Paxos and Consensus Algorithms

Relevant source files

The following files were used as context for generating this wiki page:

• specifications/Paxos/manifest.json
• specifications/MultiPaxos/README.md
• specifications/raft/README.md
• specifications/byzpaxos/README.md
• README.md
• specifications/MultiPaxos-SMR/README.md
• specifications/fastpaxos/README.md
• specifications/diskpaxos/README.md
• specifications/egalitarian-paxos/README.md
• specifications/CASPaxos/README.md
• specifications/PaxosHowToWinATuringAward/README.txt

This repository contains a collection of TLA+ specifications for various consensus algorithms, with a primary focus on Paxos and its derivatives. These specifications serve as a comprehensive example library for modeling algorithms in TLA+, a corpus for testing TLA+ tools, and a set of case studies in applying formal methods. The examples are validated using tools like TLC, Apalache, and TLAPS to check for properties such as safety, liveness, and correctness under different computation models.

The specifications cover a wide range of consensus protocols, from the classic Paxos algorithm to more advanced and specialized versions like MultiPaxos, Fast Paxos, and Byzantine Paxos. They provide formal descriptions that can be model-checked to find subtle errors or prove correctness. This is particularly valuable for distributed consensus algorithms, where concurrency and fault tolerance introduce significant complexity.

Sources: README.md:4-10, specifications/PaxosHowToWinATuringAward/README.txt:41-43

Overview of Consensus Algorithm Specifications

The repository includes TLA+ specifications for a variety of consensus algorithms. Each specification typically includes TLA+ modules (.tla), model configuration files (.cfg), and documentation. The table below summarizes some of the key Paxos variants and other consensus algorithms available.

Algorithm	Author(s)	Computation Models	Properties Checked
Paxos	Leslie Lamport	-	-
MultiPaxos	Giuliano Losa	Crashes, lost/duplicated messages	Correctness
MultiPaxos-SMR	Guanzhou Hu	-	TypeOK, Linearizability
Byzantizing Paxos	Leslie Lamport	Byzantine faults	Refinement
Fast Paxos	Leslie Lamport	Crashes, lost/duplicated messages	Nontriviality, Consistency
Simplified Fast Paxos	Lim Ngian Xin Terry, Gaurav Gandhi	Crashes, lost/duplicated messages	Nontriviality, Consistency
Disk Paxos	Leslie Lamport, Giuliano Losa	Crashes, inaccessibility	SynodSpec
Egalitarian Paxos	Iulian Moraru	Crashes	Nontriviality, Stability, Consistency
CASPaxos	Tobias Schottdorf	Crashes, lost messages	Correctness
Raft	Diego Ongaro	-	-

Sources: README.md:100, specifications/MultiPaxos/README.md:2-6, specifications/MultiPaxos-SMR/README.md:9, specifications/byzpaxos/README.md:3, specifications/fastpaxos/README.md:2-6, specifications/SimplifiedFastPaxos/README.md:5-6, specifications/diskpaxos/README.md:6-7, specifications/egalitarian-paxos/README.md:2-6, specifications/CASPaxos/README.md:2-6, specifications/raft/README.md

The following diagram illustrates the relationship and variety of Paxos-based specifications within the repository.

This diagram shows that the repository contains specifications for both Paxos and Raft, with a significant number of variants derived from or related to the classic Paxos algorithm.

Sources: README.md:100,102,105,116,118,121,123,125,137

Classic Paxos

The foundational Paxos algorithm is specified, providing a basis for understanding consensus. The specification is broken down into multiple modules that build upon each other, starting from a high-level consensus problem and refining it into the full Paxos algorithm. This approach is detailed in the "Paxos - How to Win a Turing Award" lecture materials.

The core modules for the classic Paxos specification are:

• Consensus.tla: Defines the abstract consensus problem.
• Voting.tla: Specifies a voting-based algorithm that implements consensus.
• Paxos.tla: The full Paxos algorithm specification, which implements the voting algorithm.

Sources: specifications/PaxosHowToWinATuringAward/README.txt:32-37, specifications/Paxos/manifest.json:9-58

This modular structure is designed to help users understand the algorithm progressively.

