关于无限图的外表示
On external presentations of infinite graphs
摘要：
有限状态系统的顶点通常是自然数的子集。与这些系统相关的大多数算法仅使用这一事实来选择顶点。然而，对于无限状态系统，情况就不同了：特别地，对于具有有限描述的系统，系统的每个状态都是某个机器的配置。然后，大多数算法方法依赖于这些配置的结构。这样的特征被称为内部特征。为了应用检测结构属性（如识别连通分量）的算法，可能必须首先变换系统以适合算法所需的描述。内部表征的问题在于它隐藏了结构特性，并且每个解决方案相对于配置的形式变得特别。相反，外部特征化避免了顶点的显式命名。这种特征化主要通过图形变换来定义。在本文中，我们提出了两种外部特征：确定性的图重写，这反过来又取代正则图，确定性上下文无关语言，和理性图。生成器的逆代换（如完全二叉树）为前缀可识别图、Caucal层次图和有理图提供了特征。我们说明了如何这些特性提供了一个有效的工具，无限状态系统的表示。
英文摘要：
The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in particular, for such systems having a finite description, each state of the system is a configuration of some machine. Then most algorithmic approaches rely on the structure of these configurations. Such characterisations are said internal. In order to apply algorithms detecting a structural property (like identifying connected components) one may have first to transform the system in order to fit the description needed for the algorithm. The problem of internal characterisation is that it hides structural properties, and each solution becomes ad hoc relatively to the form of the configurations. On the contrary, external characterisations avoid explicit naming of the vertices. Such characterisation are mostly defined via graph transformations. In this paper we present two kind of external characterisations: deterministic graph rewriting, which in turn characterise regular graphs, deterministic context-free languages, and rational graphs. Inverse substitution from a generator (like the complete binary tree) provides characterisation for prefix-recognizable graphs, the Caucal Hierarchy and rational graphs. We illustrate how these characterisation provide an efficient tool for the representation of infinite state systems.
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