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

TL;DR

This paper explores how iterative constant-setting can be used to establish complexity class containments, revealing the impact of gap-size bounds and ambiguity on capturing classes like UP and implications for prime-based counting classes.

Contribution

It introduces a flexible, metatheorem-based approach that broadens the conditions under which ambiguity-limited classes can be captured using iterative constant-setting.

Findings

Less restrictive gap bounds suffice for capturing ambiguity-limited classes.

The approach applies to classes like UP with logarithmic ambiguity.

The Lenstra-Pomerance-Wagstaff Conjecture implies certain NP sets are in restricted prime-based counting classes.

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 $\rm RC_{PRIMES}$.