Paper approximates neural network representations with symbolic equations across LLM tasks
- The authors report that a closed-form symbolic equation can replace a network's representation-generating process while leaving behavior largely unchanged.
- They test the approximation on small networks that manipulate lists and on LLMs in arithmetic, logic, computer code, and language.
- The symbolic approximation enables targeted interventions in LLM internal representations, and the authors say resulting behavior changes show the models rely on the identified structures.
- The 30-page arXiv preprint was submitted by R. Thomas McCoy, Paul Soulos, Tal Linzen, and Paul Smolensky on 30 August 2026.
Hacker News opinions
I read the paper as saying LLMs may encode grammar and inference in deeper, mathematically legible relations rather than irreducible layer-smeared noise. That is a pretty interesting interpretation if I have it right.
I have not read it closely yet, but the related-work section points to a rich area. Mechanistic interpretability is a good entry point, including the Transformer Circuits work at transformer-circuits.pub.
This sounds plausible, but I want to know how broad the evidence is. A neural network can approximate a function, and I am unsure whether these are a few selected examples or something more general.
I do not buy the claim that humans cannot comprehend high-dimensional spaces. We cannot visualize beyond 3D, but linear algebra lets us reason about arbitrarily many vector-space dimensions.
My first question is whether these closed-form representations are cheaper to evaluate. If they amount to analytic distillation, that could matter for deploying models outside data centers.
I do not think a symbolic form automatically makes inference cheaper or easier to understand. A huge symbolic algorithm can be as opaque as a neural network, and the tradeoff may be more storage for less GPU use.
I think the important detail is that their representation is a task-specific key-value store embedded with tensor products. If we already know a symbolic formulation for a task, recovering the model output is less surprising, though independently manipulating roles and fillers is interesting.
Page 7 says there is no guarantee DISCOVER can approximate a given network. The learned role and filler embeddings may not be smaller than the original representations, and their outer products may be larger, so this does not imply compute savings.
I dislike the introduction's claim that vectors seem inadequate for language and logic. It treats assumptions about symbolic cognition as settled when the paper should first justify that framing.
Artificial neural networks are only loosely inspired by brains. Biological neurons use mechanisms such as neurotransmitters and action potentials that standard ANNs do not model.