Markus Anders, who wrote the symmetry breaking system (satsuma) they use for SAT is pretty much a genius. His version of Kissat won the SAT Competition this year, with satsuma of course:
See slide 18. It was really nice to see him win. I have always been a huge fan of symmetry breaking, CryptoMiniSat (that I develop) had a symmetry breaking system, BreakID (which is _way_ slower than Markus' satsuma), in it for many years now.
I love SAT solver papers, always interesting to see auxiliary variable techniques, since those aren't really listed anywhere central
here for example, instead of saying {f(x,y,z)==g(x,y,z)}, authors instead make variable group a_w:=(f(x,y,z)=w||g(x,y,z)=w), and then apply "at most 1" to it. Can't be unequal if both functions only can have 1 result in total
this adds an index to iterate over, but separates internal subexpressions of f() and g(), removing 2 indixes (in this problem) and thus dropping whole power of n of clauses
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what I don't get is that they aren't searching Tarski's problem per se, but for one specific solution to it (one identity that isn't resulting from given). I'd totally look for arithmetic models that violate expectations in other ways than Wilkie
Indeed, having a different "exotic identity" that has smaller countermodels would be awesome. Unfortunately, we tried a few alternatives to Wilkies and didn't find smaller countermodels.
Note that it's not obvious at all how to search the space of possible exotic identities, so a potential direction for future work would be to have a loop in which LLMs (perhaps through something like AlphaEvolve) propose an exotic identity, and then a SAT call tries to find a small countermodel. A big issue though, is that even with our efficient encoding the SAT calls would take at least a few minutes, so we wouldn't be able to afford testing millions of candidates. But if there were 1000 candidates that could potentially be tested...
(I'm one of the authors of the paper, thanks for the coment!).
> we tried a few alternatives to Wilkies and didn't find smaller countermodels.
I wish you spent at least a couple words in the paper about that. Even negative results are worth documenting! (even if you didn't get up to size 11, I'd've loved to hear about those other alternatives)
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by the way, you mentioned that decreasing HSI6/10 from O(n^6) to O(n^5) clauses was slower - how big was the slowdown and how much less total clauses were there in that encoding? if I understand it correctly, that was still the biggest clause maker, but by how much?
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also, have you tried reordering order of operations in symmetry break? how much did it affect the search? I wonder if unique multiplication table might've been of help had it been disambiguated stronger (or weaker)
Man I wish I understood anything you said, or anything in tfa. Math has to have my personal gold medal for highest desire to learn coupled with total unwillingness to.
in a sense it's about a game. or... maybe it's better to call it a "challenge"
Tarski said "hey, what if we limit ourself only to positive integers? subtraction/division doesn't always work, so let's get rid of it. What can we say about +^? there are some obvious properties, like x+y=y+x for whatever x and y. Is that enough to describe all the equalities?".
Wilkie answered No, and a whole different hunt began - trying to find a set of pseudo "numbers" with pseudo "+^", that would have the same obvious properties, but not the non-obvious one
and SAT-solving is essentially asking computer to find things by describing what they are not - "1+2 is not invalid and can't be 2 different values", "1+(1+1) must not differ from (1+1)+1". A lot of such properties are easily expressible as "not the opposite" - but devil's in the details and some choices of opposites are faster for computer than others. Also symmetries are evil :)
They use the properties of Wilkie's counterexample to restrict the search space. So you can't just pick arbitrary identities that hold over the positive integers and repeat the process until you've found a smaller model.
I'm one of the authors of the arXiv paper. Thanks for bringing this to our attention! We were fully unaware of this repository, and it unfortunately did not come up during our literature search. Our approaches to the lower bound are pretty similar, although some technical differences make ours more efficient. For example, to show that there is no counterexample of size 10, we generate a formula with ~50k variables and ~2.7M clauses, which takes about 85 seconds to solve with Kissat. The encoder from this repository generates a formula with ~2k variables and ~33M clauses, which takes about 50 minutes to solve. We have sent an email to the authors of the Zenodo artifact to decide how to proceed!
