AI 中文总结
本文研究图中移动确定自动机的会合问题，证明固定小卵石无法实现树的RV-通用DFA，而配备单个可移动小卵石的DFA可成为树的RV-通用DFA。
AI 中文摘要
两个移动智能体被建模为相同的确定有限自动机（DFA），在同步轮次中于节点未标记的图中移动，需在某个节点相遇，图中相遇这一被广泛研究的任务称为会合。智能体从对手选定的不同节点开始，可能在不同轮次启动。会合问题的一个实例由底层图以及智能体的初始节点u和v组成；若存在一个DFA（可能仅适用于该实例），使得其相同副本从节点u和v以任意延迟启动都能完成会合，则该实例是可行的。对于一类实例，若一个DFA能保证其副本从指定节点以任意延迟启动时，该类所有可行实例都能实现会合，则该DFA是RV-通用的。本文旨在研究RV-通用DFA的存在性。首先发现，若智能体无法以任何方式标记节点，即使底层图为直线这类实例，也不存在RV-通用DFA；因此允许智能体使用相同的小卵石标记节点，考虑智能体可放置在节点上且不可拾取的固定小卵石，以及可放置和后续拾取的可移动小卵石。进一步发现，即使在更强大的可移动小卵石场景中，若智能体配备任意有限数量的小卵石，对于所有实例类也不存在RV-通用DFA；因此将研究限制在底层图为树的实例。主要贡献是两个对比结果，表明小卵石的可移动性是关键特征：首先证明对于任意有限数量的固定小卵石，不存在针对树的RV-通用DFA；随后设计了针对树的RV-通用DFA，每个智能体配备一个可移动小卵石。
英文摘要
Two mobile agents, modeled as identical deterministic finite automata (DFA) navigating in synchronous rounds in a graph with unlabeled nodes, have to meet at some node. The well-researched task of meeting in a graph is known as rendezvous. Agents start at adversarially chosen distinct nodes in possibly different rounds. An instance of the rendezvous problem is the underlying graph, together with the initial nodes $u$ and $v$ of the agents. Such an instance is feasible, if there exists a DFA (possibly working only for this instance), such that its identical copies starting at nodes $u$ and $v$, with an arbitrary delay, accomplish rendezvous. A DFA is RV-universal for a class of instances, if it guarantees rendezvous of its copies starting at the designated nodes with arbitrary delay, for all feasible instances of this class. Our goal is to investigate the existence of RV-universal DFA. We start by observing that if agents cannot mark nodes in any way then there does not exist a RV-universal DFA even for the class of instances where the underlying graph is a line. Hence we allow the use of identical pebbles to mark the nodes by the agents. We consider stationary pebbles that can be dropped by agents at nodes but cannot be picked up, and movable pebbles that can be dropped by agents and later picked up. We observe that, even in the more powerful scenario of movable pebbles, if agents are equipped with any finite number of pebbles, there is no RV-universal DFA for the class of all instances. Hence we restrict attention to instances where the underlying graph is a tree. Our main contribution are two contrasting results showing that movability of pebbles is a crucial feature. We first prove that for any finite number of stationary pebbles there is no RV-universal DFA for trees, and then we design a RV-universal DFA for trees, where each agent is equipped with a single movable pebble.
