Byzantine Generals Problem August 16, 2024 by Marcos Benevides

:ID: 68219535-efeb-4835-9670-fd2747376cf3

A summary from (Lamport, Shostak, and Pease 2019) paper.

Introduction

A reliable computer system must be able to cope with the failure of one or more of its components. A failed component may exhibit a type of behavior that is often overlooked namely, sending conflicting information to different parts of the system. The problem of coping with this type of failure is expressed abstractly as the Byzantine Generals Problem.

The Classic Problem

• Each division of Byzantine army is directed by its own general
• Generals, some of which are traitors, communicate each other by messengers

The generals must have an algorithm that guarantess the following properties:

1. All loyal generals decide upon the same plan of action.
2. A small number of traitors cannot cause the loyal generals to adopt a bad plan.

which can then be restated as:

1. All loyal generals receive the same information upon which they will somehow get to the same decision
2. The information sent by a loyal general should be used by all the other loyal generals.

This problem is then further reformulated into what into a series of one commanding general and multiple lieutenants problem.

Byzantine Generals Problem: A commanding general must send an order to his \(n - 1\) lieutenant generals such that,

• All loyal lieutenants obey the same order.
• If the commanding general is loyal, then every loyal lieutenant obeys the

order he sends.

Impossibility Results

3-General Problem with 1 Traitor

This is a special case of BGP and it’s unsolvable because a loyal lieutenant cannot distinguish betwee a loyal/traitorous officer when conflicting information arrives.

                 👑
               ┌─────┐
   ┌───────────│  L  │─────────┐
   │           └─────┘         │
   │"Attack"                   │"Attack"
┌─────┐                      ┌─────┐
│  L  │<─────────────────────│  T  │
└─────┘   "He said Retreat"  └─────┘


                 👑
               ┌─────┐
   ┌───────────│  T  │─────────┐
   │           └─────┘         │
   │"Attack"                   │"Retreat"
┌─────┐                      ┌─────┐
│  L  │<─────────────────────│  L  │
└─────┘   "He said Retreat"  └─────┘

References