They address the following concern of Zhang (2005), which contained prior work involving similar techniques:
> […] Of course, this conclusion is not proved mathematically. It is possible that the programs have some bugs, or the user (myself made some errors.
They address it as follows:
> We address this [… through] ChatGPT 5.5 Pro, through Codex, to automatically generate a Lean formalization that we then checked ourselves to confirm the statements and definitions indeed match their expected semantics. This process took multiple iterations and discussions with the model over several days, and generated over 10,000 lines of code.
This is extremely unconvincing. Manual review of 10,000 lines of AI generated code is a terrifying prospect. I’m sure the result is correct, however I am deeply uncomfortable with this being mankind’s new mathematical process. Similar concerns have been expressed since the days of the 4 color theorem, but this feels different. Perhaps it is just new.
>in a nutshell, it defines an executable function encode that takes a natural number n ≥ 5 and emits a CNF formula, which is byte-for-byte equal to the output of our Python encoding
The phrase “byte-for-byte equal” makes my eye twitch these days.
Hi! One of the authors here. Whether checking the LLM-generated Lean statements/definitions is easy or not depends heavily on the area of mathematics and the concrete definitions at play. In this case it was remarkably easy. As you can see in the repository, the definition of a countermodel is easy to parse, and the statement corresponding to their nonexistence for n <= 11 is also very easy to check. If you have any questions regarding them, I would be happy to answer them.
Personally, this is one of the cases with LLM-generated Lean that I feel most comfortable with: the proofs are all mathematically simple, and we included an example of correctness for one of the SAT constraints in the paper to show that there is nothing complicated about them. Writing them all in Lean, however, would be a very tedious task for which I see no particular benefit (i.e., it's not like some other formalization projects in which one learns something by formalizing; for an example of a human-written Lean correctness proof of a SAT encoding, you can check the paper https://arxiv.org/abs/2403.17370 in which I participated).
Regarding the byte-for-byte equality, the point is that we first had the Python code (which we wrote manually), and then asked LLMs to implement an encoder in Lean that would not only produce "equivalent" formulas but exactly equal formulas, meaning that it would use the same variable indices and the same clauses in the same order. It is well-known that such superficial factors (e.g., clause or variable ordering) can actually have large effects on solver performance (see https://www.cs.utexas.edu/~marijn/publications/pos18.pdf). The Python code is not part of the trust chain, and the Lean code does not rely on it in any way. We included the Python code in the repository since it's much shorter and easier to read.
> This is extremely unconvincing. Manual review of 10,000 lines of AI generated code is a terrifying prospect.
With https://en.wikipedia.org/wiki/Lean_(proof_assistant) (and other proof assistants), you need to review only the lines that correspond to the theorem that you want to prove and their types (I am not very experienced when it comes to lean, but I would assume that comes down to a few hundred lines of code, at most). The rest is left to typechecking (which, I would expect many in the field to agree, is at as reliable than your average peer review process in professional mathematics, and likely much more). That's the reason why Lean4 is making such a fuzz now.
That itself is not trivial too, but way easier than reviewing every function and definition used to prove that the theorems have indeed the types they claim.
If one accepts the proof of the https://en.wikipedia.org/wiki/Four_color_theorem, then there should not be new reservations these proofs; except from the maybe new additional failure scenario that the authors (still correctly!) proved theorems that don't state what they think they stated.
To sum it up: There is IMO no domain more suited for using LLMs than mathematical proofs that can be formalized using Lean4. The fact the hype-circle started earlier in software than in maths is due to the difference in monetary incentives I would assume. (Or another, rather radical and not really serious phrasing: "When it comes to Lean4 proofs that typechecks, there is no AI slop" - the theorem being proven might be uninteresting, but the proof itself is very very very very likely to be correct)
> “byte-for-byte equal”
The term itself or its association with LLMs? I would get the latter, if its the former: It's an desirable property to have, I always like seeing people going that far (assuming obviously that they indeed did so, and in the places where it matters!)