The diagram shows the refinement hierarchy of the TLA+ specifications for the Paxos algorithm.

Sources: specifications/PaxosHowToWinATuringAward/README.txt:32-37

MultiPaxos and State Machine Replication (SMR)

MultiPaxos extends Paxos to agree on a series of values, forming the basis for State Machine Replication (SMR). The repository contains two notable MultiPaxos specifications:

1. Abstract MultiPaxos: An abstract specification of Generalized Paxos.
2. Practical SMR-style MultiPaxos: A more concrete specification that closely models how a real SMR system would implement MultiPaxos for replicating a log of commands.

The SMR-style specification is particularly useful as it models a practical implementation with explicit client commands, log replication, and a client-viewed linearizability property for safety.

Sources: specifications/MultiPaxos/README.md:2, specifications/MultiPaxos-SMR/README.md:3-5,18-21

The data flow in the SMR-style MultiPaxos involves clients, a leader, and follower servers replicating a log.

This flowchart illustrates the high-level message flow in the SMR-style MultiPaxos implementation for handling client requests.

Sources: specifications/MultiPaxos-SMR/README.md:11-12

Specialized Paxos Variants

The repository contains several specialized variants of Paxos, each addressing different requirements or optimizations.

Byzantine Paxos (ByzPaxos)

This specification, authored by Leslie Lamport, adapts the Paxos algorithm to tolerate Byzantine failures, where nodes can behave arbitrarily or maliciously. The specification is developed using refinement, starting from a non-fault-tolerant version and incrementally adding the logic to handle Byzantine behavior.

Sources: specifications/byzpaxos/README.md:3, README.md:85

Fast Paxos

Fast Paxos is an extension of classic Paxos that can achieve consensus in one communication step in the absence of failures or contention, reducing latency compared to the two steps of classic Paxos.

Sources: specifications/fastpaxos/README.md:2, README.md:123

Disk Paxos

Disk Paxos is an adaptation of the Paxos algorithm for an environment where processes communicate via shared disks rather than message passing. The specification models crashes and disk inaccessibility.

Sources: specifications/diskpaxos/README.md:2,6

Egalitarian Paxos (EPaxos)

EPaxos is a leaderless replication protocol based on Paxos. It aims to provide better performance by removing the single leader bottleneck, allowing any replica to initiate commands. The TLA+ specification checks for properties like nontriviality, stability, and consistency.

Sources: specifications/egalitarian-paxos/README.md:2,5-6

CASPaxos

CASPaxos extends single-decree Paxos to implement a distributed compare-and-swap (CAS) register. This is useful for building higher-level distributed data structures. The specification models crashes and lost messages.

Sources: specifications/CASPaxos/README.md:2,5

Verification and Model Checking

A key purpose of these specifications is to enable formal verification using the TLC model checker. The .cfg files associated with each specification define models with specific constants and constraints to be checked.

Commonly checked properties include:

• Consistency/Correctness: Ensures that all nodes agree on the same value.
• Linearizability: For SMR systems, ensures that the replicated service behaves as if all operations were executed sequentially on a single copy.
• Nontriviality: Ensures that a value is actually chosen.
• Liveness: Checks for properties like termination (e.g., a value is eventually chosen), often under fairness assumptions.
• Deadlock Freedom: Ensures the system does not reach a state where no progress can be made.

The models are checked against various computation models that consider failures such as process crashes, message loss, and message duplication.

Sources: README.md:13,18, specifications/MultiPaxos-SMR/README.md:9,19, specifications/MultiPaxos/README.md:6, specifications/fastpaxos/README.md:6

Conclusion

The TLA+ Examples repository provides an invaluable resource for learning, implementing, and verifying consensus algorithms. The extensive collection of Paxos specifications, from the foundational algorithm to numerous practical and theoretical variants, demonstrates the power of formal methods in designing and analyzing complex distributed systems. By providing formally specified and model-checked examples, the repository helps developers gain a deeper, unambiguous understanding of these critical algorithms and provides a solid foundation for building reliable distributed applications.

Sources: README.md:4-7, specifications/PaxosHowToWinATuringAward/README.txt:44-46