We present quantitative logics with two-step semantics based on the framework of quantitative logics introduced by Arenas et al. (2020) and the two-step semantics defined in the context of weighted logics by Gastin & Monmege (2018). We show that some of the fragments of our logics augmented with a least fixed point operator capture interesting classes of counting problems. Specifically, we answer an open question in the area of descriptive complexity of counting problems by providing logical characterizations of two subclasses of #P, namely SpanL and TotP, that play a significant role in the study of approximable counting problems. Moreover, we define logics that capture FPSPACE and SpanPSPACE, which are counting versions of PSPACE.

翻译：我们提出了一种基于Arenas等人（2020）引入的定量逻辑框架和Gastin & Monmege（2018）在加权逻辑背景下定义的两步语义的定量逻辑。我们证明，在我们的逻辑中增加最小不动点算子后的某些片段能够刻画有趣的计数问题类。具体而言，我们回答了计数问题的描述复杂性领域中的一个开放问题，即给出了#P的两个子类——SpanL和TotP（在可近似计数问题研究中具有重要作用）的逻辑刻画。此外，我们定义了能够刻画FPSPACE和SpanPSPACE（即PSPACE的计数版本）的逻辑。