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NishMath - #logic
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Igor Konnov@Protocols Made Fun
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Igor Konnov has successfully completed full proofs of consistency for the two-phase commit (2PC) protocol using the Lean 4 theorem prover, starting from a functional specification. This work builds upon previous efforts in specifying, simulating, and property-based testing 2PC in Lean 4. The entire process, including specification and simulation from prior work, took approximately 45 hours, with the proof writing itself consuming 29 hours. The rapid proof development was attributed to a correct inductive invariant discovered using the Apalache model checker, underscoring the benefits of combining formal specification with interactive theorem proving.
The Lean 4 proofs involved a modular approach, breaking down the verification into functional, propositional, and inductive components. Statistics show that the proofs, propositional and inductive, are about 15 times longer than the system code. Specifically, Functional.lean and System.lean contained 139 lines of code, Propositional.lean had 90, PropositionalProofs.lean comprised 275, and InductiveProofs.lean reached 1077 lines of code. This ratio aligns with the empirical standard in software verification, where proofs typically range from 10 to 20 times the length of the source code. Konnov also mentioned exploring an alternative route: proving equivalence between their specification in Propositional.lean and the Veil specification. The Veil examples repository already contains a version of two-phase commit, albeit slightly different from the TLA+ version and the Lean specification. He suggests that this could be a topic for future work. The consistency of the protocol was specified using TLA+, leveraging its methodology for defining TCConsistent. The Lean 4 code for the two-phase commit protocol can be found on GitHub.
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Michael Weiss@Diagonal Argument
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Recent discussions in mathematical concepts and programming tools cover a range of topics, including theoretical foundations and practical applications. Peter Cameron highlighted the Compactness Theorem for first-order logic, explaining its consequences and connections to topology. Also, a beginner's guide to sets has been published to explain how they work and some applications.
Noel Welsh presented a talk at Imperial College on dualities in programming, exploring the relationships between data and codata, calls and returns, and ASTs and stack machines. The use of adjoints in boolean operations was justified, and Daniel Lemire published an overview of parallel programming using Go. These discussions bridge the gap between abstract mathematical principles and their concrete uses in software development and programming paradigms.
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Michael Weiss@Diagonal Argument
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Research at the intersection of logic, descent theory, and category theory is currently underway, focusing on advanced mathematical concepts. Key areas of exploration include Galois descent in algebraic contexts, a concept with applications in algebraic geometry as detailed in texts by James Milne and lecture notes by Keith Conrad and Joshua Ruiter. Additionally, researchers are investigating equivariant bicategorical shadows and traces, aiming to generalize topological Hochschild homology.
This research extends to first-order categorical logic and quantum observables, contributing to a broader understanding of these abstract mathematical structures. A recent seminar at Vanderbilt University highlighted "Equivariant Bicategorical Shadows and Traces," where the presenter discussed a new framework of equivariant bicategorical shadows and explained why twisted THH is a g-twisted shadow, also exploring g-twisted bicategorical traces. This work builds upon the foundations laid by Ponto in defining bicategorical shadows, offering potential advancements in algebraic K-theory.
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