Exploring Wedges of an Oriented Grid by an Automaton with Pebbles

TL;DR

This paper determines the minimum number of pebbles needed for a finite automaton to explore wedge-shaped regions in an infinite grid, depending on geometric and crossing constraints, providing a complete characterization.

Contribution

It provides a complete classification of the minimum pebbles required for exploration of wedges in an infinite grid, based on geometric angles and crossing abilities.

Findings

Minimum pebbles needed range from 0 to 3.

The exploration feasibility depends on the wedge's angle and crossing capabilities.

The paper offers explicit automata constructions for each case.

Abstract

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.