Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting
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.
structural complexity theory
computational complexity theory
ambiguity-limited NP
counting classes
P-printable sets
Theory of computation~Complexity classes
57:1-57:15
Regular Paper
Work supported in part by NSF Grant CCF-2006496.
We thank B. Carleton, H. Welles, and the anonymous referees.
Lane A.
Hemaspaandra
Lane A. Hemaspaandra
Department of Computer Science, University of Rochester, Rochester, NY, USA
http://www.cs.rochester.edu/u/lane/
https://orcid.org/0000-0003-0659-5204
Mandar
Juvekar
Mandar Juvekar
Department of Computer Science, University of Rochester, Rochester, NY, USA
https://orcid.org/0000-0002-1603-1843
Arian
Nadjimzadah
Arian Nadjimzadah
Department of Computer Science, University of Rochester, Rochester, NY, USA
https://orcid.org/0000-0003-0174-3271
Patrick A.
Phillips
Patrick A. Phillips
Riverside Research, Arlington, VA, USA
https://orcid.org/0000-0002-9274-022X
10.4230/LIPIcs.MFCS.2022.57
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Lane A. Hemaspaandra, Mandar Juvekar, Arian Nadjimzadah, and Patrick A. Phillips
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