Why is subtraction not part of the algebra? It’s certainly familiar to every high school math student. This omission allows the counterexample, so the reveal is a bit of a disappointment IMHO.
Subtraction is not closed over positive integers, which is untidy. The point of Tarski’s conjecture was to propose a minimal number of axioms and operations, AFAICS they define the standard semiring of positive integers (with the natural definition of exponentiation added).
(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)
I'm not sure, but maybe it is due to that the expression a - b can be replaced as a + (-b)?
Similarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...
The subtraction point is interesting but I don't think it makes the result disappointing. The whole point of Tarski's problem is what follows from that very restricted set of elementary identities so finding the exact minimum countermodel under those rules still seems like a pretty satisfying result.
No, Gödel's incompleteness theorem applies to theories that can interpret first-order arithmetic, which includes quantified statements like "for all x, there exists a prime p > x".
In this case, we have the much simpler equational theory of positive integers under addition, multiplication, and exponentiation, which does not include any quantifiers. In fact, Gurevič showed that this theory is decidable [0]. On the other hand, Gurevič later showed that this theory is not finitely axiomatizable [1], so an infinite (but still computable) set of axioms is needed to fully characterize the theory.
Markus Anders, who wrote the symmetry breaking system (satsuma) they use for SAT is pretty much a genius. His version of Kissat won the SAT Competition this year, with satsuma of course:
https://satcompetition.github.io/2026/downloads/satcomp26sli...
See slide 18. It was really nice to see him win. I have always been a huge fan of symmetry breaking, CryptoMiniSat (that I develop) had a symmetry breaking system, BreakID (which is _way_ slower than Markus' satsuma), in it for many years now.
I love SAT solver papers, always interesting to see auxiliary variable techniques, since those aren't really listed anywhere central
here for example, instead of saying {f(x,y,z)==g(x,y,z)}, authors instead make variable group a_w:=(f(x,y,z)=w||g(x,y,z)=w), and then apply "at most 1" to it. Can't be unequal if both functions only can have 1 result in total
this adds an index to iterate over, but separates internal subexpressions of f() and g(), removing 2 indixes (in this problem) and thus dropping whole power of n of clauses
---
what I don't get is that they aren't searching Tarski's problem per se, but for one specific solution to it (one identity that isn't resulting from given). I'd totally look for arithmetic models that violate expectations in other ways than Wilkie
Indeed, having a different "exotic identity" that has smaller countermodels would be awesome. Unfortunately, we tried a few alternatives to Wilkies and didn't find smaller countermodels.
Note that it's not obvious at all how to search the space of possible exotic identities, so a potential direction for future work would be to have a loop in which LLMs (perhaps through something like AlphaEvolve) propose an exotic identity, and then a SAT call tries to find a small countermodel. A big issue though, is that even with our efficient encoding the SAT calls would take at least a few minutes, so we wouldn't be able to afford testing millions of candidates. But if there were 1000 candidates that could potentially be tested...
(I'm one of the authors of the paper, thanks for the coment!).
> we tried a few alternatives to Wilkies and didn't find smaller countermodels.
I wish you spent at least a couple words in the paper about that. Even negative results are worth documenting! (even if you didn't get up to size 11, I'd've loved to hear about those other alternatives)
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by the way, you mentioned that decreasing HSI6/10 from O(n^6) to O(n^5) clauses was slower - how big was the slowdown and how much less total clauses were there in that encoding? if I understand it correctly, that was still the biggest clause maker, but by how much?
