时间粒度的时态逻辑和自动机
Temporalized logics and automata for time granularity
摘要：
在文献中已经提出了k个后继者的一元二阶理论的适当扩展，以捕获时间粒度的概念。在本文中，我们提供了一元二阶理论的向下无界分层结构，这是无限的可加细的结构，由一个粗域和无限数量的更细和更细的域，和向上无界分层结构，这是由一个最细的域和无限数量的更粗和更粗的域，与表达完全和基本可判定的时间逻辑对应。我们通过两个步骤得到这样的结果。首先，我们定义了一类新的组合自动机，称为时间自动机，它可以被证明是自动机理论对应的时间逻辑，并显示相关的属性，如关闭下的布尔运算，可判定性，并表示等价的时间逻辑，从组件自动机转移到时间的。然后，我们利用时间化逻辑和自动机之间的对应关系，以减少寻找时间粒度理论的时间逻辑对应物的任务，以更容易找到时间化自动机对应物。
英文摘要：
Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obtain such a result in two steps. First, we define a new class of combined automata, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Then, we exploit the correspondence between temporalized logics and automata to reduce the task of finding the temporal logic counterparts of the given theories of time granularity to the easier one of finding temporalized automata counterparts of them.
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