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Cai and Hemachandra used iterative constant-setting to prove that Few ⊆ ⊕ P (and thus that Few P ⊆ ⊕P). In this article, 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 “nongappiness”) 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. 

Unscoped Logical Form (ULF) of Episodic Logic is a meaning representation format that captures the overall semantic type structure of natural language while leaving certain finer details, such as word sense and quantifier scope, underspecified for ease of parsing and annotation. While a learned parser exists to convert English to ULF, its performance is severely limited by the lack of a large dataset to train the system. We present a ULF dataset augmentation method that samples type-coherent ULF expressions using the ULF semantic type system and filters out samples corresponding to implausible English sentences using a pretrained language model. Our data augmentation method is configurable with parameters that trade off between plausibility of samples with sample novelty and augmentation size. We find that the best configuration of this augmentation method substantially improves parser performance beyond using the existing unaugmented dataset. 

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 . 

We present a method of making natural logic inferences from Unscoped Logical Form of Episodic Logic. We establish a correspondence between inference rules of scope-resolved Episodic Logic and the natural logic treatment by Sánchez Valencia (1991a), and hence demonstrate the ability to handle foundational natural logic inferences from prior literature as well as more general nested monotonicity inferences.