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Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

Authors Lane A. Hemaspaandra , Mandar Juvekar , Arian Nadjimzadah , Patrick A. Phillips

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Author Details

Lane A. Hemaspaandra
  • Department of Computer Science, University of Rochester, Rochester, NY, USA
Mandar Juvekar
  • Department of Computer Science, University of Rochester, Rochester, NY, USA
Arian Nadjimzadah
  • Department of Computer Science, University of Rochester, Rochester, NY, USA
Patrick A. Phillips
  • Riverside Research, Arlington, VA, USA

Funding

Work supported in part by NSF Grant CCF-2006496.

Acknowledgements

We thank B. Carleton, H. Welles, and the anonymous referees.

Cite As Get BibTex

Lane A. Hemaspaandra, Mandar Juvekar, Arian Nadjimzadah, and Patrick A. Phillips. Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.57

Abstract

Cai and Hemachandra used iterative constant-setting to prove that Few ⊆ ⊕ P (and thus that FewP ⊆ ⊕ P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"-ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant’s unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all O(log log n)-ambiguity NP sets are in the restricted counting class RC_PRIMES.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • structural complexity theory
  • computational complexity theory
  • ambiguity-limited NP
  • counting classes
  • P-printable sets

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References

  1. M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Annals of Mathematics, 160(2):781-793, 2004. Google Scholar
  2. E. Allender. Some consequences of the existence of pseudorandom generators. Journal of Computer and System Sciences, 39(1):101-124, 1989. Google Scholar
  3. E. Allender and R. Rubinstein. P-printable sets. SIAM Journal on Computing, 17(6):1193-1202, 1988. Google Scholar
  4. R. Beigel. On the relativized power of additional accepting paths. In Proceedings of the 4th Structure in Complexity Theory Conference, pages 216-224. IEEE Computer Society Press, June 1989. Google Scholar
  5. R. Beigel, J. Gill, and U. Hertrampf. Counting classes: Thresholds, parity, mods, and fewness. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, pages 49-57. Springer-Verlag Lecture Notes in Computer Science #415, February 1990. Google Scholar
  6. L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca, NY, 1977. Google Scholar
  7. B. Borchert, L. Hemaspaandra, and J. Rothe. Restrictive acceptance suffices for equivalence problems. London Mathematical Society Journal of Computation and Mathematics, 3:86-95, 2000. Google Scholar
  8. B. Borchert and F. Stephan. Looking for an analogue of Rice’s Theorem in circuit complexity theory. Mathematical Logic Quarterly, 46(4):489-504, 2000. Google Scholar
  9. D. Bovet, P. Crescenzi, and R. Silvestri. Complexity classes and sparse oracles. Journal of Computer and System Sciences, 50(3):382-390, 1995. Google Scholar
  10. G. Brassard. Crusade for a better notation. SIGACT News, 17(1):60-64, 1985. Google Scholar
  11. G. Brassard and P. Bratley. Algorithmics: Theory & Practice. Prentice Hall, 1988. Google Scholar
  12. J.-Y. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy II: Applications. SIAM Journal on Computing, 18(1):95-111, 1989. Google Scholar
  13. J.-Y. Cai and L. Hemachandra. On the power of parity polynomial time. Mathematical Systems Theory, 23(2):95-106, 1990. Google Scholar
  14. C. Caldwell. Heuristics model for the distribution of Mersennes. The PrimePages, 2021. URL verified 2022/6/22. URL: https://primes.utm.edu/mersenne/heuristic.html.
  15. P. Chebyshev. Mémoire sur les nombres premiers. Journal de Mathématiques Pures et Appliquées. Série 1, 17:366-390, 1852. Google Scholar
  16. J. Cox and T. Pay. An overview of some semantic and syntactic complexity classes. Technical report, Computing Research Repository, June 2018. URL: http://arxiv.org/abs/1806.03501.
  17. S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116-148, 1994. Google Scholar
  18. S. Fenner, L. Fortnow, and L. Li. Gap-definability as a closure property. Information and Computation, 130(1):1-17, 1996. Google Scholar
  19. K. Ford, B. Green, S. Konyagin, J. Maynard, and T. Tao. Long gaps between primes. Journal of the American Mathematical Society, 31(1):65-105, 2018. Google Scholar
  20. K. Ford, B. Green, S. Konyagin, and T. Tao. Large gaps between consecutive prime numbers. Annals of Mathematics. Second Series, 183(3):935-974, 2016. Google Scholar
  21. J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675-695, 1977. Google Scholar
  22. D. Gillies. Three new Mersenne primes and a statistical theory. Mathematics of Computation, 18(85):93-97, 1964. Corrigendum 31(140):1051, 1977. Google Scholar
  23. L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43(1):43-58, 1986. Google Scholar
  24. J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309-335, 1988. URL: https://doi.org/10.1137/0217018.
  25. J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities. Theoretical Computer Science, 34(1-2):17-32, 1984. Google Scholar
  26. L. Hemaspaandra, M. Juvekar, A. Nadjimzadah, and P. Phillips. Gaps, ambiguity, and establishing complexity-class containments via iterative constant-setting. Technical report, Computing Research Repository, September 2021. Revised, June 2022. URL: http://arxiv.org/abs/2109.147648.
  27. L. Hemaspaandra and M. Ogihara. The Complexity Theory Companion. Springer-Verlag, 2002. Google Scholar
  28. L. Hemaspaandra and J. Rothe. A second step towards complexity-theoretic analogs of Rice’s Theorem. Theoretical Computer Science, 244(1-2):205-217, 2000. Google Scholar
  29. L. Hemaspaandra and M. Zimand. Strong forms of balanced immunity. Technical Report TR-480, Department of Computer Science, University of Rochester, Rochester, NY, December 1993. Revised, May 1994. Google Scholar
  30. K. Ko. On some natural complete operators. Theoretical Computer Science, 37(1):1-30, 1985. Google Scholar
  31. J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272-286, 1992. Google Scholar
  32. K.-J. Lange and P. Rossmanith. Unambiguous polynomial hierarchies and exponential size. In Proceedings of the 9th Structure in Complexity Theory Conference, pages 106-115. IEEE Computer Society Press, June/July 1994. Google Scholar
  33. J. Maynard. Large gaps between primes. Annals of Mathematics, Second Series, 183(3):915-933, 2016. Google Scholar
  34. C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings of the 6th GI Conference on Theoretical Computer Science, pages 269-276. Springer-Verlag Lecture Notes in Computer Science #145, January 1983. Google Scholar
  35. C. Pomerance. Recent developments in primality testing. The Mathematical Intelligencer, 3(3):97-105, 1981. Google Scholar
  36. L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976. Google Scholar
  37. L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979. Google Scholar
  38. S. Wagstaff, Jr. Divisors of Mersenne numbers. Mathematics of Computation, 40(161):385-397, 1983. Google Scholar
  39. O. Watanabe. On hardness of one-way functions. Information Processing Letters, 27(3):151-157, 1988. Google Scholar
  40. Wikipedia. Prime gap. en.wikipedia.org/wiki/Prime_gap, 2021. URL verified 2022/6/22. Google Scholar
  41. Wikipedia. Gillies' conjecture. en.wikipedia.org/wiki/Gillies%27_conjecture, 2022. URL verified 2022/6/22. Google Scholar
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