In recent years, quantitative complexity over semirings has been intensively investigated. In this context, Eiter and Kiesel (Semiring Reasoning Frameworks in AI and Their Computational Complexity, J. Artif. Intell. Res., 2023) introduced non-deterministic Turing Machines with semiring-weighted transitions (SRTMs) to capture the complexity of a manifold of semiring frameworks. Beyond computational complexity, they posed the question of how we can relate the computational power of SRTMs to logical expressiveness. While this question was partially addressed for a more limited machine model by Badia et al.\ (Logical characterizations of weighted complexity classes, MFCS, 2024), the full question remained open. To answer it, we present an improved version of Eiter and Kiesel's SRTM model of computation. First and foremost, this enables us to prove a Fagin Theorem for the SRTM model, i.e., we show that the quantitative complexity class $\text{NP}_\infty(R)$, which comprises non-deterministic polynomial time computability in the improved SRTM model over a commutative semiring $R$, is captured by a version of weighted existential second-order logic that allows for predicates interpreted as semiring-annotated relations over $R$. Furthermore, we argue that the new SRTM model is preferable over the original one and show that it reclaims some important results from Eiter and Kiesel (2023) that were flawed with respect to the latter.

翻译：近年来，半环上的量化复杂性得到了深入研究。在此背景下，Eiter与Kiesel（《人工智能中的半环推理框架及其计算复杂性》，J. Artif. Intell. Res., 2023）引入了带有半环加权转移的非确定性图灵机（SRTM），以刻画多种半环框架的复杂性。除计算复杂性外，他们还提出了如何将SRTM的计算能力与逻辑表达力相联系的问题。尽管Badia等人（《加权复杂度类的逻辑刻画》，MFCS, 2024）针对更局限的机器模型部分回答了该问题，但完整问题仍悬而未决。为解答此问题，我们提出了Eiter与Kiesel的SRTM计算模型的改进版本。首先且最重要的是，这使我们能够证明SRTM模型的法金定理，即：在改进后的、基于交换半环$R$的SRTM模型中，包含非确定性多项式时间可计算性的量化复杂度类$\text{NP}_\infty(R)$，可由一种允许将谓词解释为$R$上半环标注关系的加权存在二阶逻辑版本所刻画。此外，我们论证了新SRTM模型优于原始模型，并表明它恢复了Eiter与Kiesel（2023）中因后者而存在缺陷的一些重要结果。