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：我们提出了基于Arenas等人（2020）引入的定量逻辑框架以及Gastin与Monmege（2018）在加权逻辑语境中定义的两步语义的定量逻辑。研究表明，在我们的逻辑中增加最小不动点算子的某些片段能够刻画有趣的计数问题类。具体而言，通过提供#P的两个子类（即SpanL和TotP）的逻辑刻画，我们回答了计数问题描述复杂性领域的一个开放问题，这两个子类在可近似计数问题的研究中具有重要作用。此外，我们还定义了能够刻画FPSPACE和SpanPSPACE（即PSPACE的计数版本）的逻辑。