Before we start writing Lean. Perhaps we can start with something "dumber" like Rust or any typed program. If you want to write something correct, or you care about correctness, you should not be using dynamic languages. The most useful and used type of test is type checking.
Type errors, especially once you have designed your types to be correct by construction, is extremely, extremely useful for LLMs. Once you have the foundation correct, they just have to wiggle through that narrow gap until it figures out something that fits.
But from what I understood and read so far, I am not convinced of OP's "formal verification". A simple litmus test is to take any of your recent day job task and try to describe a formal specification of it. Is it even doable? Reasonable? Is it even there? For me the most useful kind of verification is the verification of the lower level tools i.e. data structures, language, compilers etc
For example, the type signature of Vec::operator[usize] in Rust returns T. This cannot be true because it cannot guarantee to return a T given ANY usize. To me, panic is the most laziest and worst ways to put in a specification. It means that every single line of Rust code is now able to enter this termination state.
> Perhaps we can start with something "dumber" like Rust or any typed program. If you want to write something correct, or you care about correctness, you should not be using dynamic languages. The most useful and used type of test is type checking.
Lean or TLA+ are to Rust/Java's type systems what algebraic topology or non-linear PDEs are to "one potato, two potatoes". The level of "correctness" achievable with such simple type systems is so negligible in comparison to the things you can express and prove in rich formal mathematics languages that they barely leave an impression (they do make some grunt work easier, but if we're talking about a world where a machine can do the more complicated things, a little more grunt work doesn't matter).
I once attended a talk by someone who is or was big in the node.js world. He opened with the premise, "a static type check is just a stand-in for a unit test."
I wanted to throw a shoe at him. A static type check doesn't stand in for "a" unit test; static typing stands in for an unbounded number of unit tests.
Put another way, this common misconception by users of languages like Javascript and Python that unit testing is just as good as type checking (plus more flexible) is a confusion between the "exists" and "for all" logical operators.
Plus, it is simply more enjoyable to design the types in your program than to write unit tests. The fun factor comes from operating on a higher level of abstraction and engages more of your brain’s puzzle-solving mode than just writing unit tests. Making yourself think about “for all x” rather than a concrete x forces your brain to consider deeply the properties of x being used.
> it is simply more enjoyable to design the types in your program than to write unit tests.
I have tried both and I have no idea what you're talking about.
> Making yourself think about “for all x” rather than a concrete x forces your brain to consider deeply the properties of x being used.
The entire point of dynamic typing is that you can think about interfaces rather than concrete types, which entails deep consideration of the properties of the object (semantics of the provided interface).
> A static type check doesn't stand in for "a" unit test; static typing stands in for an unbounded number of unit tests.
You have conflated "a static type check" with "static typing". Unit tests stand in, in the same way, for an unbounded number of states of real-world input. They're simply being subjected to a trial verification system rather than a proof system. It turns out that writing proofs is not very many people's idea of a good time, even in the programming world. And the concept of "type" that's normally grokked is anemic anyway.
> Put another way...
Rhetoric like this is unconvincing and frankly insulting. You pass off your taste and opinion as fact, while failing to understand opposed arguments.
Besides, it's not like types don't matter in dynamically typed languages. The (competent) programmer still needs to keep types in their head while programming. "Can this function work with a float, or must I pass an int?" "This function expects an iterable, but what happens if I pass a string?" Etc.
I started my career with JavaScript and Python, but over the years I've come to the conclusion that a language that hides types from programmers and does implicit conversion magic in the background does not deliver a better DX. It might make the language more approachable initially, and the idea of faster prototyping might be appealing, but it very quickly leads to maintenance problems and bugs. Before type hinting tools for Python became popular, I worked on many projects where `TypeError` was the #1 exception in Sentry by a large margin.
Gradual and optional typing is better than nothing, but IME if the language doesn't require it, most programmers are lazy and will do the bare minimum to properly add type declarations. Especially with things like TypeScript, which makes many declarations difficult to read, write, and understand.
I think that type inference is a solid middle ground. Types are still statically declared, but the compiler is smart enough to not bother the developer when the type is obvious.
> Before type hinting tools for Python became popular, I worked on many projects where `TypeError` was the #1 exception in Sentry by a large margin.
My experience is radically different. `ValueError` is far more common in my un-annotated Python, and the most common cause of `TypeError` anyway is the wrong order or number of arguments after a refactoring.
Good luck using static typing to model many real world unit tests for the programming languages people use most. I start with an easy example: those records should be sorted by date of birth. We can move on to more complicated scenarios.
The distinction you are trying to make is nonsensical in Python's object model. Types are inherently callable, and calling them constructs (i.e. instantiates) the type (normally; this can be overridden, by design). There is also no type->kind->category hierarchy; `type` itself is an object, which is its own type.
When you're at a level of theory where terms like "type constructor" are natural, it's unreasonable to expect any of it to be applicable to Python. This is why the Haskell people speak of dynamically-typed languages in the Python mold as "untyped" regardless of their attitude towards implicit casts.
And I love it, and have been using it for decades, and write beautiful things where the annotations hardly ever seem worth the effort — perhaps for documentation, but not for a static checker. Then I look at other, newer Pythonistas trying to figure out how to write complex generic type expressions (and sacrificing backwards compatibility as they keep up with the churn of Python figuring out how to offer useful annotation syntax) and deal with covariance vs contravariance etc. and I just smile.
You can run a third party linter on those comments, but you must hope that they're correct. There are usually some checks for that, but they're only reliable in trivial cases.
This is not static typing any more than "you can use emscripten to transpile JavaScript to C" means that JavaScript is a low level language with native assembly support. It's a huge step forward from "no system at all" and I'm thrilled it exists, but it's hardly the same thing.
Tell me more please: how does one use types in Python? Unfortunately I write Python professionally these days (it is the language that has all the libraries) and hate it with a passion.
I think it's possible to write correct systems with dynamic languages, just not the ones we commonly use like Python and JavaScript. I find Clojure, for example to be one example of a dynamic language that is pretty easy to manage and I attribute that to the immutable nature and data-centric ethos. I'm sure there are other dynamic languages that would work as well.
Now, I wouldn't necessarily use Clojure on a huge multi-organization codebase (maybe it's fine, this is outside of my experience with it), but it can be the right tool for some jobs.
Common Lisp as well. I can’t explain why, but type errors are just not something I struggle with in Common Lisp! But it is in JS and Python for sure. Maybe someone knows why it feels different?
I think it’s cause there’s less imperative code and side effects to track data transformations through.
Like any random JS/php app is probably a huge pile of loops and if statements. To track what happens to the data, you need to run the whole program in your head. “And now it adds that property to the object in the outer scope, and now that object gets sorted, now it hits the database… ok…”. Whereas in clojure most functions are either a single atomic transformation to a set of data, or batch of side effects. You still have to run it through your head, but you can do it more piece-by-piece instead of having to understand a 1,000 method with class states being auto loaded and mutated all over the place. Also you have a REPL to try stuff out as you go.
Dont get me wrong, I LOVE static types. Statically typed clojure would be the best fckin language ever. But there is definitely a wide gulf between a dynamic language like JS, and one like clojure!
I haven't done much with CL so I can only speculate, but I think stricter FP principles in general work to minimize the downsides of dynamic typing. CL, to my understanding, isn't the most "pure" when it comes to FP, but does a good job at giving the programmer a lot of power to constrain and explore systems.
The author is in the comfortable position of working on a system that does have a formal specification and a formally verified reference implementation. The post is not about how they wish things would work, but how their existing system (Cedar) works.
Regarding your point on Rust, the vast majority of software has nowhere near the amount of static guarantees provided by Rust. If you need more, use static memory allocation, that's what people do for safety critical systems. By the way, it seems that Rust aborts on OOM errors, not panics: https://github.com/rust-lang/rust/issues/43596
> To me, panic is the most laziest and worst ways to put in a specification.
This why the "existing programs don't have specs!" Hand-ringing is entirely premature. Just about every code base today has error modes the authors think won't happen.
All you have to do is start proving they won't happen. And if you do this, you will begin a long journey that ends up with a formal spec for, at least, a good part of your program.
Proving the panics are dead code is a Socratic method, between you and the proof assistant / type checker, for figuring out what your program is and what you want it to be :).
Yeah, Rust has been pretty good for formal verification so far. Hoare spec contracts I think are the way forward, especially since they fairly naturally flow from unittests. I've been using Hax to pretty good effect so far. I'm generally suspect that advances in Lean proof solving by dedicated models are that useful for program verification, compared to generalist models, though it could help lower costs a good bit.
I agree. And learning a typed language is significantly easier now that AI can explain everything. The types also help AI to write a correct code. A very positive feedback loop.
I find this discourse about AI and formal verification of software very confusing. It's like someone saying, let's assume I can somehow get a crane that would get that vintage car into my 15th floor apartment room for display in my living room, but what will I do with my suitcases?
All the problems mentioned in the article are serious. They're also easier than the problem of getting an AI to automatically prove at least hundreds of correctness properties on programs that are hundreds of thousand, if not millions of lines long. Bringing higher mathematics into the discussion is also unhelpful. Proofs of interesting mathematical theorems require ingenuity and creativity that isn't needed in proving software correct, but they also require orders of magnitude fewer lemmas and inference steps. We're talking 100-1000 lines of proof per line of program code.
I don't know when AI will be able to do all that, but I see no reason to believe that a computer that can do that wouldn't also be able to reconcile the formal statements of correctness properties with informal requirements, and even match the requirements themselves to market needs.
I think this misses a lot of reasons why learning verification is important. For instance learning the concept of invariants and their types such as loop invariants. They make reasoning about code in general easier, even if you never formally do any verification, it makes it easier to write tests or asserts(). A substantial amount of bugs are due to the program having a different state to that assumed by the programmer, and there are other tools that help with this. For example a statically typed language is a type of verification since it verifies a variable has a specific type and thus operations that can be performed on it, and limits the valid input and output range of any function. Languages like Rust are also verification in terms of memory correctness, and are also extremely useful tools.
"Beware of bugs in the above code; I have only proved it correct, not tried it." - Donald Knuth
Not that relevant in context as the code in question is used to conclude a formal proof, not the other way around. Buy hey, it is a common quote when talking about proving software and someone has to do it...
This is an aside because I agree with the author’s core point, but spelling, grammatical errors, and typos actually imply something authored by a human now. This sentence:
“It affects point number 1 because AI-assisted programming is a very natural fit fot specification-driven development.”
made me smile. Reading something hand made that hadn’t been through the filters and presses of modern internet writing.
- Lean will optimize peano arithmetic with binary bignums underneath the hood
- Property based checking and proof search already exist on a continuum, because counterexamples are a valid (dis)proof technique. This should surprise no writer of tactics.
- the lack of formal specs for existing software should become less a problem for greenfield software after these techniques go mainstream. People will be incentivized to actually figure out what they want, and successfully doing so vastly improves project management.
Finally, and most importantly, people thinking that there is a "big specification" and then "big implementation" are totally missing the mark. Remember tools like lean are just More Types. When we program with types, do we have a single big type and a single untyped term, paired together? Absolutely not.
As always, the key to productive software development is more and more libraries. Fancier types will allow writing more interesting libraries that tackle the "reusable core" of many tasks.
For example, do you want to write a "polymorphic web app" that can be instantiated with a arbitrary SQL Schema? Ideas like that become describable.
> the key to productive software development is more and more libraries
You had me until this statement. The idea that "more and more libraries" is going to solve the (rather large) quality problems we have in the software industry is .. misguided.
Don’t use a library unless you really need it. Someone recently recommended I add Zod to a project where I am only validating two different JSON objects in the entire project. I like Zod, but I already wrote the functions to progressively prove out the type in vanilla JS.
This article is underselling of how much was achieved in proof formalizing for math in the last few years and how close it is to being solved.
If we disregard programming and just look at formalizing math (Christian Szegedy has been doing it for a long time now), the length of proofs that are being formalized are exponentially growing and there's a good chance that in 2026 close to 100% of human written big/important proofs will be translated to and verified by Lean.
Just as an example for programming / modelling cache lines and cycle counts: we have quite good models for lots of architectures (even quite good reverse engineered model for NVIDIA GPUs in some papers). The problem is that calculating exact numbers for cache reads / writes is boring with lots of constants in them, and whenever we change the model a little bit the calculations have to be remade.
It's a lot of boring constraints to solve, and the main bottleneck for me when I was trying to do it by hand was that I couldn't just trust the output of LLMs.
> If you have a team of 1x f/time developer and 1x f/time tester,
Y'all have dedicated testers!? In 14 years of development, across FAANG and startup, this has never been true for me. The closest I've come is a brief period when a group of ~7 teams were able to call on the services of two testers. As you can imagine, with that ratio, the testers were not spending much time doing nothing.
I've had it at least 4 times; it's a byproduct of working in a highly regulated industry that requires the software (or goods sold) to meet a specific certification (military/munitions/EMV/etc).
In the FAANG and startup world that I worked in, there was no QA department, so I assume that FAANGs and startups don't have a dedicated and autonomous/independent QA department.
That's not the point I was making, though. The point is that we could never emit code faster than it was to deploy. Deployment (including QA) was always 2x to 4x as fast. Sometimes as much as 10x as fast.
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EDIT: Of course, I've been working for about twice the number of years as you, and back in those days it was pretty common for large companies to have dedicated QA. Even Microsoft had those :-)
I think more salient here (at term certainly) is setting up adversarial agents for testing/verification - that has been a big win for me in multi-agent workflows - when claude first released "computer use" that was a very big step in closing this loop and avoiding the manual babysitting involved in larger projects. PSA that it's not a silver bullet as the "analyzer" can still get tripped up and falsely declare something as broken (or functional), but it greatly reduces the "Hey I've done the task" when the task is not done or the output is broken.
I smell vaporware. Formal verification is easy on easy stuff like simple functions - complex functions it might be impossible. Then you most likely will get bunch of snake oil salesmen promising that you can verify full system…
There are many arguable points in this blog post, but I want to highlight just one: the need for formal specification. It is indeed a big issue. However, one must distinguish between a full specification, which is sufficient to prove functional correctness, and a specification of certain security or safety properties, which only allows us to verify those properties. For example, we can easily specify the property that "the program shall never read uninitialised memory" and prove it. That wouldn't guarantee that the program is functionally correct, but it would at least rule out a whole class of potential errors.
I agree completely with the author that AI assisted coding pushes the bottleneck to verification of the code.
But you don't really need complete formal verification to get these benefits. TDD gets you a lot of them as well. Perhaps your verification is less certain, but it's much easier to get high automated test coverage than it is to get a formally verifiable codebase.
I think AI assisted coding is going to cause a resurgence of interest in XP (https://en.wikipedia.org/wiki/Extreme_programming) since AI is a great fit for two big parts of XP. AI makes it easy to write well-tested code. The "pairing" method of writing code is also a great model for interacting with an AI assistant (much better than the vibe-coding model).
Trouble is that TDD, and formal proofs to much the same extent, assume a model of "double entry accounting". Meaning that you write both the test/proof and the implementation, and then make sure they agree. Like in accounting, the assumption is that the probability of you making the same mistake twice is fairly low, giving high confidence to accuracy when they agree. When there is a discrepancy, then you can then unpack if the problem is in the test/proof or the implementation. The fallible human can easily screw either.
But if you only fill out one side of the ledger, so to speak, an LLM will happily invent something that ensures that it is balanced, even where your side of the entry is completely wrong. So while this type of development is an improvement over blindly trusting an arbitrary prompt without any checks and balances, it doesn't really get us to truly verifying the code to the same degree we were able to achieve before. This remains an unsolved problem.
I don't fully understand what you mean by accounting expects the probability of making the same mistake twice is fairly low? Double-entry bookkeeping can only tell you if the books are balanced or not. We absolutely cannot assume that the books reflect reality just because they're balanced. You don't need to mess up twice to mess up the books in terms of truthness.
Also tests and code are independent while you always affect both sides in double-entry always. Audits exist for a reason.
With double-entry bookkeeping, the only way an error can slip through is if you make the same error on both sides, or else they wouldn’t be balanced. A similar thing is true for testing: If you make both an error in your test and in your implementation, they can cancel out and appear to be error-free.
I don’t quite agree with that reasoning, however, because a test that fails to test the property it should test for is a very different kind of error than having an error in the implementation of that property. You don’t have to make the “same” error on both sides for an error to remain unnoticed. Compared to bookkeeping, a single random error in either the tests or the implementation is more likely to remain unnoticed.
> With double-entry bookkeeping, the only way an error can slip through is if you make the same error on both sides, or else they wouldn’t be balanced. A similar thing is true for testing: If you make both an error in your test and in your implementation, they can cancel out and appear to be error-free.
Yeah but it's very different from tests vrs code though, right? Every entry has two sides at least and you do it together, they are not independent like test and code.
You can easily make a mistake if you write a wrong entry and it will still balance. Balanced books =/= accurate books is my point. And there is no difference between "code" and "tests" in double entry, it's all just "code".
So it seems like the person who made the metaphor doesn't really know how double-entry works or took maybe one accounting class.
You can use a little less snark and "high confidence" is pretty easy to understand but your metaphor makes no sense. Balanced books =/= accurate books and it is not at all a sign that the bookkeeping is accurate. The entries are also not independent like code and tests.
Naturally. Hence "high confidence" and not "full confidence". But let's not travel too far into the weeds here. Getting us back on track, what about the concept of "high confidence" is not understandable?
That sounds right in theory, but in practice my code is far, far higher quality when I do TDD than when I don't. This applies whether or not I'm using an Ai coding assistant
I don't think GP disagrees. They are (I think) stating that AI-assisted TDD is not as reliable as human TDD, because AI will invent a pointless test just to achieve a passing outcome.
> It is also painfully slow, the computational complexity of a + b, an operation so fast in CPU that it's literally an instant, is O(a + b), addition is linear in time to the added values instead of a constant operation.
To me, this reads as an insurmountably high hurdle for the application domain. We're talking about trying to verify systems which are produced very quickly by AIs. If the verification step is glacially slow (which, by any measure, a million cycles to add two integers is), I don't see how this could be considered a tractable solution.
For the verification experts: (and forgive me because I have almost zero of the math understanding of this stuff)
> This makes formal verification a prime target for AI-assisted programming. Given that we have a formal specification, we can just let the machine wander around for hours, days, even weeks.
Is this sentiment completely discounting that there can be many possible ways to write program that satisfies certain requirements that all have correct outputs? Won’t many of these be terrible in terms of performance, time complexity, etc? I know that in the most trivial case, AI doesn’t jump straight to O(n)^3 solutions or anything, but also there’s no guarantee it won’t have bugs that degrade performance as long as they don’t interfere with technical correctness.
Also, are we also pretending that having Claude spin for “even weeks” is free?
