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

TL;DR

This paper introduces a monad-based categorical framework for the ULLER language, unifying its diverse semantics and enabling modular extensions and systematic translations across neurosymbolic systems.

Contribution

It presents a novel categorical semantics for ULLER using monads, allowing modular addition of semantics and systematic translation between them.

Findings

Unified semantics for ULLER via monads

Extension of Giry monad for generalized quantification

Modular implementation of ULLER in Python and Haskell

Abstract

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.