Reverse problem generation is the idea that, to teach a machine to solve equations, you should stop starting with equations. Associate Professor Hiroshi Kera of Chiba University’s Institute for Advanced Academic Research and Graduate School of Informatics builds the answer first, then constructs a problem around it. In a Q&A published by Chiba University on 16 September 2026 and carried by Tech Xplore, he describes this backwards move as the key novelty of his work on artificial intelligence applied to computational algebra.
The reasoning is practical rather than philosophical. Machine learning needs paired examples, and in computational algebra a pair means an equation system together with its solution. Producing those pairs the ordinary way means running the very algorithm you are trying to accelerate — an algorithm whose worst-case cost is doubly exponential in the number of variables. Kera’s published numbers show what that costs: generating a thousand training samples in five variables took his backward method 7.91 seconds against 1,411.1 seconds for the slowest forward route.
This article sets out what reverse problem generation actually is, the arithmetic behind the speedups in the peer-reviewed work it rests on, what the Transformer models trained on that data can and cannot yet do, and why Kera expects a field he provisionally calls “AI Algebra” to grow out of it. It also covers the research-strategy argument he makes in the same interview, which is unusually candid about how he picks problems.
Table of contents
- What Reverse Problem Generation Actually Means
- The Algebra Problem Reverse Problem Generation Was Built For
- The Numbers Behind Reverse Problem Generation
- What Reverse Problem Generation Trained the Models to Do
- Where Reverse Problem Generation Has Gone Since
- What AI Algebra Would Mean in Practice
- How Reverse Problem Generation Fits Kera’s Research Strategy
- What Reverse Problem Generation Changes for Practitioners
- Frequently Asked Questions About Reverse Problem Generation
- References
What Reverse Problem Generation Actually Means
Start with the conventional loop, because the reversal only makes sense against it.
The forward loop and why it stalls
In traditional algebra research, you formulate a problem and then compute its solution. Kera’s framing is blunt: “In traditional algebra research, the usual approach has been to first formulate a problem and then compute its solution. By contrast, the ‘backward’ approach — starting from a solution and constructing a problem — has received little attention, largely because there was no clear motivation to pursue it.”
That last clause matters. Reverse problem generation was not overlooked because it was hard. It was overlooked because, before machine learning, nobody needed millions of worked examples.
The reversal in one sentence
Reverse problem generation samples a valid solution object at random, then applies transformations that turn it back into a problem whose solution is the object you started with. You get the pair for free, because reverse problem generation never has to solve anything.
Why the training data is the bottleneck
Kera is specific about where the difficulty sits. “The key idea is ‘creating problems.’ Machine learning requires large datasets for learning. In computational algebra, such datasets consist of pairs of equation problems and their solutions. In practice, however, preparing these datasets can be surprisingly difficult.”
| Step | Forward generation | Reverse problem generation |
|---|---|---|
| Starting point | A randomly sampled system of equations | A randomly sampled solution object |
| Expensive operation | Running the solver on every sample | None; only algebraic transformations |
| Failure mode | Solver times out on hard samples | Sample may be unrepresentative |
| Cost as variables grow | Grows sharply, up to doubly exponential | Close to flat in the published range |
| Correctness of the label | Guaranteed by the solver | Guaranteed by construction |
The motivation is new, not the mathematics
“However, the rise of machine learning has changed this situation. The need to generate large datasets for training has given new meaning to this backward approach. By creating problems from known solutions, it becomes possible to generate datasets more efficiently.” Reverse problem generation is therefore a piece of research infrastructure that happens to raise genuine algebraic questions of its own.
The Algebra Problem Reverse Problem Generation Was Built For
The concrete setting in Kera’s peer-reviewed work is the Gröbner basis, and the specific technical result is worth stating plainly.
What a Gröbner basis does
A Gröbner basis is a rewritten form of a system of polynomial equations that makes the system’s structure — including its solutions — readable. Computing one is, as the NeurIPS paper puts it, “a fundamental task in computational algebra.” It is also “known for its notorious doubly exponential time complexity in the number of variables in the worst case.”