---
also, have you tried reordering order of operations in symmetry break? how much did it affect the search? I wonder if unique multiplication table might've been of help had it been disambiguated stronger (or weaker)
Man I wish I understood anything you said, or anything in tfa. Math has to have my personal gold medal for highest desire to learn coupled with total unwillingness to.
in a sense it's about a game. or... maybe it's better to call it a "challenge"
Tarski said "hey, what if we limit ourself only to positive integers? subtraction/division doesn't always work, so let's get rid of it. What can we say about +^? there are some obvious properties, like x+y=y+x for whatever x and y. Is that enough to describe all the equalities?".
Wilkie answered No, and a whole different hunt began - trying to find a set of pseudo "numbers" with pseudo "+^", that would have the same obvious properties, but not the non-obvious one
and SAT-solving is essentially asking computer to find things by describing what they are not - "1+2 is not invalid and can't be 2 different values", "1+(1+1) must not differ from (1+1)+1". A lot of such properties are easily expressible as "not the opposite" - but devil's in the details and some choices of opposites are faster for computer than others. Also symmetries are evil :)
They use the properties of Wilkie's counterexample to restrict the search space. So you can't just pick arbitrary identities that hold over the positive integers and repeat the process until you've found a smaller model.
The lower bound had already been established in prior work: https://zenodo.org/records/18568303
I'm one of the authors of the arXiv paper. Thanks for bringing this to our attention! We were fully unaware of this repository, and it unfortunately did not come up during our literature search. Our approaches to the lower bound are pretty similar, although some technical differences make ours more efficient. For example, to show that there is no counterexample of size 10, we generate a formula with ~50k variables and ~2.7M clauses, which takes about 85 seconds to solve with Kissat. The encoder from this repository generates a formula with ~2k variables and ~33M clauses, which takes about 50 minutes to solve. We have sent an email to the authors of the Zenodo artifact to decide how to proceed!
I think you are right, it's weird that the paper doesn't mention it
They address the following concern of Zhang (2005), which contained prior work involving similar techniques:
> […] Of course, this conclusion is not proved mathematically. It is possible that the programs have some bugs, or the user (myself made some errors.
They address it as follows:
> We address this [… through] ChatGPT 5.5 Pro, through Codex, to automatically generate a Lean formalization that we then checked ourselves to confirm the statements and definitions indeed match their expected semantics. This process took multiple iterations and discussions with the model over several days, and generated over 10,000 lines of code.
This is extremely unconvincing. Manual review of 10,000 lines of AI generated code is a terrifying prospect. I’m sure the result is correct, however I am deeply uncomfortable with this being mankind’s new mathematical process. Similar concerns have been expressed since the days of the 4 color theorem, but this feels different. Perhaps it is just new.
>in a nutshell, it defines an executable function encode that takes a natural number n ≥ 5 and emits a CNF formula, which is byte-for-byte equal to the output of our Python encoding
The phrase “byte-for-byte equal” makes my eye twitch these days.
Hi! One of the authors here. Whether checking the LLM-generated Lean statements/definitions is easy or not depends heavily on the area of mathematics and the concrete definitions at play. In this case it was remarkably easy. As you can see in the repository, the definition of a countermodel is easy to parse, and the statement corresponding to their nonexistence for n <= 11 is also very easy to check. If you have any questions regarding them, I would be happy to answer them.
Personally, this is one of the cases with LLM-generated Lean that I feel most comfortable with: the proofs are all mathematically simple, and we included an example of correctness for one of the SAT constraints in the paper to show that there is nothing complicated about them. Writing them all in Lean, however, would be a very tedious task for which I see no particular benefit (i.e., it's not like some other formalization projects in which one learns something by formalizing; for an example of a human-written Lean correctness proof of a SAT encoding, you can check the paper https://arxiv.org/abs/2403.17370 in which I participated).
Regarding the byte-for-byte equality, the point is that we first had the Python code (which we wrote manually), and then asked LLMs to implement an encoder in Lean that would not only produce "equivalent" formulas but exactly equal formulas, meaning that it would use the same variable indices and the same clauses in the same order. It is well-known that such superficial factors (e.g., clause or variable ordering) can actually have large effects on solver performance (see https://www.cs.utexas.edu/~marijn/publications/pos18.pdf). The Python code is not part of the trust chain, and the Lean code does not rely on it in any way. We included the Python code in the repository since it's much shorter and easier to read.
