Symbolic Logic, 23(4):405–441, 2017.
  47. A. Razborov. On provably disjoint NP-pairs. BRICS Report Series, 36, 1994.
  48. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM, 21(2):120–126, February 1978.
  49. A. L. Selman. Promise problems complete for complexity classes. Information and Computation, 78:87–98, 1988.
  50. A. Shamir. IP=PSPACE (interactive proof=polynomial space). In Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pages 11–15 vol.1, 1990.
  51. I. Simon. On Some Subrecursive Reducibilities. PhD thesis, Stanford University, 1977.
  52. M. Sipser. On relativization and the existence of complete sets. In Proceedings 9th ICALP, volume 140 of Lecture Notes in Computer Science, pages 523–531. Springer Verlag, 1982.
  53. L. G. Valiant. Relative complexity of checking and evaluation. Information Processing Letters, 5:20–23, 1976.
  54. 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, 1991.
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