Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting
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arXiv
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| Format: | Preprint |
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2021
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| author | Hemaspaandra, Lane A. Juvekar, Mandar Nadjimzadah, Arian Phillips, Patrick A. |
| author_facet | Hemaspaandra, Lane A. Juvekar, Mandar Nadjimzadah, Arian Phillips, Patrick A. |
| contents | 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 $\rm RC_{PRIMES}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_14764 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting Hemaspaandra, Lane A. Juvekar, Mandar Nadjimzadah, Arian Phillips, Patrick A. Computational Complexity F.1.3 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 $\rm RC_{PRIMES}$. |
| title | Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting |
| topic | Computational Complexity F.1.3 |
| url | https://arxiv.org/abs/2109.14764 |