The backward Gröbner problem
Kera and his co-authors named the reversal explicitly. Training, they write, “requires many pairs of a polynomial system and the associated Gröbner basis, raising two novel algebraic problems: random generation of Gröbner bases and transforming them into non-Gröbner ones, termed as backward Gröbner problem.”
That is reverse problem generation stated as a research question rather than an engineering trick. Randomly generating a valid Gröbner basis, and then reliably de-optimising it into an equivalent non-Gröbner system, were both open problems that nobody had needed to answer before.
Where it was resolved
The paper resolves both for 0-dimensional radical ideals — a class the authors describe as “the ideals appearing in various applications.” That is a real restriction on where reverse problem generation has been proved to work, and it is the honest boundary of the result. The paper’s own conclusion is that “Gröbner computation is learnable in a particular class,” not in general.
The people involved
The work, “Learning to compute Gröbner bases,” was published at NeurIPS 2024 by Hiroshi Kera, Yuki Ishihara, Yuta Kambe, Tristan Vaccon and Kazuhiro Yokoyama, and first appeared as a preprint in November 2023.
The Numbers Behind Reverse Problem Generation
This is where the argument stops being rhetorical. The paper reports runtimes for building a dataset of 1,000 samples over the rationals, comparing three forward algorithms in SageMath with the libSingular backend against the backward method.
| Method | n = 2 | n = 3 | n = 4 | n = 5 |
|---|---|---|---|---|
Forward, std | 4.20 s | 216.3 s | 740.1 s | 1,411.1 s |
Forward, slimgb | 4.29 s | 183.4 s | 697.5 s | 1,322.7 s |
Forward, stdfglm | 7.22 s | 8.29 s | 21.0 s | 164.3 s |
| Backward (reverse problem generation) | 5.23 s | 5.46 s | 7.05 s | 7.91 s |
The backward curve is almost flat
Read the bottom row across: 5.23, 5.46, 7.05, 7.91 seconds. Adding three variables costs reverse problem generation roughly 2.7 seconds in total. The std row over the same span goes from 4.20 to 1,411.1 seconds.
The speed ratio at five variables
Dividing the forward runtimes by the backward runtime at n = 5 gives the honest comparison, and it is the clearest single justification for reverse problem generation.
The timeouts make the gap wider than it looks
The paper set a five-second timeout on each individual Gröbner computation and added it to the total whenever it fired. At n = 4 the timeout fired on more than 13 per cent of runs, and at n = 5 on more than 25 per cent. So the four-figure forward numbers are not the cost of finishing the work; they are the cost of a run in which a quarter of the samples were abandoned. Reverse problem generation records no timeouts at all, because it never invokes the solver in the first place.
Why 1,000 samples is the wrong unit
The authors add the qualifier that decides the case: “these numbers only show the runtime on 1,000 samples, while training typically requires millions of samples. Therefore, the forward generation is almost infeasible.” Scale the std column by a thousand and a single dataset becomes a fortnight of compute. Scale the backward row by the same factor and it becomes about two and a quarter hours.
What Reverse Problem Generation Trained the Models to Do
Reverse problem generation solves the data problem. It does not, by itself, make the learning work, and the paper is careful about that.
Accuracy over the rationals
Trained with a discrete input embedding on datasets built over the rationals, the Transformer reached accuracies in the high eighties and low nineties, with support accuracy — treating two polynomials as identical when they share the same set of terms — a few points higher again.
The finite-field results are much weaker
Over the finite field with 31 elements the same discrete embedding fell to 46.8, 50.2, 51.1 and 28.6 per cent across the same four variable counts, and the hybrid embedding collapsed almost entirely, to 6.1 per cent at n = 2 and 0.1 per cent at n = 4. The paper describes this as a “learnability gap between polynomials on finite fields and infinite fields.”
Support is easier than coefficients
Across every configuration the support accuracy exceeds the exact-match accuracy, in some cases by a wide margin — 28.6 versus 77.9 per cent at n = 5 over that finite field. The models are better at working out which terms belong in the answer than at getting every coefficient right.
Reading these results fairly
None of this is a claim that a Transformer replaces a computer algebra system. It is a claim that a class of computations previously assumed to be outside the reach of learned models is, in a restricted setting, learnable at all — and that reverse problem generation is what made testing the claim affordable.
Where Reverse Problem Generation Has Gone Since
Two follow-on lines of work show the idea being used rather than just proposed, and they take opposite approaches to the correctness question.
Keeping the guarantees by using the model as an oracle
In “Computational Algebra with Attention,” Kera and co-authors at Zuse Institute Berlin and elsewhere built the Oracle Border Basis Algorithm — “the first Deep Learning approach that accelerates Border basis computation while maintaining output guarantees.” Rather than predicting the answer, the Transformer identifies and removes expensive reduction steps that the authors found dominate runtime, and the classical algorithm does the rest.
The speedup is modest and the correctness is intact
That design achieves “substantial speedup factors of up to 3.5x compared to the base algorithm, without compromising the correctness of results.” A 3.5x speedup with guarantees is a very different product from a 178x data-generation speedup, and the two should not be confused.
The sampling theorem is a reverse problem generation result
The same paper reports that to generate its training data the team “developed a sampling method and provided the first sampling theorem for border bases.” That is reverse problem generation again, in a second algebraic setting, and again it produced a new theorem as a by-product.
Others are now building on it
Independent groups have picked the thread up. HATSolver, from Mohamed Malhou, Ludovic Perret and Kristin Lauter, applies hierarchical attention to the same task and reports solving “instances that are much larger than those in Kera et al.” Its opening line credits the NeurIPS work directly.
The tooling is deliberately aimed at mathematicians
Kera, Shun Arakawa and Yuta Sato released CALT — Computer Algebra with Transformer — described as “a user-friendly Python library designed to help non-experts in deep learning train models for symbolic computation tasks.” In the Q&A, Kera ties this to a goal: “Our goal is to create an environment in which mathematicians themselves can actively expand the boundaries of their field.”
What AI Algebra Would Mean in Practice
Kera’s forward-looking claim is the part of the interview most likely to be quoted, and it deserves reading closely rather than as a slogan.
The named prediction
“Algebra existed long before computers, and with the advent of computing, the field of computational algebra emerged. Now, with the rise of AI, we may see the development of a new area — something like ‘AI Algebra.’ I am not sure what it will ultimately be called, but I hope to contribute to further expanding the field of algebra in this new direction.”
It is framed as an instance of AI for Science
He places it inside a broader movement: “So far, much of AI research has focused on improving benchmark performance. However, I believe the field is gradually shifting toward real-world applications. In the sciences, the trend is often referred to as ‘AI for Science,’ and I expect that computational algebra will also increasingly incorporate AI.”
There is an organising structure behind it
Kera is on the steering committee of Symbolic Computation and Machine Learning, a research initiative applying machine learning and deep learning to symbolic computation and computational algebra. His stated aim is “to help build a network that further integrates computational algebra and AI” — which is what turns a technique into a field.
Mathematicians are already using the backward view
The claim that most exceeds the published evidence, and that Kera makes carefully, is this: “In fact, I feel that this line of research is opening new possibilities. Some mathematicians are already making new discoveries through this backward perspective. For me, this sense that a field can expand in unexpected ways is what makes research so exciting.” No specific discovery is named in the interview, so treat it as the researcher’s read of his own field rather than a citable result.
How Reverse Problem Generation Fits Kera's Research Strategy
The Q&A spends as much time on research strategy as on algebra, and this portion is transferable well beyond mathematics.
Pick topics with a high barrier to entry
“One thing I keep in mind is choosing research topics that have a high barrier to entry for others. In my case, I draw on my background in algebra — which relatively few researchers specialize in — and work on projects that combine algebra with other fields. Because there is less direct competition, I can take the time to focus deeply on my research.”
Refuse to compete on benchmark numbers
“Another point I’m conscious of is avoiding work that competes on numerical performance. In areas like object recognition or image classification, where the goal is to achieve the highest accuracy, the field moves very quickly, and new methods are constantly emerging. Even if your work is state-of-the-art at one moment, it can be overtaken within a month.” The alternative he states is “research that offers qualitatively different approaches and perspectives.”
