First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates $conn_k (x, y, z_1 ,\ldots, z_k)$ that hold true in a graph if there exists a path between (the valuations of) $x$ and $y$ after (the valuations of) $z_1,\ldots,z_k$ have been deleted, we obtain separator logic $FO + conn$. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic $FO + DP$ by adding the atomic predicates $disjoint-paths_k [(x_1, y_1 ),\ldots , (x_k , y_k )]$ that evaluate to true if there are internally vertex disjoint paths between (the valuations of) $x_i$ and $y_i$ for all $1 \le i \le k$. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Finally, we compare the expressive power of the new logics with that of transitive closure logics and monadic second-order logic.
翻译:第一顺序逻辑( FO) 可以表达图表上的许多算法问题, 比如独立设置和主导设置问题, 以解决方案大小为参数 。 另一方面, FO 无法表达两个脊椎是否连接的非常简单的算法问题 。 我们用连接性上游使FO 丰富, 以参数化算法通常研究的算法属性。 通过添加原子上游 $con_k (x, y, z_ 1,\ldots, z_k) $, 以图中保留真实路径。 如果在( 美元和美元估值) 之后有一条路径( ) 美元和 美元之间的路径, 美元和 美元 美元( k) 的路径, z_k 美元( 美元, 美元), 我们得到更强烈的线性逻辑, 美元- 和 美元( 美元) 电流流 。 我们用更强的 数字- 和 美元( 美元) 电流流, 我们用更强的电流 解的, 和 美元- 美元- 电流化的 解算 。