This directory contains proofs for Basic Paxos for single valued consensus.

1. PaxosLam.tla: TLA+ specification of Basic Paxos as in https://github.com/tlaplus/v1-tlapm/blob/master/examples/paxos/Paxos.tla. Following modifications have been made to the file:

   1. Moved lemmas QuorumNonEmpty and NoneNotAValue after definition of Spec. This is done to get true specification line count easily.

   2. Commented the following:

      1. chosenBar, line 541
      2. C, line 543
      3. Theorem Refinement, line 549---567

      This is done because our goal is to prove Safety of the specification, not to prove that it also refines some other specification.

2. PaxosHistVarNFM18.tla: TLA+ specification and proof of Basic Paxos using History variables as described in [3].

This directory contains proofs for Multi-Paxos for multi valued consensus.

1. MultiPaxosFM16.tla: TLA+ specification and proof of Multi-Paxos as described in [1].

2. MultiPaxos.tla: An improved TLA+ specification and proof of Multi-Paxos compared to MultiPaxosFM16.tla. These improvements are described in [2].

3. MultiPaxosHistVarNFM18.tla: TLA+ specification and proof of Multi-Paxos using History variables as described in [3].

This directory contains proofs for Multi-Paxos with Preemption. Preemption specifies when and how proposers abandon their current proposal number and pick a new one.

1. MultiPaxosPreemptionFM16.tla: TLA+ specification and proof of Multi-Paxos with Preemption as described in [1].

2. MultiPaxosPreemption.tla: An improved TLA+ specification and proof of Multi-Paxos with Preemption compared to MultiPaxosPreemptionFM16.tla. These improvements are described in [2].

3. MultiPaxosPreemptionHistVarNFM18.tla: TLA+ specification and proof of Multi-Paxos with Preemption using History variables as described in [3].

The MultiPaxos proofs in files *FM16.tla were checked using TLAPS version 1.5.3 and 1.5.6. The improved MultiPaxos proofs in other files were checked using version 1.5.6. A summary of the proof statistics are reported in [2].

[1] Chand, S., Liu, Y. A., & Stoller, S. D. (2016, November). Formal Verification of Multi-Paxos for Distributed Consensus. In International Symposium on Formal Methods (pp. 119-136). SpringerLink

[2] Chand, S., Liu, Y. A., & Stoller, S. D. (2019, January). Formal Verification of Multi-Paxos for Distributed Consensus. arXiv preprint arXiv:1606.01387. arXiv

[3] Chand, S., & Liu, Y. A. (2018, April). Simpler specifications and easier proofs of distributed algorithms using history variables. In NASA Formal Methods Symposium (pp. 70-86). SpringerLink