NeSyCat: A Monad-Based Categorical Semantics of the Neurosymbolic ULLER Framework
For researchers in neurosymbolic AI, this provides a principled way to combine and translate between different semantics, but the work is incremental as it applies existing categorical concepts to a specific framework.
The paper unifies three seemingly disparate semantics (classical, fuzzy, probabilistic) of the ULLER neurosymbolic framework under a single categorical framework based on monads, enabling modular addition of new semantics and systematic translations. It demonstrates the approach by extending the Giry monad to add generalised quantification in Logic Tensor Networks to arbitrary domains.
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.