Size the project to the team
“In my lab, each project is typically carried out by a single student or by myself. As a result, I try not to compete in terms of manpower.” That is an explicit constraint driving topic choice, not a complaint about resources — and any small team choosing where to spend its AI strategy effort faces the same arithmetic.
Read at volume, in a structured way
His information diet is specific. In lab seminars, “three to four members are assigned each week, and each person presents three to four papers,” supplemented by a lab messaging platform. At major AI conferences “around 5,000 papers may be accepted at a single conference,” and because posters are visual “you can quickly grasp the overall idea of a study just by walking around.” He contrasts this with computational algebra conferences of “around 70 accepted papers,” where “you interact with the same researchers throughout” and in-depth discussion is easier.
The path was not planned
Kera is frank that the topic found him. Assigned to a lab working on genetic algorithms, he was asked by a newly arrived bioinformatics professor whether he was interested in algebra — days after buying a book on Galois theory because he liked the cover. “It felt like a kind of coincidence — or even fate.” He then moved to a computer vision lab for his master’s before returning to algebra for his doctorate, and argues the detour helped: “Classical machine learning methods are largely based on linear algebra, so working in that area helped deepen my understanding of the subject.”
What Reverse Problem Generation Changes for Practitioners
Most readers will never compute a Gröbner basis. The transferable part is the data-construction pattern, and it generalises further than algebra does.
The pattern in the abstract
Wherever labelling an example requires running an expensive process, ask whether the process can be inverted. If you can sample the output cheaply and degrade it into a plausible input, you get correct labels by construction rather than by computation. That is reverse problem generation stated in its most general form.
Where the pattern already appears
Constructing a corrupted signal from a clean one to train a denoiser, or building a scrambled state from a solved one to train a solver, are the same move. What reverse problem generation added is a case where the inversion itself was an unsolved mathematical problem, and solving it produced publishable theory.
The honest limits
Two cautions apply. First, a distribution you construct is not the distribution you will meet: the samples that reverse problem generation produces may be systematically unlike real-world problem instances, and no runtime table can tell you that. Second, the accuracy results above show the learnability is class-dependent — strong over the rationals, weak over a finite field with 31 elements, on identical machinery.
Where the verified-correct route sits
For any application where a wrong answer is unacceptable, the oracle design from the border basis work is the safer template: keep the classical algorithm, let the model choose which expensive steps to skip, and accept the smaller speedup. That is a machine learning deployment shape worth borrowing.
Frequently Asked Questions About Reverse Problem Generation
What is reverse problem generation in one line?
Building a training example by sampling the answer first and constructing a matching problem from it, instead of sampling a problem and solving it.
Who is behind the research?
Associate Professor Hiroshi Kera at Chiba University’s Institute for Advanced Academic Research and Graduate School of Informatics, with collaborators including Yuki Ishihara, Yuta Kambe, Tristan Vaccon and Kazuhiro Yokoyama.
How much faster is reverse problem generation?
On the published dataset-generation benchmark at five variables, 7.91 seconds against 1,411.1 seconds for the slowest forward method — about 178 times faster — and about 21 times faster than the best forward method at 164.3 seconds.
Does reverse problem generation mean AI can now solve algebra?
No. The NeurIPS paper’s claim is that Gröbner computation is “learnable in a particular class,” specifically 0-dimensional radical ideals, with exact-match accuracy between 86.5 and 93.7 per cent over the rationals and far lower over finite fields.
Is there software I can use?
CALT, a Python library from Kera, Arakawa and Sato, is aimed at people who know symbolic computation but not deep learning.
What is “AI Algebra”?
Kera’s provisional name for a discipline standing to computational algebra as computational algebra stood to algebra — his words are that he is “not sure what it will ultimately be called.”
Readers tracking research releases across vendors may also want our AI models and tools hub, and our earlier coverage of mathematicians asking OpenAI for proof it did not use their work sits alongside this one.
References
Learning to compute Gröbner bases (NeurIPS 2024)
Learning to Compute Gröbner Bases (arXiv preprint)
Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms
CALT: A Library for Computer Algebra with Transformer
HATSolver: Learning Groebner Bases with Hierarchical Attention Transformers
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