Minimum firing times of firing squad synchronization problems for paths in grid spaces

TL;DR

This paper investigates the minimal firing times for synchronization problems in grid paths, highlighting the current knowledge gaps and proposing directions for future research to establish their existence or nonexistence.

Contribution

The paper provides insights into the complexity of minimal firing times for grid paths and suggests research directions to determine their existence.

Findings

Minimal-time solutions are unknown and likely do not exist.

No proof currently exists for their nonexistence.

The paper proposes a framework for future studies.

Abstract

We consider the firing squad synchronization problems for paths in the two and the three-dimensional grid spaces. Minimal-time solutions of these problems are not known and are unlikely to exist. However, at present we have no proofs of their nonexistence. In this paper we show one result that suggests what type of study is necessary in order to prove their nonexistence.