AWS 用 Lean 4 数学证明 Aurora Grover 数据修复协议的鲁棒性

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Automated reasoning

Proving the robustness of AWS Aurora’s data distribution protocol in Lean4

Large language models write both the formal specifications and proofs of the protocol, and the Lean 4 theorem prover checks every step, accelerating protocol development.

BySamuel Tenka,Kanna Shimizu

November 2, 2026

3 min read

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Key takeaways

Amazon's Automated Reasoning group mathematically proved in Lean4 that the core data-repair-and-migration protocol of Aurora's Grover keeps its promise. At the default sixfold redundancy, that promise is robustness to simultaneous failure of any two of six boxes.
The proof does not assume a particular number of boxes, so it covers any Grover redundancy setting, not just the default six. It shows that once data is safely stored, it stays safely stored through every later round of repair and reshuffling. An LLM proposed the Lean 4 specification and attempted a proof or counterexample for each proposal, and Lean 4 checked the result.
Formalizing Grover in Lean 4 exposed many ambiguities and typos in its documentation that, read literally, describe a protocol that can lose data. One example is an unrecorded hidden assumption whose violation could cause data loss at higher-than-default redundancies.

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Much of our digital data — our family photos, novel drafts, medical histories — exists as electrical patterns in metal boxes subject to grid outages and disasters natural and artificial. Any given box has an intolerably high chance of failure, so we copy our data into several metal boxes in different geographical regions. Redundancy protects data.

But what if we want to change our data — to edit that novel or add a new blood test to that medical record? We send our desired edits to all the boxes, but we can’t assume reliable communication: some box’s received edits may have gaps; some boxes might miss out on other boxes’ updates; and different boxes may hear about an edit at different times.

Keeping redundant data synchronized and up to date requires a delicate dance of distributed updates. AWS Aurora’s distributed protocol Grover choreographs that dance for database storage. For example, Grover’s default setting has sixfold redundancy and promises to be robust to simultaneous failure of any two out of six boxes.

Image 54: Redundant data distribution.png

A simple example of repairing redundant data distributed across servers. Six redundant boxes (green) start in agreement about the data they contain (left). Then two boxes fail (red). We recruit two spare boxes (grey) as replacements, then copy data from any of the healthy boxes to the replacements (now green). The result is restored redundancy (right).

In Amazon’s Automated Reasoning group, we have mathematically proven that Grover’s core data-repair-and-migration protocol keeps that promise. The proof, like a Sudoku solution, can be checked without trusting the process that found it (in our case, generation by large language models). To enable automatic proof checking, we express our claims and their proofs in Lean 4, a language and toolkit for expressing and proving claims with mathematical rigor.

Step 1: Initial specification of the Grover protocol in Lean 4

Our first challenge was to precisely define the Grover protocol in Lean 4. We wrote a Lean 4 template by hand, a template general enough for any distributed protocol, then had LLMs fill it in for Grover from the existing natural-language documentation.

During this process, we found many ambiguities and typos in the Grover documentation that, if taken literally, would describe a protocol that can lose data. One example is a hidden assumption whose violation would sometimes cause data loss when Grover scaled to higher-than-default redundancies. The documentation authors were aware that Grover makes that assumption but forgot to record it.

Our work surfaced such safety requirements and recorded them for those who will maintain and modify Grover. Formalization has therefore not only produced a verified reference algorithm on which to base future implementations but has also helped improve existing documentation.

Step 2: Coevolving the proof and the specification

Using the Lean 4 version of the Grover protocol, we proved that once a piece of data is safely stored, it stays safely stored through every later round of repair and reshuffling. The proof doesn't assume a particular number of boxes, so it covers whatever redundancy setting Grover runs with, not just the default six.

We used an LLM to iteratively propose the Lean 4 specification of the protocol and the same LLM to attempt a proof or a counterexample for each proposal. The search usually produced a counterexample, in which case we made a minimal, most-faithful-to-documentation change to the specification to rule it out, then repeated the process. We converged to a formal protocol plus correctness proof, both in Lean 4. To check that every deviation from the documentation was indeed necessary, we automatically constructed counterexamples showing that leaving any single change out would break the associated theorems.