Verified software should satisfy the liveness property; otherwise, an infinite loop that never returns would pass as "correct."
Verifying realtime software goes even further and enforces an upper bound on the maximum number of ticks it takes to complete the algorithm in all cases.
You can. Naturals are the default because the kinds of people who write and use proof assistants think usually in terms of natural numbers first rather than the 2-adic numbers mod 2^N we use in programming, or even the plain 2-adics used for algorithmics. It's like mathematicians and 1-based indexing. It violates programming conventions, but the people using it find it easier.
I lack the level of education and eloquence of the author, but I have my own notion that I think agrees with them: Specification is difficult and slow, and bugs do not care whether they are part of the official specification or not.
Some software needs formal verification, but all software needs testing.
On another subject...
> Tests are great at finding bugs ... but they cannot prove the absence of bugs.
Unless people therefore decide that testing unnecessary... Which has happened a lot in academia. One of the reasons testing is not being taught that well on some universities...
Doesn't this run into the same bottleneck as developing AI first languages? AI need tons of training material for how to write good formal verification code or code in new AI first languages that doesn't exist. The only solution is large scale synthetic generation which is hard to do if humans, on some level, can't verify that the synthetic data is any good.
Formal verification is a nice idea but it's a big hill to climb from where we're at. Most people can't even get agents to robustly E2E QA code, which is a much smaller hill to climb for (probably) larger benefits. I'm sure this area will improve over time though, since it is an eventual unlock for fully autonomous engineering.
I think for most complex systems, robust E2E QA is a waste of money. A small handful of E2E smoke tests and thoughtful application of smaller tests is usually enough. Though to be fair, agent aren't good at that either.
I think the section on AI from Zero to QED (a proofs in Lean/lang guide) gives a sober path forward from the perspective of market-makers and trading:
"Imagine market infrastructure where agents must prove, before executing, that their actions satisfy regulatory constraints, risk limits, fairness properties, and eventually machine-checkable proofs of Pareto efficiency of market mechanisms. This is a big, hairy, ambitious goal. Not “we reviewed the code” but “the system verified the proof.” The agent that cannot demonstrate compliance cannot act."
I dream of a future where before any software is released we can predict 100 years into the future what effect it will have on every living thing and not release it if unhappiness delta for some living thing falls below a certain threshold.
See also Regehr's example[1] where a formally verified C compiler generates incorrect output because of an inconsistent value in <limits.h> (TL;DR: The compiler can pick whether "char" is signed or unsigned. Compcert picked one, but the linux system header used the other for CHAR_MIN and CHAR_MAX).
We won't be formally verifying millions of LOC anytime soon, don't get your hopes that high up.
...but we will be modelling those 5-10kLOC modules across multiple services doing critical business logic or distributed transactions. This has been unthinkable a couple months ago and today is a read-only-Friday experiment away (try it with a frontier model and you'll be surprised).
Thanks for the article. Perhaps you could write a follow-up article or tutorial on your favored approach, Verification-Guided Development? This is new to most people, including myself, and you only briefly touch on it after spending most of the article on what you don't like.
Good luck with your degree!
P.S. Some links in your Research page are placeholders or broken.
I'll add some links for the original VGD paper and related articles, that should help in short term. Thank you! I'll look into writing something on VGD itself in the next few weeks.
The fact that we're reading about it here today and have read about it in the past weeks is one piece of evidence. Another is that we hadn't been reading about it in the past months before November. Opus 4.5 and GPT 5.2 have crossed an usefulness frontier.
Anecdotally, I've been having some success (guiding LLMs) writing Alloy models in the past month and ensuring conformance with code. Making these would've been unjustifiable from ROI perspective fairy tales just this summer. The landscape has changed qualitatively.
Formal verification is going mainstream as watercooler weakend project fodder. As someone that has been well-versed in functional programming and depedent types for over a decade, this is a vast improvement.
The hobby project to day job methodology pipeline is real.
LLM-style AI isn't great for formal verification, not so far as I understand. And the recent advances in AI didn't do much for the kind of AI that is useful for formal verification.
Before we start writing Lean. Perhaps we can start with something "dumber" like Rust or any typed program. If you want to write something correct, or you care about correctness, you should not be using dynamic languages. The most useful and used type of test is type checking.
Type errors, especially once you have designed your types to be correct by construction, is extremely, extremely useful for LLMs. Once you have the foundation correct, they just have to wiggle through that narrow gap until it figures out something that fits.
But from what I understood and read so far, I am not convinced of OP's "formal verification". A simple litmus test is to take any of your recent day job task and try to describe a formal specification of it. Is it even doable? Reasonable? Is it even there? For me the most useful kind of verification is the verification of the lower level tools i.e. data structures, language, compilers etc
For example, the type signature of Vec::operator[usize] in Rust returns T. This cannot be true because it cannot guarantee to return a T given ANY usize. To me, panic is the most laziest and worst ways to put in a specification. It means that every single line of Rust code is now able to enter this termination state.
> Perhaps we can start with something "dumber" like Rust or any typed program. If you want to write something correct, or you care about correctness, you should not be using dynamic languages. The most useful and used type of test is type checking.
Lean or TLA+ are to Rust/Java's type systems what algebraic topology or non-linear PDEs are to "one potato, two potatoes". The level of "correctness" achievable with such simple type systems is so negligible in comparison to the things you can express and prove in rich formal mathematics languages that they barely leave an impression (they do make some grunt work easier, but if we're talking about a world where a machine can do the more complicated things, a little more grunt work doesn't matter).
I once attended a talk by someone who is or was big in the node.js world. He opened with the premise, "a static type check is just a stand-in for a unit test."
I wanted to throw a shoe at him. A static type check doesn't stand in for "a" unit test; static typing stands in for an unbounded number of unit tests.
Put another way, this common misconception by users of languages like Javascript and Python that unit testing is just as good as type checking (plus more flexible) is a confusion between the "exists" and "for all" logical operators.
Plus, it is simply more enjoyable to design the types in your program than to write unit tests. The fun factor comes from operating on a higher level of abstraction and engages more of your brain’s puzzle-solving mode than just writing unit tests. Making yourself think about “for all x” rather than a concrete x forces your brain to consider deeply the properties of x being used.
> it is simply more enjoyable to design the types in your program than to write unit tests.
I have tried both and I have no idea what you're talking about.
