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：移动代理器, 以确定性有限自动标尺为模型, 以无限的匿名定位网格 $\ mathbb ⁇ \ time\ mathb ⁇ $ 。 它必须通过访问所有节点来探索一个给定的网格无限的子图 。 我们的焦点是简单的底线, 称为 em wedges}, 由位于平面两条半线之间, 具有共同来源的网格的所有节点所覆盖。 许多网格无法由无法标记网格节点的自动标点来探索 。 因此, 我们研究以下问题 : 以 W$ 为单位, 什么是网格网格中最小的 $p$ (movable) ebbles 最小的 $ (movbbles) 。 我们关注的顶点是 $ $ $( embge) $ ( $ w) $)? 我们的主要贡献是解决这个问题的完整解决方案 。 对于每个 We $ ( ) 确定这个最小的 $ $ 美元, 我们确定这个最小的 $ ppline 美元, 表示一个自动标无法探索它用 $ ($bbbbbbless) 。
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