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

Abstract

Cai and Hemachandra used iterative constant-setting to prove that Few⊆⊕P (and thus that Few p ⊆⊕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 \((\mathcal {O}(1) + \log \log n) \) -ambiguity NP sets are in the restricted counting class RC PRIMES .