Use of Information, Memory and Randomization in Asynchronous Gathering

TL;DR

This paper explores how initial information, unbounded memory, and randomization enable mobile agents to gather on a grid, demonstrating that all three are necessary for guaranteed gathering of all configurations.

Contribution

It introduces a probabilistic state machine with initial input, unbounded memory, and randomization that can gather all configurations, and proves the necessity of these features together.

Findings

Probabilistic machine can gather all configurations with probability 1.

Machines lacking any one of the three features cannot guarantee gathering.

Deterministic Turing Machines and finite automata can gather specific types of configurations.

Abstract

We investigate initial information, unbounded memory and randomization in gathering mobile agents on a grid. We construct a state machine, such that it is possible to gather, with probability 1, all configurations of its copies. This machine has initial input, unbounded memory, and is randomized. We show that no machine having any two of these capabilities but not the third, can be used to gather, with high probability, all configurations. We construct deterministic Turing Machines that are used to gather all connected configurations, and we construct deterministic finite automata that are used to gather all contractible connected configurations.