AI 中文总结
本文研究基于图的自动机，将其语言与图的路径对应，证明树语言存在唯一最小图及同态最小图，并引入剩余语言以捕获图的结构信息如色数。
AI 中文摘要
我们研究基于图的自动机：由边着色或有向图通过将每个顶点同时作为初始状态和接受状态、每条边作为一对相反转移而得到的非确定性有限自动机。这些自动机的语言对应于映射到其相应图的边着色或有向路径的集合。我们为树发展了类似的概念，并刻画了这些模型所识别的语言。对于树语言，我们利用对偶方法证明了唯一的最小规模图的存在性，更一般地，对于词语言和树语言均证明了同态最小图的存在性。为了进一步激励这些模型，我们展示了图论与自动机理论交叉领域的一些结果：我们将基于图的自动机与可逆自动机联系起来，给出了基于图的语言分解为可逆语言的方法，并引入了无向图的剩余语言作为其所有定向的交集。该剩余语言捕获了图的结构信息，如色数。
英文摘要
We study graph-based automata: nondeterministic finite automata obtained from edge-colored or oriented graphs by taking every vertex as both initial and accepting, and every edge as a pair of opposite transitions. The language of these automata corresponds to the set of edge-colored or oriented paths mapping to their corresponding graphs. We develop an analogous notion for trees and characterise the languages recognised by these models. For tree languages we prove the existence of a unique size- minimal graph and, more generally, a homomorphism-minimal graph for both word and tree languages using duality methods. In order to further motivate these models, we showcase a few results at the inter- section between graph theory and automata theory: We relate graph-based automata to reversible automata, give a decomposition of graph-based languages into reversible languages, and introduce the remanent language of an undirected graph as the inter- section of all its orientations. This remanent language captures structural information on the graph such as chromatic number.
