NeSyCat Torch: Differentiable categorical semantics for NeSy

NeSyCat Torch is a differentiable tensor implementation of categorical semantics for neurosymbolic learning, published to arXiv on 17 Jun 2026 as arXiv:2606.19279 by Daniel Romero Schellhorn, Till Mossakowski and Björn Gehrke. The project embeds neural predicates and functions into the NeSyCat framework and supplies HaskTorch, JAX, and PyTorch implementations, evaluated on an MNIST addition task where it outperforms LTN and DeepProbLog in speed and accuracy and achieves nearly the accuracy of DeepStochLog.

What is NeSyCat Torch?

NeSyCat Torch is an implementation that completes NeSyCat by giving predicates and functions a neural-network interpretation and making the categorical semantics differentiable. The paper presents a unified inductive definition of truth that subsumes classical, fuzzy, probabilistic and neural systems, and it supplies concrete tensor-based backends so those semantics can be trained with gradient methods.

The authors describe the construction as parametric in a strong monad and an aggregation structure on truth-values. They provide code-level axioms written in monad-based do-notation, so the semantics and computation align: "The axioms are the source code." The implementation variants include HaskTorch, JAX and PyTorch backends.

How does NeSyCat Torch implement semantics and training?

NeSyCat Torch implements symbols via neural networks and uses multiple monads for semantics and stable training: the distribution monad is used for reference semantics and metric evaluation, while a lazy log-tensor monad over the log-semiring supports numerically stable, differentiable training; a batch monad enables efficient batched training. Monadic bind performs marginalisation and the lazy monad prunes unneeded branches during computation.

Practically, the paper supplies tensor-based and probabilistic-programming backends so the same monadic axioms drive different implementations. The construction is explicitly parametric in the monad, which the authors note makes it extensible: instantiating the framework with the Giry monad would extend it to continuous probability, though working out a neural representation for that case is left for future work.

What did the MNIST addition experiments show?

On the MNIST addition benchmark the paper states its HaskTorch, JAX, and PyTorch implementations outperform Logic Tensor Networks (LTN) and DeepProbLog in both speed and accuracy, while achieving nearly the accuracy of DeepStochLog. The authors use the distribution monad for reference semantics and the lazy log-tensor monad for training stability, and they attribute the efficiency gains to the monad-based implementation and lazy marginalisation.

The paper does not publish numeric accuracy or timing values in the abstract; the comparative claims given are qualitative: faster and more accurate than LTN and DeepProbLog, and nearly matching DeepStochLog on accuracy.

Why it matters

NeSyCat Torch closes a gap in neurosymbolic research by giving a single categorical semantics a concrete neural implementation and differentiable training route. Because the framework is parametric in the monad, the same core semantics can target varied probabilistic choices from discrete distributions to continuous Giry-style measures. That makes the approach a potential common substrate for many first-order neurosymbolic systems, rather than a collection of ad hoc integrations.

Demonstrated performance gains on MNIST addition versus LTN and DeepProbLog show the idea can be competitive in practice, and the near parity with DeepStochLog on accuracy suggests the abstraction does not force a performance tradeoff compared with specialized systems.

What to watch

Look for a neural representation of the Giry monad, which the authors identify as future work; that will determine whether the parametric construction can handle continuous probability in practice. Also watch for full numeric results and code releases from the HaskTorch, JAX and PyTorch implementations to verify the paper's claims about speed and accuracy on MNIST addition.

References: arXiv:2606.19279, submitted 17 Jun 2026 by Daniel Romero Schellhorn, Till Mossakowski and Björn Gehrke.