> Making yourself think about “for all x” rather than a concrete x forces your brain to consider deeply the properties of x being used.
The entire point of dynamic typing is that you can think about interfaces rather than concrete types, which entails deep consideration of the properties of the object (semantics of the provided interface).
> A static type check doesn't stand in for "a" unit test; static typing stands in for an unbounded number of unit tests.
You have conflated "a static type check" with "static typing". Unit tests stand in, in the same way, for an unbounded number of states of real-world input. They're simply being subjected to a trial verification system rather than a proof system. It turns out that writing proofs is not very many people's idea of a good time, even in the programming world. And the concept of "type" that's normally grokked is anemic anyway.
> Put another way...
Rhetoric like this is unconvincing and frankly insulting. You pass off your taste and opinion as fact, while failing to understand opposed arguments.
> I wanted to throw a shoe at him.
You should have!
Agreed.
Besides, it's not like types don't matter in dynamically typed languages. The (competent) programmer still needs to keep types in their head while programming. "Can this function work with a float, or must I pass an int?" "This function expects an iterable, but what happens if I pass a string?" Etc.
I started my career with JavaScript and Python, but over the years I've come to the conclusion that a language that hides types from programmers and does implicit conversion magic in the background does not deliver a better DX. It might make the language more approachable initially, and the idea of faster prototyping might be appealing, but it very quickly leads to maintenance problems and bugs. Before type hinting tools for Python became popular, I worked on many projects where `TypeError` was the #1 exception in Sentry by a large margin.
Gradual and optional typing is better than nothing, but IME if the language doesn't require it, most programmers are lazy and will do the bare minimum to properly add type declarations. Especially with things like TypeScript, which makes many declarations difficult to read, write, and understand.
I think that type inference is a solid middle ground. Types are still statically declared, but the compiler is smart enough to not bother the developer when the type is obvious.
> Before type hinting tools for Python became popular, I worked on many projects where `TypeError` was the #1 exception in Sentry by a large margin.
My experience is radically different. `ValueError` is far more common in my un-annotated Python, and the most common cause of `TypeError` anyway is the wrong order or number of arguments after a refactoring.
Good luck using static typing to model many real world unit tests for the programming languages people use most. I start with an easy example: those records should be sorted by date of birth. We can move on to more complicated scenarios.
The comment didn’t claim that types are a stand in for tests either! IMO, they are orthogonal.
The comment explicitly set out to refute the idea "...that unit testing is just as good as type checking" by describing the former as simply inferior.
No one claims that types are a stand in for all unit tests.
They stand in for the banal unit tests.
If you care about the type of a parameter you can just add an assertion in the method /s
Hundreds of unit tests replace a type.
Start using properties and it is in the thousands.
Most code should be typed. Python is great for prototypes, but once the prototype gels, you need types.
I've always hated Python. Could never enjoy it at all. Pretty much the same poor DX as PHP, Javascript, Ruby, etc.
Finally set up neovim with pyright; use types on every single fucking thing, and now I love Python[1].
Can't wait to see TC39 become a reality (learned about it just this past week on HN, actually). Maybe I'll enjoy Javascript too.
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[1] Within reason; the packaging experience is still extremely poor!
Didn't get the bit about TC39 - it is just a group. It has nothing to do with types in JS.
Sorry, I should have been clearer about which proposal I was talking about:
https://github.com/tc39/proposal-type-annotations
I've never seen types used correctly in Python. A `tuple` is not even a type! It's a type constructor!
The distinction you are trying to make is nonsensical in Python's object model. Types are inherently callable, and calling them constructs (i.e. instantiates) the type (normally; this can be overridden, by design). There is also no type->kind->category hierarchy; `type` itself is an object, which is its own type.
When you're at a level of theory where terms like "type constructor" are natural, it's unreasonable to expect any of it to be applicable to Python. This is why the Haskell people speak of dynamically-typed languages in the Python mold as "untyped" regardless of their attitude towards implicit casts.
And I love it, and have been using it for decades, and write beautiful things where the annotations hardly ever seem worth the effort — perhaps for documentation, but not for a static checker. Then I look at other, newer Pythonistas trying to figure out how to write complex generic type expressions (and sacrificing backwards compatibility as they keep up with the churn of Python figuring out how to offer useful annotation syntax) and deal with covariance vs contravariance etc. and I just smile.
Most of the types are "not even types". It's why you can pass a type as a value (not a string) to a function.
If the type was a type, you'd not be able to use it as a value.
Good that Python supports types then
Python has formalized magic comments.
You can run a third party linter on those comments, but you must hope that they're correct. There are usually some checks for that, but they're only reliable in trivial cases.
This is not static typing any more than "you can use emscripten to transpile JavaScript to C" means that JavaScript is a low level language with native assembly support. It's a huge step forward from "no system at all" and I'm thrilled it exists, but it's hardly the same thing.
Tell me more please: how does one use types in Python? Unfortunately I write Python professionally these days (it is the language that has all the libraries) and hate it with a passion.
> Good that Python supports types then
"Optional typing" is not the same as "Static typing".
Great, my program will crash, because I forgot to opt-in to typing :-/
C has void pointers.
> C has void pointers.
And? Void pointers are not the default type :-/
With Python I have to do extra work to get type errors.
With C I have to do extra work to hide the type errors.
I am battling to understand the point you are making.
Poorly, though, and with lots of edge cases and foot guns to make you miserable once your code calls out to numpy or gets some JSON.
I think it's possible to write correct systems with dynamic languages, just not the ones we commonly use like Python and JavaScript. I find Clojure, for example to be one example of a dynamic language that is pretty easy to manage and I attribute that to the immutable nature and data-centric ethos. I'm sure there are other dynamic languages that would work as well.
Now, I wouldn't necessarily use Clojure on a huge multi-organization codebase (maybe it's fine, this is outside of my experience with it), but it can be the right tool for some jobs.
Common Lisp as well. I can’t explain why, but type errors are just not something I struggle with in Common Lisp! But it is in JS and Python for sure. Maybe someone knows why it feels different?
I think it’s cause there’s less imperative code and side effects to track data transformations through.
Like any random JS/php app is probably a huge pile of loops and if statements. To track what happens to the data, you need to run the whole program in your head. “And now it adds that property to the object in the outer scope, and now that object gets sorted, now it hits the database… ok…”. Whereas in clojure most functions are either a single atomic transformation to a set of data, or batch of side effects. You still have to run it through your head, but you can do it more piece-by-piece instead of having to understand a 1,000 method with class states being auto loaded and mutated all over the place. Also you have a REPL to try stuff out as you go.
