# The Evolution and Optimization Strategies of a PBFT Consensus Algorithm for Consortium Blockchains
## 3 An Introduction to Mainstream Consensus Algorithms
### The Mainstream Consensus Algorithms of Private Blockchains
#### 3.2.1 Paxos
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Author** & **Year** & **Improvement Method** & **Core Mechanism** & **Limitations** & **Reward/Penalty Mechanism** \\ \hline [PERSON] [39] & 2022 & CHOST Attack Defense & Defends against long-range attacks and validator collusion. & May require additional overhead for detecting and preventing attacks. & No explicitly defined incentive mechanism. \\ \hline [PERSON] [35] & 2023 & Ethereum Decentralization Analysis & Quantifies the distribution of validation nodes and staking concentration. & Focuses on analysis rather than introducing a solution to decentralization issues. & No reward or penalty mechanism. \\ \hline [PERSON] [33] & 2023 & \begin{tabular}{c} Ransomware Attack Game Theory Model \\ \end{tabular} & \begin{tabular}{c} Models revenue matrices to reduce the success rate of \\ \end{tabular} & \begin{tabular}{c} Effectiveness depend on accurate threat modeling. \\ \end{tabular} & \begin{tabular}{c} No explicitly defined incentive mechanism. \\ \end{tabular} \\ \hline [PERSON] [34] & 2023 & \begin{tabular}{c} Validator Exit \\ Verification Tool \\ \end{tabular} & \begin{tabular}{c} Uses TLA+ modeling for exit \\ processes to ensure \\ state consistency. \\ \end{tabular} & \begin{tabular}{c} TLA+ modeling requires \\ high computational effort \\ \end{tabular} & No reward or penalty mechanism. \\ \hline [PERSON] [38] & 2024 & \begin{tabular}{c} EigenLayer and \\ Lido Optimization \\ \end{tabular} & \begin{tabular}{c} Resstaking and liquidity tokens to \\ lower staking thresholds. \\ \end{tabular} & \
\begin{tabular}{c} Increased risk of liquidity \\ fragmentation and potential \\ security vulnerabilities in \\ restaking mechanisms. \\ \end{tabular} &
\begin{tabular}{c} Provides liquidity \\ incentives and lower \\ staking requirements. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(Cont.\)[PERSON] et al. [44] are pioneers in using structural invariants derived from [PERSON]'s original specification to apply automated proof techniques, simplifying the verification of this historically complex algorithm. [PERSON] et al. [45] proposed an automatic alert method based on Paxos algorithm for anomalous data in power grid reports. The grid report data is automatically collected using RPA technique, and then the isolated forest algorithm with incremental learning of subforest gradual update is used to detect anomalies in the collected data. [PERSON] et al. [46], in \"Relaxed Paxos\", revisited the quorum requirements, proposing a two-phase consensus model that reduces synchronization overhead while maintaining correctness. [PERSON] et al. [47] introduced Paxos, a leaderless variant of Multi-Paxos, which employs a random backoff mechanism to mitigate adversarial attacks. [PERSON] et al. [48] combined Multi-Paxos with modular state machine replication in Mandator and Sporades, achieving high-throughput consensus in geographically distributed systems. [PERSON] et al. [49] conducted empirical evaluations using NS3 simulations to quantify the performance trade-offs between Paxos, Raft, and PBFT, emphasizing the relevance of consensus mechanisms for delay-sensitive applications such as blockchain systems. Theoretical advancements continue to shape the trajectory of the field. [PERSON] et al. [50] contributed a declarative framework using three-valued modal logic to formally specify distributed algorithms like Paxos, enabling a rigorous analysis of system behavior through mathematical abstraction. This work reflects the growing intersection between formal methods and practical consensus engineering. As shown in Table 5, on the theoretical level, some studies focus on the development of formal verification tools. By using automated theorem proving or modal logic frameworks, these studies shift the validation of Paxos correctness from empirical derivation to mathematical provability, significantly enhancing the rigor of protocol design.