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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 AsGet 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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