The aim of this work is to provide a family of qualitative theories for spatial change in general, and for motion of spatial scenes in particular. To achieve this, we consider a spatio-temporalisation MTALC(Dx), of the well-known ALC(D) family of Description Logics (DLs) with a concrete domain: the MTALC(Dx) concepts are interpreted over infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest; the roles split into m+n immediate-successor (accessibility) relations, which are serial, irreflexive and antisymmetric, and of which m are general, not necessarily functional, the other n functional; the concrete domain Dx is generated by an RCC8-like spatial Relation Algebra (RA) x, and is used to guide the change by imposing spatial constraints on objects of the "followed" spatial scene, eventually at different time points of the input trees. In order to capture the expressiveness of most modal temporal logics encountered in the literature, we introduce weakly cyclic Terminological Boxes (TBoxes) of MTALC(Dx), whose axioms capture the decreasing property of modal temporal operators. We show the important result that satisfiability of an MTALC(Dx) concept with respect to a weakly cyclic TBox can be reduced to the emptiness problem of a Buchi weak alternating automaton augmented with spatial constraints. In another work, complementary to this one, also submitted to this conference, we thoroughly investigate Buchi automata augmented with spatial constraints, and provide, in particular, a translation of an alternating into a nondeterministic, and an effective decision procedure for the emptiness problem of the latter.

中文翻译：

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ALC(D) 的时空化及其转换为空间约束增强的交替自动机

这项工作的目的是为一般的空间变化，特别是空间场景的运动提供一系列定性理论。为了实现这一点，我们考虑了具有具体域的众所周知的 ALC(D) 描述逻辑 (DL) 家族的时空化 MTALC(Dx)：MTALC(Dx) 概念在无限 k 元 Sigma 上进行解释-trees，节点代表时间点，Sigma 除了在经典 k-ary Sigma-trees 中的使用外，还包括对感兴趣的 n 对象空间场景的快照的描述；角色分裂为 m+n 个直接后继（可访问性）关系，它们是连续的、非自反的和反对称的，其中 m 是通用的，不一定是函数式的，另外 n 个函数式的；具体域 Dx 由类似 RCC8 的空间关系代数 (RA) x 生成，并用于通过对“跟随”空间场景的对象施加空间约束来引导变化，最终在输入树的不同时间点。为了捕捉文献中遇到的大多数模态时间逻辑的表达能力，我们引入了 MTALC(Dx) 的弱循环术语框 (TBoxes)，其公理捕获了模态时间算子的递减特性。我们展示了一个重要的结果，即 MTALC(Dx) 概念对于弱循环 TBox 的可满足性可以简化为带有空间约束的 Buchi 弱交替自动机的空性问题。在另一项工作中，作为对这一工作的补充，也提交给了这次会议，我们彻底研究了空间约束增强的 Buchi 自动机，并特别提供了，