One of the oldest arguments in AI is about whether neural networks and symbolic systems are fundamentally different things or whether one secretly subsumes the other. A new paper from R. Thomas McCoy, Paul Soulos, Tal Linzen, and Paul Smolensky comes down firmly on the side of “not as different as you think” — with concrete empirical evidence.
The central claim: the vector representations of neural networks, which live in continuous high-dimensional spaces and are typically treated as opaque numeric blobs, can be closely approximated by symbolic structures. Close enough that the authors can “replace the network’s entire representation-generating process with a closed-form equation instantiating a symbolic structure” while preserving the network’s behavior.
They show this across two regimes: small-scale networks trained to manipulate lists, where the ground truth symbolic structure is known, and large language models working in four domains — arithmetic, logic, computer code, and natural language. In both cases, they identify the implicit symbolic structure and confirm it’s not just a post-hoc gloss: they can intervene on an LLM’s internal representations through those identified structures and reliably shift its behavior in targeted ways. If the symbolic structure were a superficial approximation that didn’t capture anything real, those interventions would fail. They don’t.
This work is rooted in Paul Smolensky’s tensor product representations, a framework for encoding discrete symbolic structures in continuous vector spaces that he has been developing since the 1980s. The framework decomposes a representation into a sum of tensor products between fillers (the values) and roles (the structural positions), yielding a continuous object that nonetheless supports structure-sensitive operations. The new contribution is showing empirically, at scale, that trained neural networks arrive at essentially this structure without being told to — it emerges from learning.
The implications for interpretability are interesting. A lot of interpretability work treats neural network representations as arbitrary learned features that need to be reverse-engineered through probing, activation patching, or circuit analysis. This paper suggests a complementary framing: maybe the representations have a principled algebraic structure that can be identified analytically rather than inductively. If you can write down the closed-form equation for a network’s representations, interventions become significantly more precise than “ablate this attention head and see what breaks.”
There is also a consequence for generalization theory. Symbolic structures support systematic generalization in ways that purely statistical associations do not — if a network has internalized a symbolic role/filler decomposition, it can potentially apply that structure to new inputs in the way a rule-following system would, not just in the way a nearest-neighbor lookup would. Whether this explains LLM generalization behavior is an open empirical question, but the paper gives a mechanism that was previously missing.
Smolensky’s theoretical framework has long been somewhat ahead of the experimental evidence that would validate it at scale. This paper looks like that validation, and it arrives at a moment when interpretability has become a serious engineering and policy concern, not just a research curiosity.