Dont get me wrong, I LOVE static types. Statically typed clojure would be the best fckin language ever. But there is definitely a wide gulf between a dynamic language like JS, and one like clojure!
> Statically typed clojure
Well, if you also like Common Lisp, there's Coalton, which is Common Lisp with a Haskell-like type system: https://coalton-lang.github.io/
I haven't done much with CL so I can only speculate, but I think stricter FP principles in general work to minimize the downsides of dynamic typing. CL, to my understanding, isn't the most "pure" when it comes to FP, but does a good job at giving the programmer a lot of power to constrain and explore systems.
The author is in the comfortable position of working on a system that does have a formal specification and a formally verified reference implementation. The post is not about how they wish things would work, but how their existing system (Cedar) works.
Regarding your point on Rust, the vast majority of software has nowhere near the amount of static guarantees provided by Rust. If you need more, use static memory allocation, that's what people do for safety critical systems. By the way, it seems that Rust aborts on OOM errors, not panics: https://github.com/rust-lang/rust/issues/43596
> To me, panic is the most laziest and worst ways to put in a specification.
This why the "existing programs don't have specs!" Hand-ringing is entirely premature. Just about every code base today has error modes the authors think won't happen.
All you have to do is start proving they won't happen. And if you do this, you will begin a long journey that ends up with a formal spec for, at least, a good part of your program.
Proving the panics are dead code is a Socratic method, between you and the proof assistant / type checker, for figuring out what your program is and what you want it to be :).
Yeah, Rust has been pretty good for formal verification so far. Hoare spec contracts I think are the way forward, especially since they fairly naturally flow from unittests. I've been using Hax to pretty good effect so far. I'm generally suspect that advances in Lean proof solving by dedicated models are that useful for program verification, compared to generalist models, though it could help lower costs a good bit.
I agree. And learning a typed language is significantly easier now that AI can explain everything. The types also help AI to write a correct code. A very positive feedback loop.
I find this discourse about AI and formal verification of software very confusing. It's like someone saying, let's assume I can somehow get a crane that would get that vintage car into my 15th floor apartment room for display in my living room, but what will I do with my suitcases?
All the problems mentioned in the article are serious. They're also easier than the problem of getting an AI to automatically prove at least hundreds of correctness properties on programs that are hundreds of thousand, if not millions of lines long. Bringing higher mathematics into the discussion is also unhelpful. Proofs of interesting mathematical theorems require ingenuity and creativity that isn't needed in proving software correct, but they also require orders of magnitude fewer lemmas and inference steps. We're talking 100-1000 lines of proof per line of program code.
I don't know when AI will be able to do all that, but I see no reason to believe that a computer that can do that wouldn't also be able to reconcile the formal statements of correctness properties with informal requirements, and even match the requirements themselves to market needs.
I think this misses a lot of reasons why learning verification is important. For instance learning the concept of invariants and their types such as loop invariants. They make reasoning about code in general easier, even if you never formally do any verification, it makes it easier to write tests or asserts(). A substantial amount of bugs are due to the program having a different state to that assumed by the programmer, and there are other tools that help with this. For example a statically typed language is a type of verification since it verifies a variable has a specific type and thus operations that can be performed on it, and limits the valid input and output range of any function. Languages like Rust are also verification in terms of memory correctness, and are also extremely useful tools.
"Beware of bugs in the above code; I have only proved it correct, not tried it." - Donald Knuth
Not that relevant in context as the code in question is used to conclude a formal proof, not the other way around. Buy hey, it is a common quote when talking about proving software and someone has to do it...
Context: https://staff.fnwi.uva.nl/p.vanemdeboas/knuthnote.pdf
This is an aside because I agree with the author’s core point, but spelling, grammatical errors, and typos actually imply something authored by a human now. This sentence:
“It affects point number 1 because AI-assisted programming is a very natural fit fot specification-driven development.”
made me smile. Reading something hand made that hadn’t been through the filters and presses of modern internet writing.
He's not a native English speaker, for starters. But I like that he didn't fill that gap with an LLM.
To my surprise, I recently found a typo in AI-generated code. It's rare though
This blog post is out of its depth
- Lean will optimize peano arithmetic with binary bignums underneath the hood
- Property based checking and proof search already exist on a continuum, because counterexamples are a valid (dis)proof technique. This should surprise no writer of tactics.
- the lack of formal specs for existing software should become less a problem for greenfield software after these techniques go mainstream. People will be incentivized to actually figure out what they want, and successfully doing so vastly improves project management.
Finally, and most importantly, people thinking that there is a "big specification" and then "big implementation" are totally missing the mark. Remember tools like lean are just More Types. When we program with types, do we have a single big type and a single untyped term, paired together? Absolutely not.
As always, the key to productive software development is more and more libraries. Fancier types will allow writing more interesting libraries that tackle the "reusable core" of many tasks.
For example, do you want to write a "polymorphic web app" that can be instantiated with a arbitrary SQL Schema? Ideas like that become describable.
> the key to productive software development is more and more libraries
You had me until this statement. The idea that "more and more libraries" is going to solve the (rather large) quality problems we have in the software industry is .. misguided.
see:
https://www.folklore.org/Negative_2000_Lines_Of_Code.html
https://caseymuratori.com/blog_0031
Don’t use a library unless you really need it. Someone recently recommended I add Zod to a project where I am only validating two different JSON objects in the entire project. I like Zod, but I already wrote the functions to progressively prove out the type in vanilla JS.
Less is more, including other people’s libraries.
Agreed.
> People will be incentivized to actually figure out what they want
That's the AGI I want to see.
This article is underselling of how much was achieved in proof formalizing for math in the last few years and how close it is to being solved.
If we disregard programming and just look at formalizing math (Christian Szegedy has been doing it for a long time now), the length of proofs that are being formalized are exponentially growing and there's a good chance that in 2026 close to 100% of human written big/important proofs will be translated to and verified by Lean.
Just as an example for programming / modelling cache lines and cycle counts: we have quite good models for lots of architectures (even quite good reverse engineered model for NVIDIA GPUs in some papers). The problem is that calculating exact numbers for cache reads / writes is boring with lots of constants in them, and whenever we change the model a little bit the calculations have to be remade.
