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：我们基于Arenas等人（2020）提出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，提出了具有两步语义的定量逻辑。我们证明了，在添加最小不动点算子后，我们逻辑的某些片段能够捕捉到有趣的计数问题类。具体而言，我们通过为#P的两个子类（即SpanL和TotP）提供逻辑描述，回应了计数问题的描述复杂性领域中的一个开放性问题，这两个子类在可近似计数问题的研究中扮演着重要角色。此外，我们定义了能够捕获FPSPACE和SpanPSPACE的逻辑，它们是PSPACE的计数版本。