Finally, we simplified the formal protocol from an explicit style — the style that automatic reasoning works with most easily, which spells out every transition in full — into a human-readable style, a compact notation we designed for distributed protocols, in which the whole protocol fits in one file an expert can read. We formally proved the two versions equivalent. This allows human experts on the Grover team, as ongoing work, to check that the proven protocol is indeed the intended protocol.

Image 55: Web of consistency.png

A “web of consistency” can help with the design and checking of enhancements to a protocol. Green blocks are formal artifacts, such as the robustness proof for Grover’s data repair subsystem. The blue blob is our quasi-automatic workflow for translating natural-language documentation into Lean4, consisting of templates, macros, and prompts.

Our proof of concept illustrates a new working method that may accelerate Amazon’s broader work on distributed protocols and strengthen the end results. Armed with automatic correctness checkers, developers can optimize aggressively and tailor to complicated customer uses, without fear that new complexity invites bugs. If we succeed in applying our automated-reasoning methods to general protocols, then developers can enhance distributed protocols at a rate that scales with their appetite for enhancement, not their appetite for risk.

Acknowledgements: Alexey Gotsman, Mark Tuttle

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Principal Applied Scientist, Ads Measurement Science

US, NY, New York

The Measurement Science team in the Measurement, Ad Tech, and Data Science (MADS) organization of Amazon Ads serves a centralized role developing solutions for a multitude of performance measurement products. We create solutions which measure the comprehensive impact of their ad spend, including sales impacts both online and offline and across timescales, and provide actionable insights that enable our advertisers to optimize their media portfolios. We leverage a host of scientific technologies to accomplish this mission, including Generative AI, classical ML, Causal Inference, Natural Language Processing, and Computer Vision. This Principal Applied Scientist leads the technical vision and execution for Amazon Advertising's performance measurement. Our measurement products and tools help customers understand the holistic impact of their spend on Amazon Ads and optimize their portfolios. This is a complex, multi-disciplinary scientific space spanning machine learning, causal inference, statistics and experimentation, natural language processing, and generative AI. As a GenAI-first organization, we build foundational and agentic models that power advertiser use cases across Ads, while empowering our Applied Scientists to directly build and ship products. You will guide a team of scientists with expertise across different domains to come together and innovate on some of the hardest problems in advertising, made more challenging by evolving privacy regulations and technological changes. In measuring the comprehensive impact of advertising, key challenges you will tackle include capturing impact: 1) both for identified and increasingly unidentified web traffic in light of an evolving global privacy landscape; 2) for outcomes across retailers, including both within and beyond Amazon; and 3) for both short- and long-term impacts of advertising. Key job responsibilities * Driving the scientific vision of the teams in your organization and you advise and influence its technical leadership on performance measurement models and products. * Identifying, tackling, and proposing innovative solutions to intrinsically hard, previously unsolved problems. You solve problems pragmatically, applying judgment and experience to balance trade-offs between competing interests. * Bringing clarity to complex problems, probing assumptions, illuminating pitfalls, fostering shared understanding, and guiding towards effective solutions. * Serving and being recognized by internal and external peers as a thought leader in modeled measurement in Ads. You will engage routinely with the broader scientific community, e.g. as a reviewer, program committee member, workshop organizer, or chair at peer-reviewed internal/external conferences. * Leading by example, with your papers, analyses, and code setting the standard in your organization for scientific engineering excellence. * Leading science and design reviews, aligning teams across your organization around coherent algorithmic strategies. * Influencing your team’s science and business strategy by driving one or more team roadmaps contributing to the organization’s roadmap and taking responsibility for some organizational goals. You drive multiple new product features from inception to production launch. * Guiding the career development of others, actively mentoring and educating the larger applied science community on trends, technologies, and best practices. You drive the documentation of scientific innovation. You attract bar-raising talent from academia and industry. * Staying abreast of academic and industry trends, knowing the state of the art, and building on top of existing solutions instead of reinventing when possible. About the team We are a team of scientists across Applied, Research, Data Science and Economist disciplines. You will work with colleagues with deep expertise in ML, NLP, CV, Gen AI, and Causal Inference with a diverse range of backgrounds. We partner closely with top-notch engineers, product managers, sales leaders, and other scientists with expertise in the ads industry and on building scalable modeling and software solutions.

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Image 60: View from space of a connected network around planet Earth representing the Internet of Things.

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