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

2026-04-27

AI Summary

This work addresses the fragmentation among classical, fuzzy, and probabilistic semantics in the ULLER framework, which lacks a unified formal foundation and thus hinders modular extension or mutual transformation. To resolve this, the paper introduces monadic structures from category theory into neuro-symbolic computation, proposing a unified semantic framework wherein diverse logical semantics are treated as instances of a common categorical construction. This enables systematic semantic translation and modular composition. By extending the Giry monad, the framework supports generalized quantification over arbitrary domains. A prototype implementation in Python and Haskell demonstrates its practical applicability, successfully integrating with Logic Tensor Networks to handle generalized quantifiers over infinite domains.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Neuro-Symbolic Learning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Neurosymbolic
• Categorical semantics
• Monad
• ULLER
• Unified logic

Innovation

• monad
• categorical semantics
• neurosymbolic
• ULLER
• Giry monad