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Hacker News Hacker NewsI spent some time exploring this topic. Here's my thesis: Formal verification was expensive. 20x expensive compared to just developing the software, as the author notes. The cost of finding and developing exploits also was high. That creates an incentive to put software verification aside, since it solves a relatively small problem, at an extremely high cost.
We've seen how Mythos has found more vulnerabilities than the rest of the security industry combined. (you can argue about the quality and what counts as a vulnerability, but not the point) So the cost of finding and developing exploits has dropped dramatically.
On the other hand, formal verification is now much easier, since LLMs can automate the proof. You don't even need to worry about hallucinations, you merely need to trust Lean core. If the LLM is wrong, the proof will get rejected!
The problem of exploits gets bigger, and the solution of formal verification gets cheaper. As a result, the needle is now moving in the direction of "more formal verification".
I, personally, think that it is ridiculous that ~none of the software we use is known to work correctly. It just happens to work correctly, most of the time.
My (ambitious) goal is to have a self-hosting, formally verified compiler, which allows proof transfer from source code down to assembly. I have not achieved that goal yet.
What I have so far:
- one (non-optimized) compiler step which is formally verified
- three simple functions (hex, hex with labels, strtoull) formally verified, against RISC-V assembly, and against a custom IR
https://github.com/m1el/riscv-fv-bootstrap
The project is in quite a bad shape, and I am trying to improve my skills in that direction.
Verus (https://github.com/verus-lang/verus) is a good start for the rust ecosystem, but it's essentially a standalone language today (with custom syntax and type system).
Suppose that you have managed to write a non-trivial piece of software with dependent types encoding all sorts of properties everywhere. The actual computation and proof are intermingled. Think of the author's innocent bound-check proof in the zstd decoder, but across the whole program, with more elaborate properties and longer proofs.
Suddenly, you realize that you need to prove a new property of your program. Can you keep your existing work and build on top of it? In general, no, you have to refine every dependent type everywhere by adding a new conjunct expressing a new invariant, and adapt every proof, as the new property is threaded in the existing program.
That is because dependent types (and other staples of naive approaches to proving program properties, like a unique invariant per loop) structure the program along the wrong dimension: they encourage grouping everything that concerns a value ("put this value in a dependent type that encodes everything known about it") or a program point ("write the precondition for this function as a big conjunction mixing all the concerns"), where it works much better, for long-term maintenance, to structure the development along concerns: computational parts of the program, basic functional properties and absence of UB, termination, other functional properties, security, etc., where each layer builds on top of the previous ones without requiring them to change.
That is not to say that dependent types do not have their use. Where they shine is at module boundaries. Consider a library that exposes an opaque type and operations on it. Users of the library can only build and modify values of that type through the library's API. This type should be a dependent type. If it needs to be refined at any point to encode a new invariant, this will have no impact on the library's users, since all they do is pass around values without interpreting them. However, inside the library, I would recommend unpacking/repacking the dependent type at the library's entry points and handling the concerns separately.
What's more interesting to me is, how do we prove that the business intent is accurately encoded in the code? This is a different problem. Not "is the logic internally coherent," but "is the logic going to give us what we want?"
And how do we make sure that the logic keeps giving us what we want, iteration after iteration?
I've open source a tool, Drift, for this purpose. So far, biased and anecdotally as its builder, this has been the missing layer. It works because the spec is treated as a duplicate of the code, written and committed in the repo and kept in sync with hashes. The LLM then checks for diffs manually. There are ways to use perplexity here as well. Here is an article about how to get started. https://monarchwadia.medium.com/how-to-make-your-agent-more-...
Doubtless, many more automations will come. We live in very exciting times. It has been a while since a new technology affected every corner of software engineering with such depth of change. This might be unprecedented.
Eventually I did take a look at all the new code, and found that one of the main features had been implemented completely backwards, in a way that was pointless and completely defeated the purpose.
(An LLM also pointed that out, so I guess it must have been a different one which implemented it? I'm trying to get them to sign their names in the commits...)
