We consider the bimodal language, where the first modality is interpreted by a binary relation in the standard way, and the second is interpreted by the relation of inequality. It follows from Hughes (1990), that in this language, non-$k$-colorability of a graph is expressible for every finite $k$. We show that modal logics of classes of non-$k$-colorable graphs (directed or non-directed), and some of their extensions, are decidable.
翻译:我们考虑双模态语言,其中第一个模态按标准方式由二元关系解释,而第二个模态则由不等式关系解释。根据Hughes(1990)的结果,在该语言中,对于每个有限的$k$,图的非$k$-可着色性是可以表达的。我们证明了非$k$-可着色图类(有向或非有向)及其一些扩展的模态逻辑是可判定的。