A mobile agent, modeled as a deterministic finite automaton, navigates in the infinite anonymous oriented grid $\mathbb{Z} \times \mathbb{Z}$. It has to explore a given infinite subgraph of the grid by visiting all of its nodes. We focus on the simplest subgraphs, called {\em wedges}, spanned by all nodes of the grid located between two half-lines in the plane, with a common origin. Many wedges turn out to be impossible to explore by an automaton that cannot mark nodes of the grid. Hence, we study the following question: Given a wedge $W$, what is the smallest number $p$ of (movable) pebbles for which there exists an automaton that can explore $W$ using $p$ pebbles? Our main contribution is a complete solution of this problem. For each wedge $W$ we determine this minimum number $p$, show an automaton that explores it using $p$ pebbles and show that fewer pebbles are not enough. We show that this smallest number of pebbles can vary from 0 to 3, depending on the angle between half-lines limiting the wedge and depending on whether the automaton can cross these half-lines or not.

翻译：移动代理（以确定性有限自动机建模）在无限匿名定向网格 $\mathbb{Z} \times \mathbb{Z}$ 中导航。该代理需通过访问所有节点来探索网格的给定无限子图。本文聚焦于一类称为"楔形区域"的最简子图——该子图由平面内两条共起点半直线之间所有网格节点构成。研究表明，若自动机无法标记网格节点，则多数楔形区域无法被探索。因此，我们研究如下问题：给定楔形区域 $W$，使自动机能利用 $p$ 颗（可移动）卵石成功探索 $W$ 的最小数量 $p$ 是多少？我们的主要贡献在于对该问题的完整解答：针对每个楔形区域 $W$，我们确定最小卵石数 $p$，构造使用 $p$ 颗卵石的探索自动机，并证明更少的卵石无法完成任务。结果表明，该最小卵石数取值范围为 0 至 3，具体数值取决于限定楔形区域的两条半直线间夹角，以及自动机能否穿越这两条半直线。