Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted formulas output are evaluated. Gastin and Monmege (2018) gave abstract semantics for a version of weighted monadic second-order logic to give a more general and modular proof of the equivalence of the logic with weighted automata. We focus on the abstract semantics of the logic and we give a complete axiomatization both for the full logic and for a fragment without general sum, thus giving a more fine-grained understanding of the logic. We discuss how common decision problems for logical languages can be adapted to the weighted setting, and show that many of these are decidable, though they inherit bad complexity from the underlying first- and second-order logics. However, we show that a weighted adaptation of satisfiability is undecidable for the logic when one uses the abstract interpretation.

翻译：加权月度第二阶逻辑是蒙亚迪第二阶逻辑的加权延伸,它精确地反映了加权自动数据的行为。其语义学的参数是用来评估加权公式输出值的分点。Gastin和Monmege(2018年)为加权月度第二阶逻辑的版本提供了抽象语义学,以更笼统和模块化地证明该逻辑与加权自动数据等同。我们侧重于逻辑的抽象语义,我们给出了完全的逻辑和零散的分解,从而对逻辑有了更精细的理解。我们讨论了逻辑语言的共同决定问题如何能够适应加权环境,并表明其中许多问题是可以分解的,尽管它们从基本的一阶和二阶逻辑中继承了不良的复杂程度。然而,我们表明,当一个人使用抽象解释时,对可变性的加权调整对于逻辑是不可分化的。