It's a lot of boring constraints to solve, and the main bottleneck for me when I was trying to do it by hand was that I couldn't just trust the output of LLMs.
> AI-assisted programming pushes the limits of programming from what you can implement to what you can specify and what you can verify.
This really resonates. We can write code a lot faster than we can safely deploy it at the moment.
> We can write code a lot faster than we can safely deploy it at the moment.
We always could. That has been true since the days we programmed computers by plugging jumper wires into a panel.
> We always could. That has been true since the days we programmed computers by plugging jumper wires into a panel.
That's news to me, and I'm an ancient greybeard in development.
If you have a team of 1x f/time developer and 1x f/time tester, the tester would be spending about half their day doing nothing.
Right now, a single developer with Claude code can very easily overwhelm even a couple of testers with new code to test.
> If you have a team of 1x f/time developer and 1x f/time tester,
Y'all have dedicated testers!? In 14 years of development, across FAANG and startup, this has never been true for me. The closest I've come is a brief period when a group of ~7 teams were able to call on the services of two testers. As you can imagine, with that ratio, the testers were not spending much time doing nothing.
I've had it at least 4 times; it's a byproduct of working in a highly regulated industry that requires the software (or goods sold) to meet a specific certification (military/munitions/EMV/etc).
In the FAANG and startup world that I worked in, there was no QA department, so I assume that FAANGs and startups don't have a dedicated and autonomous/independent QA department.
That's not the point I was making, though. The point is that we could never emit code faster than it was to deploy. Deployment (including QA) was always 2x to 4x as fast. Sometimes as much as 10x as fast.
=================
EDIT: Of course, I've been working for about twice the number of years as you, and back in those days it was pretty common for large companies to have dedicated QA. Even Microsoft had those :-)
> We can write code a lot faster than we can safely deploy it at the moment.
This has always been the case?
I think more salient here (at term certainly) is setting up adversarial agents for testing/verification - that has been a big win for me in multi-agent workflows - when claude first released "computer use" that was a very big step in closing this loop and avoiding the manual babysitting involved in larger projects. PSA that it's not a silver bullet as the "analyzer" can still get tripped up and falsely declare something as broken (or functional), but it greatly reduces the "Hey I've done the task" when the task is not done or the output is broken.
So it is not reliable enough to run automatically?
I smell vaporware. Formal verification is easy on easy stuff like simple functions - complex functions it might be impossible. Then you most likely will get bunch of snake oil salesmen promising that you can verify full system…
There are many arguable points in this blog post, but I want to highlight just one: the need for formal specification. It is indeed a big issue. However, one must distinguish between a full specification, which is sufficient to prove functional correctness, and a specification of certain security or safety properties, which only allows us to verify those properties. For example, we can easily specify the property that "the program shall never read uninitialised memory" and prove it. That wouldn't guarantee that the program is functionally correct, but it would at least rule out a whole class of potential errors.
I agree completely with the author that AI assisted coding pushes the bottleneck to verification of the code.
But you don't really need complete formal verification to get these benefits. TDD gets you a lot of them as well. Perhaps your verification is less certain, but it's much easier to get high automated test coverage than it is to get a formally verifiable codebase.
I think AI assisted coding is going to cause a resurgence of interest in XP (https://en.wikipedia.org/wiki/Extreme_programming) since AI is a great fit for two big parts of XP. AI makes it easy to write well-tested code. The "pairing" method of writing code is also a great model for interacting with an AI assistant (much better than the vibe-coding model).
Trouble is that TDD, and formal proofs to much the same extent, assume a model of "double entry accounting". Meaning that you write both the test/proof and the implementation, and then make sure they agree. Like in accounting, the assumption is that the probability of you making the same mistake twice is fairly low, giving high confidence to accuracy when they agree. When there is a discrepancy, then you can then unpack if the problem is in the test/proof or the implementation. The fallible human can easily screw either.
But if you only fill out one side of the ledger, so to speak, an LLM will happily invent something that ensures that it is balanced, even where your side of the entry is completely wrong. So while this type of development is an improvement over blindly trusting an arbitrary prompt without any checks and balances, it doesn't really get us to truly verifying the code to the same degree we were able to achieve before. This remains an unsolved problem.
I don't fully understand what you mean by accounting expects the probability of making the same mistake twice is fairly low? Double-entry bookkeeping can only tell you if the books are balanced or not. We absolutely cannot assume that the books reflect reality just because they're balanced. You don't need to mess up twice to mess up the books in terms of truthness.
Also tests and code are independent while you always affect both sides in double-entry always. Audits exist for a reason.
With double-entry bookkeeping, the only way an error can slip through is if you make the same error on both sides, or else they wouldn’t be balanced. A similar thing is true for testing: If you make both an error in your test and in your implementation, they can cancel out and appear to be error-free.
I don’t quite agree with that reasoning, however, because a test that fails to test the property it should test for is a very different kind of error than having an error in the implementation of that property. You don’t have to make the “same” error on both sides for an error to remain unnoticed. Compared to bookkeeping, a single random error in either the tests or the implementation is more likely to remain unnoticed.
> With double-entry bookkeeping, the only way an error can slip through is if you make the same error on both sides, or else they wouldn’t be balanced. A similar thing is true for testing: If you make both an error in your test and in your implementation, they can cancel out and appear to be error-free.
Yeah but it's very different from tests vrs code though, right? Every entry has two sides at least and you do it together, they are not independent like test and code.
You can easily make a mistake if you write a wrong entry and it will still balance. Balanced books =/= accurate books is my point. And there is no difference between "code" and "tests" in double entry, it's all just "code".
So it seems like the person who made the metaphor doesn't really know how double-entry works or took maybe one accounting class.
What about the concept of "high confidence" is not understandable?
You can use a little less snark and "high confidence" is pretty easy to understand but your metaphor makes no sense. Balanced books =/= accurate books and it is not at all a sign that the bookkeeping is accurate. The entries are also not independent like code and tests.
> Balanced books =/= accurate books
Naturally. Hence "high confidence" and not "full confidence". But let's not travel too far into the weeds here. Getting us back on track, what about the concept of "high confidence" is not understandable?
That sounds right in theory, but in practice my code is far, far higher quality when I do TDD than when I don't. This applies whether or not I'm using an Ai coding assistant
I don't think GP disagrees. They are (I think) stating that AI-assisted TDD is not as reliable as human TDD, because AI will invent a pointless test just to achieve a passing outcome.
