I've always been intrigued by formal methods but when I try to think of an application of TLA+ my problems are either too trivial to need formal analysis or too complicated for formal analysis to appear feasible.
The team I was on in AWS used it for the implementation of a particular component that needed to operate at large scale and had absolutely zero tolerance of failure. Two engineers learned TLA+, started writing up the logic and discovered various ways in which the logic would work on small scale but wouldn't scale up to parallel operation across multiple servers. It's fair to say TLA+ saved our bacon there before we'd written a single line of service code.
In the end we actually got the component out to production faster than predicted. The TLA+ code gave solid shape for the actual production code, and we were quickly able to prove that the code was valid and that things were working correctly.
In my current role in Oracle Cloud Infrastructure, there is actually a small dedicated team of engineers that help services use TLA+ for components, headed by Chris Newcombe, one of the authors of the TLA+ whitepaper that came out of Amazon (https://lamport.azurewebsites.net/tla/formal-methods-amazon....)
My current team has just started exploring it for use for a component that similarly has little tolerance for error. It's not in the same territory of "if it fails it's catastrophic", thankfully, but if it fails it's a major inconvenience for customers.
My experience is that, using TLA+, or other tools like it, encourages you to think precisely about the problem and at the same time exclude unnecessary details. That alone I find useful in terms of the insights it gives e.g. surfacing interesting invariants, or alternatively, invalidating them. In any case, you use the model-checker (TLC) to test these assumptions quite easily, and then bring these invariants into your code, either in the form of assertions, contracts or perhaps in property-based testing. In any case, I would say the process of writing such a specification yields benefits. Perhaps this can best be summed up by something I think Leslie Lamport once said: "If you're not writing when you're thinking, you only think you're thinking."
I should add that despite claims about TLA+ being particularly well-suited to concurrency problems, my implementation targets have never been 'concurrent' in the 'multi-threaded' sense. Rather, they have been single-threaded event-driven state machines, and I personally find that it excels in this space. However, as you will find, in TLA+, concurrency is just a matter of abstraction...
I have found that, because TLA+ doesn't 'generate code', many colleagues struggle to see the tangible benefit. I personally think that there is a gap between the whiteboard and the editor which tools like TLA+ fill very nicely. It's not a panacea or a magic bullet, and like any tool you must still exercise judgement about when to use it, and how to use it effectively, which includes understanding the limitations of the tooling (TLC cannot check arbitrary specifications).
But if I find myself faced with a potentially tricky algorithm, or indeed want to understand an existing algorithm, I reach for TLA+.
* https://andy.hammerhartes.de/finding-bugs-in-systems-through...
* http://roscidus.com/blog/blog/2019/01/01/using-tla-plus-to-u...
The service was essentially a key/value store. If one region didn't know about a key, it would recursively ask the other regions in parallel for the data.
Eventually we needed to add support for deleting data. We deleted data that hadn't been read in a few months.
We used TLA+ to prove our region-sync code and deletion code interacted properly along with regular get/put operations.
A few years back I had an interesting discussion with one of its authors and was shown the real model and code. Interestingly, they generated C from the TLA+ spec and this code generator was "unproven". However, the software itself had high coverage requirements and needed to be tested extensively (100% code/branch/MCD coverage, I don't quite remember).
One of the points he made was that the upfront cost pays for itself by avoiding bugs that are hard to debug.
[0]: https://drive.google.com/file/d/1rAn3N5hViv3xNe2E55lMzpFFym1...
In general it is quite useful exercise to even just model existing code line-by-line and see what possible states it can have. I'm still learning though, next step would be to try TLAPS.
Defects surfaced in the model may not imply defects in the software e.g. obscure race conditions in the model that cannot actually occur in implementation due to details not captured in the model. But it forces you to explicitly acknowledge and investigate that.
This would have saved me tons of work in hindsight.