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DOI: 10.4230/LIPIcs.MFCS.2022.57
Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

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

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

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.
Cite as

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)

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@InProceedings{hemaspaandra_et_al:LIPIcs.MFCS.2022.57,
  author =	{Hemaspaandra, Lane A. and Juvekar, Mandar and Nadjimzadah, Arian and Phillips, Patrick A.},
  title =	{{Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.57},
  URN =		{urn:nbn:de:0030-drops-168552},
  doi =		{10.4230/LIPIcs.MFCS.2022.57},
  annote =	{Keywords: structural complexity theory, computational complexity theory, ambiguity-limited NP, counting classes, P-printable sets}
}
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