We describe two formalisms for defining graph languages, and prove that they are equivalent: 1. Separator logic. This is first-order logic on graphs which is allowed to use the edge relation, and for every $n \in \{0,1,\ldots \}$ a relation of arity $n+2$ which says that "vertex $s$ can be connected to vertex $t$ by a path that avoids vertices $v_1,\ldots,v_n$". 2. Star-free graph expressions. These are expressions that describe graphs with distinguished vertices called ports, and which are built from finite languages via Boolean combinations and the operations on graphs with ports used to construct tree decompositions. Furthermore, we prove a variant of Sch\"utzenberger's theorem (about star-free languages being those recognized by a periodic monoids) for graphs of bounded pathwidth. A corollary is that, given $k$ and a graph language represented by an \mso formula, one can decide if the language can be defined in either of two equivalent formalisms on graphs of pathwidth at most $k$.
翻译:我们描述用于定义图形语言的两种形式主义, 并证明它们是等效的 : 1. 分隔逻辑 。 这是允许使用边际关系的图表的第一阶逻辑 。 这是允许使用边际关系的图表的第一阶逻辑, 以及每一个 $ $ $ +2 的 美元 + 美元 。 这表明“ 顶点$ 可以通过一条路径连接到顶点 $v_ 1,\ ldots,v_ n$ 。 ” 2. 无星图表达式。 这些表达式描述了以不同的顶点命名为端端口的图表, 并且用有限的语言通过布林组合和图上用于构建树分解位置的操作来构建。 此外, 我们证明, “ 没有恒点语言是被定期单点识别的无星语言 ” 。 必然结果是, 以 $k 和\ mso 公式代表的图表语言代表着不同的边框 。 在两种正等量的路径上,, 我们可以决定大多数语言是否在两种正态路径上定义 $ 。