# NeSyCat: A Monad-Based Categorical Semantics of the Neurosymbolic ULLER Framework

arXiv: https://arxiv.org/abs/2604.24612 · HTML: https://mlsanity.com/paper/2604.24612/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 44 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Built on by (most cited first, 1 of 1)

- [NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning](https://mlsanity.com/paper/2606.19279/) (2026)

## Abstract (arXiv)

ULLER (Unified Language for LEarning and Reasoning) offers a unified first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics: classical, fuzzy, and probabilistic, each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in functional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalised quantification in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular implementation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.