These new models are very thorough and responsible however, so of course it had written a copious number of tests, and of course all the tests passed! I found this very amusing. It had diligently proven the correctness of the incorrect functionality!*
I've been thinking about how cool it is the AI can assist with writing proofs now, and how we can use this new ability to boost the correctness of our code. It later occurred to me, however, that formal verification would not have helped in this case. It would have just formally verified that the wrong thing was correct!
--
*Yeah I know tests prove the presence of bugs but not their absence. Couldn't think of a better way to phrase that though.
To make a shameless plug, I'm working on a Python package called OpenATP [1] to make it easy to benchmark different models/harnesses for automated theorem proving. It supports running agents in Docker containers or Modal out of the box. If you try it out, I'd love to get your feedback!
I recently wrote about the surprisingly good performance I saw from Grok [2]. On more challenging proofs, Grok doesn't keep up with Opus/Fable and GPT 5.6. I was recently blown away by GPT 5.6 Sol. It's persistence in closing out proofs is unparalleled from what I've seen so far. OpenATP also supports Kimi and Leanstral [3], among others.
[1] https://github.com/henryrobbins/open-atp
When I've used interactive proof systems like Roq, I'd often kinda code myself into a hole by cutting along the wrong axes and not specifying my problem in a way that's easy to prove. After all, this is just like in math: you really want to cut at a problem the right way to get to the easy proof.
I think people are discounting how important that decision making is. It's not just about whether an LLM can churn through specific proof strategies on a problem, but also about how to pose the problem etc.
I'm not saying LLMs can't help, but I think it's less that "proof engineering is not needed" and more that "when these tools are used in the right way, proof engineering is easier". Because at the end of the day these tools work well when they have the right kind of foundations in the first place
> AWS made LNSym: a semantics and simulator for AArch64. That's cool. Perhaps we could use it to show equivalence between an optimised assembly implementation of some functions, and their Lean counterparts, and then use the assembly code at run-time? Then we could let LLMs rip at optimisation and they couldn't introduce any functional bugs. Verified assembly is well-trodden in crypto implementations, but perhaps now it could be cheap?
Trying to one-shot compcert might be hard! Thinking about it and planning it out might make it easier though...
The dimensions to notice are: Research speedup, practitioner expertize, labour dynamics, junior entrants, world impact.
In that sense it's a canary, whatever happens to Math, will in order flow to other sciences in-order of purity: Computer science, physics, chem, bio, social sciences.
Or, as Wolfram proposes in another link on the front page today, the irreducibility of computation means we cannot generally get all bugs out of systems.
To my mind, one of the risks of going all in on verification is that you just move the difficulty to a harder, more abstract layer - the specification. In my experience, this typically looks appealing to a 'type' -- very high IQ, very little real world engineering experience -- it's a dream that hits the inevitable grittiness of the real world and .. the real world keeps winning.
This isn't new, it is actually part of how Dafny came to be, another formal verification programming language.
"Safe to the Last Instruction: Automated Verification of a Type-Safe Operating System"
https://www.microsoft.com/en-us/research/publication/safe-to...
However so far hardly anyone in mainstram cared about verified assembly, maybe now with LLMs.
"Programming Language Design and Implementation in the Era of Machine Learning"
The biggest issue was always cost. But there's still an alignment problem. Without human supervision, things might drift away from the original specification and intent.
From my own experience, what works best is some kind of Hoare/separation logic (contracts), as these are quite easy to follow and decompose.
Even something as simple as a minimal Haskell subset, plus a bit of LiquidHaskell, can get you really far if you are pragmatic.
Whatever happened to Idris? Is that still being developed?
Edit: Looks like they are working on Idris 2 https://github.com/idris-lang/Idris2
I used HOL Light about 15 years ago. The vast majority of the proof details are done by a machine, generally split between term rewriting and then offloads to a generic automated prover for first-order logic. Your job then is formalising the theorem statements and orchestrating the automation, and the latter still takes an enormous amount of labour.
Around this time, we were getting excited about recommender systems for lemma selection. The idea was that you turn every theorem into a bunch of features and train a recommender system against the lemma needed to prove it. Then when you face a new problem, you use its features to recommend which lemmas are likely to be needed. You then throw those lemmas and your conjecture at a bunch of industrial strength automated provers, get the provers to come back with a minimal set of necessary lemmas, and then use your verified automated prover to do the real proof with a tractable set of inputs. It first got implemented in Isabelle/HOL as Sledgehammer, and was a massive improvement to tooling.
