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：基于康托尔导子算子的模态逻辑拓扑语义产生了导子逻辑（亦称d-逻辑）。与基于拓扑闭包算子的逻辑不同，d-逻辑此前未在动力系统框架下得到研究。动力系统由拓扑空间X与连续函数f: X→X构成的对(X,f)所定义。我们引入逻辑系统wK4C、K4C和GLC，证明它们均具有有限克里普克模型性质，并在该动力设定下关于d-语义具有可靠性和完全性。具体而言，我们证明wK4C是所有动态拓扑系统的d-逻辑，K4C是所有T_D动态拓扑系统的d-逻辑，而GLC是基于离散空间的动态拓扑系统的d-逻辑。此外，我们针对f为同胚的情形给出了一个一般性结论，从而导出了相应系统wK4H、K4H和GLH的可靠性与完全性。本研究的主要贡献在于为动态拓扑d-逻辑的有限模型性质和完全性提供了一种通用证明方法。进一步地，关于GLC的结论迈出了用有限公理化证明三模态拓扑-时态语言完全性的第一步——而此前已知这在所有空间类上是不可能的。