# Exploring an Infinite Space with Finite Memory Scouts

## TL;DR

This paper investigates the minimal number of finite automaton-controlled scouts needed to reliably find a hidden target in an infinite grid, proving that for one and two dimensions, the known upper bound is tight.

## Contribution

It proves that the upper bound of d+1 scouts for guaranteeing finite mean hitting time is tight for dimensions 1 and 2.

## Key findings

- For d=1, d+1 scouts are necessary and sufficient.
- For d=2, d+1 scouts are necessary and sufficient.
- The bound is not necessarily tight for higher dimensions.

## Abstract

Consider a small number of scouts exploring the infinite $d$-dimensional grid with the aim of hitting a hidden target point. Each scout is controlled by a probabilistic finite automaton that determines its movement (to a neighboring grid point) based on its current state. The scouts, that operate under a fully synchronous schedule, communicate with each other (in a way that affects their respective states) when they share the same grid point and operate independently otherwise. Our main research question is: How many scouts are required to guarantee that the target admits a finite mean hitting time? Recently, it was shown that $d + 1$ is an upper bound on the answer to this question for any dimension $d \geq 1$ and the main contribution of this paper comes in the form of proving that this bound is tight for $d \in \{ 1, 2 \}$.