Efficient Self-stabilizing Simulations of Energy-Restricted Mobile Robots by Asynchronous Luminous Mobile Robots

TL;DR

This study introduces protocols that simulate LUMI robots in Rsynch using 4k colors in Ssynch and 5k colors in Asynch, for algorithms that employ k colors, and establishes that for n=2, Rsynch can be optimally simulated in Asynch using a minimal number of colors.

Abstract

In this study, we explore efficient simulation implementations to demonstrate computational equivalence across various models of autonomous mobile robot swarms. Our focus is on Rsynch, a scheduler designed for energy-restricted robots, which falls between Fsynch and Ssynch. We propose efficient protocols for simulating n(>=2) luminous (LUMI) robots operating in Rsynch using LUMI robots in Ssynch or Asynch. Our contributions are twofold: (1) We introduce protocols that simulate LUMI robots in Rsynch using 4k colors in Ssynch and 5k colors in Asynch, for algorithms that employ k colors. This approach notably reduces the number of colors needed for Ssynch simulations of Rsynch, compared to previous efforts. Meanwhile, the color requirement for Asynch simulations remains consistent with previous Asynch simulations of Ssynch, facilitating the simulation of Rsynch in Asynch. (2) We establish that for n=2, Rsynch can be optimally simulated in Asynch using a minimal number of colors. Additionally, we confirm that all our proposed simulation protocols are self-stabilizing, ensuring functionality from any initial configuration.

Figures (9)

• Figure 1: Transition diagram representations of protocol $\textup{SIM}^{RS}_{S}$((a)), and self-stabilizing protocol ss-$\textup{SIM}^{RS}_{S}$((b)). The label in the nodes represents the color of the light of the executing robot. The label of an edge expresses the condition on the lights of all the other robots that must be satisfied for the transition to occur. The notation "$\forall A,B$" means: "$\{\textup{Light\lbrack{r}\rbrack}\ |\ \forall r \in R\} = \{A,B\}$", "$\exists A$" means: "$\exists r \in R, \textup{Light\lbrack{r}\rbrack} \in \{\textup{A}\}$", "$\nexists A$" means: "$\{\textup{Light\lbrack{r}\rbrack}\ |\ \forall r \in R\} \cap \{A\} = \emptyset$". Conditions, colored red in (b) are newly added.
• Figure 2: Transition diagram of configurations (protocol $\textup{SIM}^{RS}_{S}$(the left part of this figure) and self-stabilizing ss-$\textup{SIM}^{RS}_{S}$).
• Figure 3: Transition diagram representations of (a) protocol $\textup{SIM}^{RS}_{A}$ and (b) self-stabilizing protocol ss-$\textup{SIM}^{RS}_{A}$. The condition colored red in (b) is newly added to $\textup{SIM}^{RS}_{A}$ to achieve self-stabilization, but not newly added in ss-$\textup{SIM}^{RS}_{A}$ are omitted.
• Figure 4: Transition diagram of configurations (protocol $\textup{SIM}^{RS}_{A}$(the left part of this figure) and self-stabilizing ss-$\textup{SIM}^{RS}_{A}$).
• Figure 5: (a) Transition diagram of protocol SIM-$2^{RS}_A$. (b) Transition diagram of state transitions (protocol SIM-$2^{RS}_A$)

Theorems & Definitions (41)

• theorem thmcountertheorem
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary