Gathering Semi-Synchronously Scheduled Two-State Robots

TL;DR

The paper addresses the minimal memory (lights) and conditions required for deterministic Gathering of $n$ anonymous mobile robots under $SSYNCH$, focusing on 2-color ${\mathcal{FST\!A}}$ and ${\mathcal{FCOM}}$. It establishes a lower bound showing 2-color gathering is impossible in $SSYNCH$ for these models under standard assumptions, including rigidity and chirality, and extends the discussion to self-stabilizing ${\mathcal{FST\!A}}$ where unlimited colors do not help; crucially, it demonstrates that by excluding two specific initial patterns, a 2-color ${\mathcal{FST\!A}}$ algorithm can achieve Gathering in $SSYNCH$ with Non-Rigid$(+\delta=)$. The main contribution is a detailed three-phase constructive algorithm for ${\mathcal{FST\!A}}$ using two colors: (i) transform to an $OnLDS$ configuration with $|\,\overline{pq}\,|\ge 2d$, (ii) reduce to a 2-point state with $2d-\epsilon \le |\overline{pq}| < 2d$, and (iii) reach Gathering by moving endpoints to the midpoint while toggling colors from $A$ to $B$, guided by the $A3P$ and $A4P$ predicates. Together, these results delineate the boundary between possibility and impossibility for color-limited gathering in semi-synchronous robotics and inform future work on unconditional gathering and asynchronous settings.

Abstract

We study the problem \emph{Gathering} for $n$ autonomous mobile robots in synchronous settings with a persistent memory called \emph{light}. It is well known that Gathering is impossible in the basic model ($OBLOT$) where robots have no lights, even if the system is semi-synchronous (called SSYNCH). Gathering becomes possible, however, if each robot has a light of some type that can be set to a constant number of colors. In the $FCOM$ model, the robots can only see the lights of other robots. In the $FSTA$ model, each robot can only observe its own light. In the $LUMI$ model, all robots can see all lights. This paper focuses on $FSTA$ robots with 2-colored lights in synchronous settings. We show that 2-color $FSTA$ and $FCOM$ robots cannot solve Gathering in SSYNCH without additional conditions, even with rigid movement and agreement of chirality and the minimum moving distance. We also improve the condition of the previous gathering algorithm for $FSTA$ robots with 2-color working in SSYNCH.

Figures (3)

• Figure 1: The prohibited initial configurations
• Figure 2: The transition of configurations in Gathering-${\mathcal{FST\!A}}$-Robots($r_i$).
• Figure 3: Configurations in $A3P$ and $A4P$.

Theorems & Definitions (11)

• theorem thmcountertheorem: 3 colors of ${\mathcal{FCOM}}$ and ${\mathcal{FST\!A}}$ robots are necessary
• proof
• theorem thmcountertheorem
• lemma thmcounterlemma
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof