Search by a Metamorphic Robotic System in a Finite 3D Cubic Grid

TL;DR

This paper investigates the minimal number of modules needed for a metamorphic robotic system to perform search tasks in a finite 3D cubic grid under various knowledge assumptions, extending prior 2D results.

Contribution

It establishes the necessary and sufficient number of modules for search in 3D grids under different common knowledge conditions, revealing richer structures than in 2D.

Findings

Three modules suffice with a common compass.

Four modules are needed when only the vertical axis is common.

Five modules are required without any common compass.

Abstract

We consider search in a finite 3D cubic grid by a metamorphic robotic system (MRS), that consists of anonymous modules. A module can perform a sliding and rotation while the whole modules keep connectivity. As the number of modules increases, the variety of actions that the MRS can perform increases. The search problem requires the MRS to find a target in a given finite field. Doi et al. (SSS 2018) demonstrate a necessary and sufficient number of modules for search in a finite 2D square grid. We consider search in a finite 3D cubic grid and investigate the effect of common knowledge. We consider three different settings. First, we show that three modules are necessary and sufficient when all modules are equipped with a common compass, i.e., they agree on the direction and orientation of the $x$, $y$, and $z$ axes. Second, we show that four modules are necessary and sufficient when all modules agree on the direction and orientation of the vertical axis. Finally, we show that five modules are necessary and sufficient when all modules are not equipped with a common compass. Our results show that the shapes of the MRS in the 3D cubic grid have richer structure than those in the 2D square grid.