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.
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.
> 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.
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.
Thank you for the explanation, makes it a lot more digestible
Just nit picking a bit:
> Training any model to express some truth table is very hard
What kind of models are you including here? Truth tables can be modeled in regular code very easily and reliably. And I’m sure there are many deterministic models that could do the same. Are you talking about LLMs in particular or a certain category/type of models?
Reduced Ordered BDDs are likely not as succinct as an arbitrary BDD.
The famous Hidden Weight Bit problem can be more succinctly expressed in arbitrary BDDs (changing the order and revisiting nodes), but are provably EXPSPACE in ROBDDs.
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We study ROBDDs because they uniquely reduce to a canonical form. All functions have exactly one (!!!!) ROBDD.
BDDs in general however are really arbitrary, as arbitrary as any codebase can basically get. That makes BDDs in general to difficult to study or do math upon.
Results based on the studies of arbitrary BDDs do NOT apply to the simpler, easier to understand world of ROBDDs. And vice versa.
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.
No, this is in the same ballpark as ideas like big-O notation. The paper is saying that transformers can recognize a language with exponentially fewer symbols than other kinds of systems, i.e. they're more succinct.
It's exactly as related to real models as computer science is to real computers.
That's a budget thing. Claude is suffering from huge demand and they're pulling out all the stops to try to keep the lights on: terse tokens, lobotomizing Claude six ways from Sunday, aggressive batching, the works.
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?
No. We have an infinitely more succinct formalism, it's Turing machines. Succinctness is not necessarily a desirable property, it just says where on the capability-tractability tradeoff something is. Turing machines can express literally anything computable, but in exchange we can't use computers to reason about them in general (Rice's Theorem). Regexes are much more limited, they famously can't even recognize HTML, but we get to automatically prove things about them.
Transformers are Markov chains [1]. Somewhere around this fascinating site [2] I read that stateful models have an advantage. Author provided an example, a state machine with two states A and B, where at state A transitions are to state A (output 0) and to state B (output 1) with equal probability and at state B the transition is always to state A and output is always 1.
For this state machine just one bit of memory can make an optimal prediction that ones always go in pairs, whereas Markov chain will approximate this prediction and never reach optimality.
Markov chains are themselves a kind of state machine, namely a probabilistic deterministic finite automaton (PDFA), albeit where state is solely governed by the N most recent symbols. (Deterministic means that given a sequence, we can always infer the associated state transitions unambiguously). I believe the example in the reference you provide represents the more general case of PDFA, which is not representable as a finite order Markov processs.
Not quite an answer to your question, but you might find this interesting. The Olmo Hybrid paper has some results on relative complexity of problems that can be solved by transformers and RNNs. They don't look at size, just solvability, and find that the sets of problems solvable by the two architectures are incomparable. They actually use these results to inform their architecture design, which includes both attention and state space layers. (Specifically, they choose gated delta-nets with negative eigenvalues, which they show have greater expressivity than those without.)
Proposition 16. UHATs have polynomially bounded expansion over LTL. In particular, given an
LTL formula φ, one can construct in polynomial time a UHAT T such that L(T) = L(φ).
i.e. the blowup is only exponential in one direction.
That says every LTL formula can be compiled into UHAT w/ polynomial overhead. It doesn't say that all languages recognizable w/ UHATs necessarily do not have succinct recgonizers in LTLs or RNNs.
Edit: Actually nevermind. If UHAT could be compiled into LTL w/ polynomial overhead then that would also work for the languages that have exponential overhead in LTL but since they don't there is a strict separation.
> 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.
> 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.
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.
Just nit picking a bit:
> Training any model to express some truth table is very hard
What kind of models are you including here? Truth tables can be modeled in regular code very easily and reliably. And I’m sure there are many deterministic models that could do the same. Are you talking about LLMs in particular or a certain category/type of models?
Reduced Ordered BDDs are likely not as succinct as an arbitrary BDD.
The famous Hidden Weight Bit problem can be more succinctly expressed in arbitrary BDDs (changing the order and revisiting nodes), but are provably EXPSPACE in ROBDDs.
-------
We study ROBDDs because they uniquely reduce to a canonical form. All functions have exactly one (!!!!) ROBDD.
BDDs in general however are really arbitrary, as arbitrary as any codebase can basically get. That makes BDDs in general to difficult to study or do math upon.
Results based on the studies of arbitrary BDDs do NOT apply to the simpler, easier to understand world of ROBDDs. And vice versa.
Transformers Are Inherently Succinct (2025) - https://news.ycombinator.com/item?id=48014197 - May 2026 (9 comments)
It's exactly as related to real models as computer science is to real computers.
Transformers are Markov chains [1]. Somewhere around this fascinating site [2] I read that stateful models have an advantage. Author provided an example, a state machine with two states A and B, where at state A transitions are to state A (output 0) and to state B (output 1) with equal probability and at state B the transition is always to state A and output is always 1.
For this state machine just one bit of memory can make an optimal prediction that ones always go in pairs, whereas Markov chain will approximate this prediction and never reach optimality.
If you include all the information the LLM uses to produce the next token as part of the state, then of course the LLM is a Markov chain.
So would be any other process for sampling continuations of a text, with finite memory.
https://arxiv.org/abs/2604.03444
i.e. the blowup is only exponential in one direction.
Edit: Actually nevermind. If UHAT could be compiled into LTL w/ polynomial overhead then that would also work for the languages that have exponential overhead in LTL but since they don't there is a strict separation.