I had no idea that LLMs (or the transformer architecture) were within reach of complexity theory. But if transformers "can be" exponentially more succinct than RNNs, doesn't that mean we're approaching optimality?
Paper went over my head but is this in any way related to my experience of Claude Opus 4.8 using increasingly terse language with very short, overloaded words? Lately I've been having trouble parsing the things it writes about my own code, it's using the kind of compressed language that you see typically in git commit message subject lines but relentless, always on.
The last line of the abstract has the most important takeaway.
> As a consequence of this succinctness, we show that basic
verification problems for transformers, such as emptiness and equivalence, are
provably intractable: specifically, EXPSPACE-complete.
If you were hoping to formally prove the correctness of a large transformer, it turns out that you're going to need an exponentially larger amount of space to do your verification, more than you could possibly afford.
How does it follow that there is no point in trying for formal correctness? In many problems there is an interesting subset that is quickly solvable even when the general case is not.
SAT solvers in practice are quick on just about everything.
SAT solvers being programs that solve the original NP-compete problem.
Authors used LTL (linear temporal logic) to express, basically, non-reduced non-ordered binary decision diagrams. Or just binary decision diagrams, BDDs.
BDDs are almost guaranteed to have exponential size because they do not employ reduction (sharing of common expressions). Reduced BDDs are more succinct and reduced ordered BDDs are even more succinct.
Also, transformers in the paper are constructed, not trained. Training any model to express some truth table is very hard. They also did not perform comparison with, say, Kolmogorov-Arnold representation, which is also universal approximator.
So this paper is not as deep as one may think it is.
This is a truly important paper. It formalizes the intuition that many in the field have. We can stop wasting time doing formal analysis of LLMs. If you have a problem that requires formal verification, don't use an LLM. You can use an LLM to help you build such a system, but the LLM can't be the system.
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[ 3.7 ms ] story [ 37.5 ms ] threadTransformers Are Inherently Succinct (2025) - https://news.ycombinator.com/item?id=48014197 - May 2026 (9 comments)
> As a consequence of this succinctness, we show that basic verification problems for transformers, such as emptiness and equivalence, are provably intractable: specifically, EXPSPACE-complete.
If you were hoping to formally prove the correctness of a large transformer, it turns out that you're going to need an exponentially larger amount of space to do your verification, more than you could possibly afford.
SAT solvers in practice are quick on just about everything.
SAT solvers being programs that solve the original NP-compete problem.
Authors used LTL (linear temporal logic) to express, basically, non-reduced non-ordered binary decision diagrams. Or just binary decision diagrams, BDDs.
BDDs are almost guaranteed to have exponential size because they do not employ reduction (sharing of common expressions). Reduced BDDs are more succinct and reduced ordered BDDs are even more succinct.
Also, transformers in the paper are constructed, not trained. Training any model to express some truth table is very hard. They also did not perform comparison with, say, Kolmogorov-Arnold representation, which is also universal approximator.
So this paper is not as deep as one may think it is.