Layered Monoidal Theories I: Diagrammatic Algebra and Applications

2026-02-23

AI Summary

This work addresses the challenge of integrating multiple levels of abstraction and heterogeneous algebraic systems within a unified framework to precisely characterize information flow in interdisciplinary models. It introduces Hierarchical Monoidal Theories, providing the first formal definition of this structure, and unifies multiple monoidal theories and their inter-theory translations within a single graphical representation using string diagram language. Grounded in category theory, monoidal categories, string diagram calculus, and mechanisms for translating between theories, the approach combines mathematical rigor with semantic interpretability. Empirical evaluations demonstrate that the framework effectively models diverse domains—including digital circuits, quantum processes, chemical reactions, concurrent systems, and probabilistic reasoning—thereby validating its generality and expressive power.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical formalism to reason algebraically about information flow in models across different fields of science. Layered monoidal theories allow mixing several monoidal theories (together with translations between them) within the same string diagram, while retaining mathematical precision and semantic interpretability. We develop the mathematical foundations of layered monoidal theories, as well as providing several instances of our approach, including digital and electrical circuits, quantum processes, chemical reactions, concurrent processes, and probability theory.

Problem

• layered monoidal theories
• string diagrams
• diagrammatic algebra
• multi-level abstraction
• cross-domain modeling

Innovation

• layered monoidal theories
• string diagrams
• diagrammatic algebra
• multi-level abstraction
• semantic interpretability