Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting
Lane A. Hemaspaandra, Mandar Juvekar, Arian Nadjimzadah, Patrick A. Phillips
TL;DR
This paper develops two metatheorems that connect the nongappiness of target sets defining restricted counting classes ${\rm RC}_{T}$ to the ambiguity bounds of nondeterministic classes ${\rm UP}_{\leq f(n)}$, enabling containment ${\rm UP}_{\leq j(n)} \subseteq {\rm RC}_{T}$ under broad conditions. By formalizing ${\cal O}(n)$-, ${n^{k}}$-, ${n\log n}$-, and even ${2^{n}}$-nongappy notions via the new ${\rm F}$-nongappy framework, the authors extend iterative constant-setting far beyond constant or polynomial ambiguity, providing concrete containment results for ${\rm UP}_{\leq {\cal O}(\log n)}$, ${\rm UP}_{\leq {\cal O}(1)+\log\log n}$ (including ${\rm PRIMES}$ under LPW-like assumptions), and other growth regimes. The work demonstrates a fundamental trade-off between gap density in the target set and the allowed nondeterministic ambiguity, offering a pathway to relate infinite, P-printable prime subsets to superconstant ambiguity analogues of UP. It also shows how conjectures about prime distribution, notably LPW, can impact the power of primes as restricted counting acceptors. Overall, the results illuminate the limits and potential of iterative constant-setting as a unifying framework for ambiguity-bounded counting and prime-based reductions.
Abstract
Cai and Hemachandra used iterative constant-setting to prove that Few $\subseteq$ $\oplus$P (and thus that FewP $\subseteq$ $\oplus$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(1) + loglogn)-ambiguity NP sets are in the restricted counting class $\rm RC_{PRIMES}$.