Now LLMs are here and the game looks completely different. I went back over some verification I spent a week on about 5 years ago, in a pretty obscure formal verifier called HOL Light, on a problem of my own making. I'd seen how terrible ChatGPT was at propositional logic a few years back, so this is the sort of thing I'd had in my back-pocket as an "impossible benchmark" for LLMs. So as a half-joke, I gave Claude the main theorem I wanted proving, hoping to watch it embarrass itself.
In a minute or two, it has come up with the same proof strategy I had used, proving four lemmas to get to the main theorem. This was impressive, but that's not the laborious part, and I was pretty confident it would die trying to prove just the first lemma. It has its four lemmas enumerated and goes to thinking, while I go off for a cup of tea.
I come back five minutes later and it has scratched off the first goal. I'm pretty shaken, and go off again trying to process that. Come back, the second goal has gone. And then the third. And then the fourth. And then, after just fifteen minutes, it's pulled off a week of my work, done it more efficiently, and near enough one-shotted each proof. That took me a while to process. I realised that if I had this 15 years ago, it would have done 95% of my PhD, which is to say that 15 years ago, I would have done a 20x more ambitious thesis.
I contacted my old supervisor, who said the theorem proving community are all on top of this, including the creator of HOL Light, now at AWS. HOL Light, incidentally, was used on what I still believe is the most ambitious mathematical theorem proving project to date, the verification of the Kepler Conjecture. The lead on that project recruited a team to get the proof through over about 5 years. Today, I wouldn't be surprised if he could have solo'd it in 6 months. The same goes for another extremely ambitious project, the verification of the seL4 microkernel. And for another open problem, I know there are people seriously wanting to get a verification of Fermat's Last Theorem, which sounded delusional 15 years ago, but sounds pretty plausible now.
Exactly where this goes, I am not sure. I suspect we can now start on verification projects that would have been insane to contemplate. But a few things concern me. One is that Lean seems to have all the mind-share. Maybe it deserves it, but there are very major and mature theorem proving technologies such as Coq (now Rocq), Isabelle, HOL Light, ACL2 and Mizar that I believe still win in terms of having the largest verification libraries and the biggest verified projects. These should not be forgotten about, since they will also probably have the most training data, having been going for many decades now.
The second is that verifying our crappy human specifications probably isn't going to fly. As the creator of HOL Light says, representation (how you formalise your problem domain) is still where it all matters, and the LLMs aren't very good at this. And neither are the writers of our current specs. Verifying that a piece of software meets the HTTP spec will be considered an achievement, but it's the wrong goal.
We'll need specifications that are modular and composable, that mesh together so that each piece is sanity checking the others. This is how we build mathematics, and it's how we'd need to build software. Specs need to be short and comprehensible, so that a human can verify them. If your spec is as long and complex as the implementation, it's worthless.
But I believe it is possible to design software from the ground up where the motivation is specification engineering rather than code engineering, and thereby the specifications become the only human-facing understandable part of software. The implementation is just some artifact that an LLM generates that nobody looks at unless they are curious. And I think LLMS today mean that "build the world over again" isn't as mad a thought as it used to be.
Seriously, please review Curry Howard Isomorphism if you’re getting pulled down this rabbit hole.
Programs are proofs. Proofs are programs.
So if you can formally describe the correct output for every input, you can have an LLM loop automatically fill in the gaps of how to get there. Congrats, that sounds at least as hard as writing the correct program in most cases.
Don’t get me wrong, I do think there are useful tools combining formal methods and llms. Let’s just not get carried away.
LLMs have a VERY different set of things which are easy and which are hard and from software security to math proofs we're seeing very early impressive results but those will run out and new classes of what is difficult will show themselves.
The "this new tech will cause everything to be easy from now on!" feeling has happened many times before and always new limitations were found.
Industrialization was thought to be basically the end of the need for money or significant labor during the industrial revolution (this optimism is where communism came from) and that future never came.
The same will be here: rapid change but also new limitations which are still hard to imagine.
But still no P=NP.