Note that the claims are about the kernel that computes the mesh intersection, not the web demo: While the kernel is formally verified, the UI and glue code (that call the kernel) are not. I once hit a bug that looked like there were holes in the resulting mesh: I found that it was an overflow in the glue code (now fixed) that converts the exact rationals coordinates of the output mesh, for which we prove the specification, to floats before sending them to the GPU. All geometry is happening inside the kernel, for which we know the specification holds.
This is really neat. I think CSG is one of the more intuitive ways to build 3d environments (especially 'indoor' ones). No geometry is ever destroyed in a proper CSG workflow. A hierarchy of brushes makes it possible to very rapidly iterate parameters like how long a hallway is or the time it takes to get from one objective to another. Often, you only have to apply a transform to a single brush to affect something that would take a non-CSG workflow an entire afternoon to recover from.
The current story with CSG around commercial game engines is pretty awful. Unity has some 3rd party options that can work well but they're also glitchy and don't integrate well with the other parts of the ecosystem. For a while I thinking about building my own CSG implementation but the exact things this project solved were what was stopping me. The EditorWindow integration stuff is not as scary to me (although it is painful). The advanced geometry and set operations is what kept me from trying. It appears that dragon has been (at least partially) slain.
> A human reviewer only needs to read 93 lines of formal specification and run the Lean checker to certify the correctness of the kernel, skipping the intricate 1000+ lines of AI-written implementation. To prove correctness, AI autonomously wrote over 60,000 lines of Lean proofs, which also never have to be inspected by a human.
I am curious about the approach as I find claims like this hard to believe. There is a theory behind proof kernels that they must be small because they have to be trusted and verifiable by a human. Otherwise the whole system breaks.
How does one trust an LLM generated kernel is proving the right things?
Maybe what you mean is that kernels of proof assistants must be small. Here I am referring to a geometry processing kernel (that is formally verified by a proof assistant). The implementation of the algorithm can be very long, the proof that it conforms to the spec can be very long. But lean checks the proof. And so you only have to trust the spec and that the lean proof assistant is correct.
In the 93 lines I assumed a reviewer already trusts that the kernel of the Lean proof assistants is correct. We have to trust somethings.
The runtime performance is most probably worse than yours. The goal of this project is a minimal specification that a reviewer can trust. See section in readme on performance and how it could be improved while still keeping formal guarantees, but would mean a reviewer having to trust axioms on floating point numbers:
https://github.com/schildep/verified-3d-mesh-intersection#pe...
On a readme style note, please spend more time explaining the algorithm and less time bragging on how amazing vibe coding is or how Lean has a zero trust proof.
I presume the new advance is claim that verified-3d-mesh-intersection is verified and not the claim that vibe coding can replace hand proofs or that the claim that lean4 can be correct without trusting llms.
This seems like very cool work. I'm sorry if I missed it: I'm still not sure how we go from verified proof (Great!) to an implementation (by LLM or by Human) which you're sure actually conforms to the proof? How do we know that the implementation maps precisely to the description within the proof?
Both the implementation and the proof are written in Lean.
In the files I reference here https://github.com/schildep/verified-3d-mesh-intersection#mi...
there are theorems about the function that does the mesh intersection computation.
The implementation is in the CSG/Impl folder. The proof is in the CSG/Proof folder. They are imported and tied together in the human reviewed file. Lean checks that these theorems in the human reviewed file are proven via the proofs.
The issue I have with all these formally verified numerical algorithms is that, at the end of the day, I need this implemented using actual floating point. It's great that it's verified, but not for the case that is practically meaningful. Unless it's implemented using hardware accelerated floating point, most of these algorithms are not sufficiently performant to be practically useful.
> The issue I have with all these formally verified numerical algorithms is that, at the end of the day, I need this implemented using actual floating point.
You can go a very long way using discrete math though. I'm pretty sure it's doable to implement, say, a 3D slicer (to prepare layers/paths for 3D printing I mean) without using any floating point operation.
P.S: FWIW I was coding intros and demos using 3D effects on the Commodore Amiga and then on 386 PCs without using the 387 (the FP co-processor of the 386, when present). I'm not saying it's useful to play games in 4K: what I'm saying is that there are certainly applications where formally verified algos using integer math do make sense.
I think this would need to be reimplemented for performance. But maybe the spec can be reused. As agents get better we might reimplement the same spec for performance with minimal human effort. (See notes regarding performance and floats in readme.)
Honestly I think I'd have an easier time understanding 1000s of lines of slop than 93 lines of proofs lol.
Does anyone have any recommended learning resources for this type of thing? Skimming the repo, it looks like a lot of unicode and math terminology, but this project seems really compelling to me.
I learned working with the Isabelle proof assistant in a university course. There were weekly exercises and a group project at the end. (Actually proving things by hand, no LLM.) That really helped. Before that course it was also hard for me to get into this.
Note that the claims are about the kernel that computes the mesh intersection, not the web demo: While the kernel is formally verified, the UI and glue code (that call the kernel) are not. I once hit a bug that looked like there were holes in the resulting mesh: I found that it was an overflow in the glue code (now fixed) that converts the exact rationals coordinates of the output mesh, for which we prove the specification, to floats before sending them to the GPU. All geometry is happening inside the kernel, for which we know the specification holds.
