论文标题
在适当的次要闭合图类中，具有不相交路径的一阶逻辑检查模型检查
Model-Checking for First-Order Logic with Disjoint Paths Predicates in Proper Minor-Closed Graph Classes
论文作者
论文摘要
逻辑逻辑（fol+dp）是一阶逻辑（fol）的扩展，其额外的原子谓词$ \ mathsf {dp} _k（x_1，y_1，y_1，\ ldots，x_k，y_k），表达内部角度的$ x____i $ y__i $ y__i $ y__i $ y__i $ y__i $ $ i \ in \ {1，\ ldots，k \} $。这种逻辑可以表达各种各样的问题，以逃避人们的表现潜力。我们证明，对于每个适当的次要闭合图类别，可以在二次时间内完成对FOL+DP的模型检查。我们还介绍了FOL+DP的扩展，即分散的分离路径逻辑，FOL+SDP，我们进一步考虑了原子谓词$ s {\ sf -sdp} _k（x_1，y_1，y_1，\ ldots，\ ldots，x__k，y_k，y_k，y_k，y_k，y_k，y_k，y_k，y_k，y_k，$ nive nive nive distion decords solide decords solide solding solde decord soldy soldy soss indies soldy nives soss indies soldy $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $。使用相同的技术，我们证明，可以在具有界面Euler属的图表上以二次时间的时间进行fol+SDP的模型检查。
The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in\{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every proper minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate $s{\sf -sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus.