> This is extremely unconvincing. Manual review of 10,000 lines of AI generated code is a terrifying prospect.
With https://en.wikipedia.org/wiki/Lean_(proof_assistant) (and other proof assistants), you need to review only the lines that correspond to the theorem that you want to prove and their types (I am not very experienced when it comes to lean, but I would assume that comes down to a few hundred lines of code, at most). The rest is left to typechecking (which, I would expect many in the field to agree, is at as reliable than your average peer review process in professional mathematics, and likely much more). That's the reason why Lean4 is making such a fuzz now.
That itself is not trivial too, but way easier than reviewing every function and definition used to prove that the theorems have indeed the types they claim.
If one accepts the proof of the https://en.wikipedia.org/wiki/Four_color_theorem, then there should not be new reservations these proofs; except from the maybe new additional failure scenario that the authors (still correctly!) proved theorems that don't state what they think they stated.
To sum it up: There is IMO no domain more suited for using LLMs than mathematical proofs that can be formalized using Lean4. The fact the hype-circle started earlier in software than in maths is due to the difference in monetary incentives I would assume. (Or another, rather radical and not really serious phrasing: "When it comes to Lean4 proofs that typechecks, there is no AI slop" - the theorem being proven might be uninteresting, but the proof itself is very very very very likely to be correct)
> “byte-for-byte equal”
The term itself or its association with LLMs? I would get the latter, if its the former: It's an desirable property to have, I always like seeing people going that far (assuming obviously that they indeed did so, and in the places where it matters!)
Why is subtraction not part of the algebra? It’s certainly familiar to every high school math student. This omission allows the counterexample, so the reveal is a bit of a disappointment IMHO.
Subtraction is not closed over positive integers, which is untidy. The point of Tarski’s conjecture was to propose a minimal number of axioms and operations, AFAICS they define the standard semiring of positive integers (with the natural definition of exponentiation added).
(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)
Well, yes, but negative numbers are also well known to every high school math student.
Sure. But "High School Algebra (Excluding Subtraction) Problem" isn’t as catchy a name.
They subtracted the subtraction exclusion in the name of simplicity?
Because subtraction is not a total operation on positive integers. Negative numbers leave the domain.
Why was it important to Tarski to limit the domain to positive integers? That seems pointlessly arbitrary.
Truly, you could say that about many conjectures, especially more “fun” classical ones.
I'm not sure, but maybe it is due to that the expression a - b can be replaced as a + (-b)?
Similarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...
We’re in the semiring of positive integers, so there are no additive (or multiplicative) inverses.
The subtraction point is interesting but I don't think it makes the result disappointing. The whole point of Tarski's problem is what follows from that very restricted set of elementary identities so finding the exact minimum countermodel under those rules still seems like a pretty satisfying result.
Isn't the underlying question proved impossible by Godel's incompletness theorem?
No, Gödel's incompleteness theorem applies to theories that can interpret first-order arithmetic, which includes quantified statements like "for all x, there exists a prime p > x".
In this case, we have the much simpler equational theory of positive integers under addition, multiplication, and exponentiation, which does not include any quantifiers. In fact, Gurevič showed that this theory is decidable [0]. On the other hand, Gurevič later showed that this theory is not finitely axiomatizable [1], so an infinite (but still computable) set of axioms is needed to fully characterize the theory.
[0] R. Gurevič, Equational theory of positive numbers with exponentiation, 1985, https://doi.org/10.2307/2044966
[1] R. Gurevič, Equational theory of positive numbers with exponentiation is not finitely axiomatizable, 1990, https://doi.org/10.1016/0168-0072(90)90049-8
I thought SAT like the high school admissions test