This is really neat. I think CSG is one of the more intuitive ways to build 3d environments (especially 'indoor' ones). No geometry is ever destroyed in a proper CSG workflow. A hierarchy of brushes makes it possible to very rapidly iterate parameters like how long a hallway is or the time it takes to get from one objective to another. Often, you only have to apply a transform to a single brush to affect something that would take a non-CSG workflow an entire afternoon to recover from.
The current story with CSG around commercial game engines is pretty awful. Unity has some 3rd party options that can work well but they're also glitchy and don't integrate well with the other parts of the ecosystem. For a while I thinking about building my own CSG implementation but the exact things this project solved were what was stopping me. The EditorWindow integration stuff is not as scary to me (although it is painful). The advanced geometry and set operations is what kept me from trying. It appears that dragon has been (at least partially) slain.
That's why I integrated manifold into godot engine for constructive solid geometry!
https://github.com/godotengine/godot-proposals/issues/9711
> A human reviewer only needs to read 93 lines of formal specification and run the Lean checker to certify the correctness of the kernel, skipping the intricate 1000+ lines of AI-written implementation. To prove correctness, AI autonomously wrote over 60,000 lines of Lean proofs, which also never have to be inspected by a human.
I am curious about the approach as I find claims like this hard to believe. There is a theory behind proof kernels that they must be small because they have to be trusted and verifiable by a human. Otherwise the whole system breaks.
How does one trust an LLM generated kernel is proving the right things?
Maybe what you mean is that kernels of proof assistants must be small. Here I am referring to a geometry processing kernel (that is formally verified by a proof assistant). The implementation of the algorithm can be very long, the proof that it conforms to the spec can be very long. But lean checks the proof. And so you only have to trust the spec and that the lean proof assistant is correct.
In the 93 lines I assumed a reviewer already trusts that the kernel of the Lean proof assistants is correct. We have to trust somethings.
How does it compare against https://github.com/elalish/manifold in performance and zero corruption?
I spent a lot of time making elalish/manifold work in Godot Engine and it is now a method in Blender too.
https://manifoldcad.org
If it helps, feel free to use the apache2 licensed unit tests that were generated in manifold development https://github.com/elalish/manifold/tree/master/test
The runtime performance is most probably worse than yours. The goal of this project is a minimal specification that a reviewer can trust. See section in readme on performance and how it could be improved while still keeping formal guarantees, but would mean a reviewer having to trust axioms on floating point numbers: https://github.com/schildep/verified-3d-mesh-intersection#pe...
> Why we cannot have manifold output meshes in general
As far as I know manifold is not verified software, but it does promise that manifold inputs become manifold outputs.
I wonder how manifold will break in that condition or would it snap to the nearest manifold.
https://github.com/schildep/verified-3d-mesh-intersection#wh...
On a readme style note, please spend more time explaining the algorithm and less time bragging on how amazing vibe coding is or how Lean has a zero trust proof.
I presume the new advance is claim that verified-3d-mesh-intersection is verified and not the claim that vibe coding can replace hand proofs or that the claim that lean4 can be correct without trusting llms.
This seems like very cool work. I'm sorry if I missed it: I'm still not sure how we go from verified proof (Great!) to an implementation (by LLM or by Human) which you're sure actually conforms to the proof? How do we know that the implementation maps precisely to the description within the proof?
Both the implementation and the proof are written in Lean. In the files I reference here https://github.com/schildep/verified-3d-mesh-intersection#mi... there are theorems about the function that does the mesh intersection computation. The implementation is in the CSG/Impl folder. The proof is in the CSG/Proof folder. They are imported and tied together in the human reviewed file. Lean checks that these theorems in the human reviewed file are proven via the proofs.
The issue I have with all these formally verified numerical algorithms is that, at the end of the day, I need this implemented using actual floating point. It's great that it's verified, but not for the case that is practically meaningful. Unless it's implemented using hardware accelerated floating point, most of these algorithms are not sufficiently performant to be practically useful.
It is possible to prove things involving floats, you just have to addionally trust a small set of axioms on floats. See https://github.com/schildep/verified-3d-mesh-intersection#pe... Don't know if there is something like this in Lean. But there is https://flocq.gitlabpages.inria.fr
> The issue I have with all these formally verified numerical algorithms is that, at the end of the day, I need this implemented using actual floating point.
You can go a very long way using discrete math though. I'm pretty sure it's doable to implement, say, a 3D slicer (to prepare layers/paths for 3D printing I mean) without using any floating point operation.
P.S: FWIW I was coding intros and demos using 3D effects on the Commodore Amiga and then on 386 PCs without using the 387 (the FP co-processor of the 386, when present). I'm not saying it's useful to play games in 4K: what I'm saying is that there are certainly applications where formally verified algos using integer math do make sense.
Congrats with this achievement. Do you see it possible that this project can be integrated into, for example, FreeCAD?
I think this would need to be reimplemented for performance. But maybe the spec can be reused. As agents get better we might reimplement the same spec for performance with minimal human effort. (See notes regarding performance and floats in readme.)
Honestly I think I'd have an easier time understanding 1000s of lines of slop than 93 lines of proofs lol.
Does anyone have any recommended learning resources for this type of thing? Skimming the repo, it looks like a lot of unicode and math terminology, but this project seems really compelling to me.
I learned working with the Isabelle proof assistant in a university course. There were weekly exercises and a group project at the end. (Actually proving things by hand, no LLM.) That really helped. Before that course it was also hard for me to get into this.