Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as $d$-logics. Unlike logics based on the topological closure operator, $d$-logics have not previously been studied in the framework of dynamical systems, which are pairs $(X,f)$ consisting of a topological space $X$ equipped with a continuous function $f\colon X\to X$. We introduce the logics $\bf{wK4C}$, $\bf{K4C}$ and $\bf{GLC}$ and show that they all have the finite Kripke model property and are sound and complete with respect to the $d$-semantics in this dynamical setting. In particular, we prove that $\bf{wK4C}$ is the $d$-logic of all dynamic topological systems, $\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic topological systems, and $\bf{GLC}$ is the $d$-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where $f$ is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems $\bf{wK4H}$, $\bf{K4H}$ and $\bf{GLH}$. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological $d$-logics. Furthermore, our result for $\bf{GLC}$ constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -- something known to be impossible over the class of all spaces.

翻译：基于康托尔导数算子的模态逻辑拓扑语义衍生出导数逻辑，亦称$d$-逻辑。与基于拓扑闭包算子的逻辑不同，$d$-逻辑此前尚未在动力系统框架下得到研究——动力系统是指由拓扑空间$X$及其连续映射$f\colon X\to X$构成的二元组$(X,f)$。本文引入逻辑系统$\bf{wK4C}$、$\bf{K4C}$和$\bf{GLC}$，证明它们均具有有限克里普克模型性质，并在该动力学设定下关于$d$-语义具有可靠性与完全性。具体而言，我们证明$\bf{wK4C}$是所有动态拓扑系统的$d$-逻辑，$\bf{K4C}$是所有$T_D$动态拓扑系统的$d$-逻辑，而$\bf{GLC}$是所有基于散布空间的动态拓扑系统的$d$-逻辑。针对$f$为同胚的情形，我们给出了一般性结论，进而得到对应系统$\bf{wK4H}$、$\bf{K4H}$和$\bf{GLH}$的可靠性与完全性。本文的主要贡献在于为动态拓扑$d$-逻辑的有限模型性质与完全性建立了一种通用证明方法。此外，关于$\bf{GLC}$的结果标志着在三模态拓扑时间语言关于有限公理化系统方面迈出了关键第一步——这在所有空间类上已被证明是不可能的。