Category Theory

Today’s frontier models imitate computation without reliably executing it. Andrew Dudzik’s test is surgical: change one digit in a patterned addition problem and ChatGPT stops propagating the carry correctly, while Petar Veličković notes that some current frontier models perform “hundreds of billions of multiplications” per token yet cannot reliably multiply small numbers. For investors, capability scale and computational reliability remain distinct thesis variables.

Tool use helps, but it does not remove the architectural bottleneck. A model must still predict the calculator’s correct inputs, and longer reasoning chains may require repeated calls, rethinking, and model-tool transitions. Dudzik argues that internalized basic computation “has a chance to be a lot more stable,” with potentially meaningful inference-efficiency gains.

Taco Cohen rejects the strongest version of the claim that neural networks cannot learn algorithms. Properly trained models might learn long addition up to memory limits and occasional carrying mistakes, much like humans; neural machinery can execute something symbolic while retaining world knowledge and tolerance for vague concepts. The disagreement is over reliable extrapolation, not whether neural computation can ever resemble a procedure.

Geometric deep learning demonstrates the practical value of correct architectural priors. Translation and permutation equivariance restrict the hypothesis space without bias when the symmetry is valid, cutting required data “almost exponentially”; transformers themselves exploit permutation equivariance after token and position embeddings are attached. But group symmetry only describes reversible transformations, while real programs routinely destroy information.

Category theory is proposed as the generalization from spatial symmetry to computation itself. Moving from groups to monoids removes invertibility; moving to categories also removes the assumption that every operation composes with every other operation. The ambition is a “periodic table for neural networks” from which architectures can be derived rather than discovered through ad hoc knobs and tweaks.

The framework’s concrete targets are recursion, principled weight sharing, and neural implementations of carrying. In a 2-category, weight tying becomes a coherent reparameterization rather than an unexplained coding convention, and the same abstraction can express relationships beyond literal copying. Dudzik’s harder target is a neural adder that transports the information hidden in a state change—potentially a route toward “actual CPUs in neural networks.”

Veličković envisions a hybrid rather than a purely neural system. Neural networks would translate noisy reality into an abstract space, with computation handled through learned priors, carefully designed losses, or tool and algorithmic components offering correctness or convergence guarantees. His standard is not 100% accuracy, but systems that know the effort a problem requires and can say, “the problem…is too computationally large for my capabilities.”

🔗 Original source & video: Making deep learning perform real algorithms with Category Theory (Andrew Dudzik, Petar Velichkovich, Taco Cohen, Bruno Gavranović, Paul Lessard)