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A Machine-checked Proof of Correctness of Pastry


Azmy,  Noran
Automation of Logic, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

Weidenbach,  Christoph
Automation of Logic, MPI for Informatics, Max Planck Society;

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Azmy, N. (2016). A Machine-checked Proof of Correctness of Pastry. PhD Thesis, Universität des Saarlandes, Saarbrücken.

A distributed hash table (DHT) is a peer-to-peer network that offers the function of a classic hash table, but where different key-value pairs are stored at different nodes on the network. Like a classic hash table, the main function provided by a DHT is key lookup, which retrieves the value stored at a given key. Examples of DHT protocols include Chord, Pastry, Kademlia and Tapestry. Such DHT protocols certain correctness and performance guarantees, but formal verification typically discovers border cases that violate those guarantees. In his PhD thesis, Tianxiang Lu reported correctness problems in published versions of Pastry and developed a model called LuPastry, for which he provided a partial proof of correct delivery of lookup messages assuming no node failure, mechanized in the TLA+ Proof System. In analyzing Lu's proof, I discovered that it contained unproven assumptions, and found counterexamples to several of these assumptions. The contribution of this thesis is threefold. First, I present LuPastry+, a revised TLA+ specification of LuPastry. Aside from needed bug fixes, LuPastry+ contains new definitions that make the specification more modular and significantly improve proof automation. Second, I present a complete TLA+ proof of correct delivery for LuPastry+. Third, I prove that the final step of the node join process of LuPastry/LuPastry+ is not necessary to achieve consistency. In particular, I develop a new specification with a simpler node join process, which I denote by Simplified LuPastry+, and prove correct delivery of lookup messages for this new specification. The proof of correctness of Simplified LuPastry+ is written by reusing the proof for LuPastry+, which represents a success story in proof reuse, especially for proofs of this size. Each of the two proofs amounts to over 32,000 proof steps; to my knowledge, they are currently the largest proofs written in the TLA+ language, and---together with Lu's proof---the only examples of applying full theorem proving for the verification of DHT protocols