> It is also painfully slow, the computational complexity of a + b, an operation so fast in CPU that it's literally an instant, is O(a + b), addition is linear in time to the added values instead of a constant operation.
To me, this reads as an insurmountably high hurdle for the application domain. We're talking about trying to verify systems which are produced very quickly by AIs. If the verification step is glacially slow (which, by any measure, a million cycles to add two integers is), I don't see how this could be considered a tractable solution.
Related discussion from last week:
AI will make formal verification go mainstream
https://news.ycombinator.com/item?id=46294574
For the verification experts: (and forgive me because I have almost zero of the math understanding of this stuff)
> This makes formal verification a prime target for AI-assisted programming. Given that we have a formal specification, we can just let the machine wander around for hours, days, even weeks.
Is this sentiment completely discounting that there can be many possible ways to write program that satisfies certain requirements that all have correct outputs? Won’t many of these be terrible in terms of performance, time complexity, etc? I know that in the most trivial case, AI doesn’t jump straight to O(n)^3 solutions or anything, but also there’s no guarantee it won’t have bugs that degrade performance as long as they don’t interfere with technical correctness.
Also, are we also pretending that having Claude spin for “even weeks” is free?
Verified software should satisfy the liveness property; otherwise, an infinite loop that never returns would pass as "correct."
Verifying realtime software goes even further and enforces an upper bound on the maximum number of ticks it takes to complete the algorithm in all cases.
Yup, I've already spent like $20k using Claude to verify things, so like there's probably some room for cost cutting.
> proof assistants, traditionally, don't use our classic two's complement integers packed into words in our memory, they use Peano numbers
Why can't we just prove theorems about the standard two's complement integers, instead of Nat?
You can. Naturals are the default because the kinds of people who write and use proof assistants think usually in terms of natural numbers first rather than the 2-adic numbers mod 2^N we use in programming, or even the plain 2-adics used for algorithmics. It's like mathematicians and 1-based indexing. It violates programming conventions, but the people using it find it easier.
I lack the level of education and eloquence of the author, but I have my own notion that I think agrees with them: Specification is difficult and slow, and bugs do not care whether they are part of the official specification or not.
Some software needs formal verification, but all software needs testing.
On another subject...
> Tests are great at finding bugs ... but they cannot prove the absence of bugs.
I wish more people understood this.
Unless people therefore decide that testing unnecessary... Which has happened a lot in academia. One of the reasons testing is not being taught that well on some universities...
people cant even bother to write code and you expect them to test it?
Doesn't this run into the same bottleneck as developing AI first languages? AI need tons of training material for how to write good formal verification code or code in new AI first languages that doesn't exist. The only solution is large scale synthetic generation which is hard to do if humans, on some level, can't verify that the synthetic data is any good.
Formal verification is a nice idea but it's a big hill to climb from where we're at. Most people can't even get agents to robustly E2E QA code, which is a much smaller hill to climb for (probably) larger benefits. I'm sure this area will improve over time though, since it is an eventual unlock for fully autonomous engineering.
I think for most complex systems, robust E2E QA is a waste of money. A small handful of E2E smoke tests and thoughtful application of smaller tests is usually enough. Though to be fair, agent aren't good at that either.
I think the section on AI from Zero to QED (a proofs in Lean/lang guide) gives a sober path forward from the perspective of market-makers and trading:
"Imagine market infrastructure where agents must prove, before executing, that their actions satisfy regulatory constraints, risk limits, fairness properties, and eventually machine-checkable proofs of Pareto efficiency of market mechanisms. This is a big, hairy, ambitious goal. Not “we reviewed the code” but “the system verified the proof.” The agent that cannot demonstrate compliance cannot act."
https://sdiehl.github.io/zero-to-qed/20_artificial_intellige...
I dream of a future where before any software is released we can predict 100 years into the future what effect it will have on every living thing and not release it if unhappiness delta for some living thing falls below a certain threshold.
That would require making the behavior of the living things part of the formal specification, which isn’t really possible.
Test, just don’t verify.
How I learned to deploy faster.
At the end of the day, you either validate every line of code manually, or you have the agent write tests. Automate your review.
See also Regehr's example[1] where a formally verified C compiler generates incorrect output because of an inconsistent value in <limits.h> (TL;DR: The compiler can pick whether "char" is signed or unsigned. Compcert picked one, but the linux system header used the other for CHAR_MIN and CHAR_MAX).
1: https://blog.regehr.org/archives/482 there were many issues here, not just with compcert
We won't be formally verifying millions of LOC anytime soon, don't get your hopes that high up.
...but we will be modelling those 5-10kLOC modules across multiple services doing critical business logic or distributed transactions. This has been unthinkable a couple months ago and today is a read-only-Friday experiment away (try it with a frontier model and you'll be surprised).
Alperen,
Thanks for the article. Perhaps you could write a follow-up article or tutorial on your favored approach, Verification-Guided Development? This is new to most people, including myself, and you only briefly touch on it after spending most of the article on what you don't like.
Good luck with your degree!
P.S. Some links in your Research page are placeholders or broken.
I'll add some links for the original VGD paper and related articles, that should help in short term. Thank you! I'll look into writing something on VGD itself in the next few weeks.
> AI is making formal verification go mainstream.
This nonsense again. No. No it isn’t.
I’m sure the people selling it wish it was, but that doesn’t make it true.
You haven't been paying attention.
The fact that we're reading about it here today and have read about it in the past weeks is one piece of evidence. Another is that we hadn't been reading about it in the past months before November. Opus 4.5 and GPT 5.2 have crossed an usefulness frontier.
Anecdotally, I've been having some success (guiding LLMs) writing Alloy models in the past month and ensuring conformance with code. Making these would've been unjustifiable from ROI perspective fairy tales just this summer. The landscape has changed qualitatively.
Formal verification is going mainstream as watercooler weakend project fodder. As someone that has been well-versed in functional programming and depedent types for over a decade, this is a vast improvement.
The hobby project to day job methodology pipeline is real.
LLM-style AI isn't great for formal verification, not so far as I understand. And the recent advances in AI didn't do much for the kind of AI that is useful for formal verification.
You don’t use AI to perform formal verification. You give the agent access to verification tools whose output can then be fed back to the model.
It’s the same design as giving LLMs the current time, since they can’t tell time themselves, either.
my user should get upvotes for this :)