Tape Diagrams for Monoidal Monads

2025-03-28

AI Summary

Tape diagrams were originally restricted to rig categories with biproducts, limiting their applicability in semantics of nonlinear, probabilistic, and quantum computation. Method: We generalize tape diagrams to arbitrary symmetric monoidal categories by first characterizing symmetric monads algebraically and then constructing a semantic interpretation and graphical calculus for their Kleisli categories. Contribution/Results: (1) We introduce a syntax and graphical reasoning system for tape diagrams applicable to any symmetric monad—not only those arising from biproducts; (2) we uniformly model additive structures prevalent in semantics of non-linear, probabilistic, and quantum programming; (3) we provide a compositional, category-theoretically sound visual formalism for monadic computational models. This extension substantially enhances the expressivity and practical utility of tape diagrams in programming language semantics and quantum information theory.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models

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: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, $oplus$ and $otimes$, where $otimes$ distributes over $oplus$. However, their applicability is limited to categories where $oplus$ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.

Problem

• Extend tape diagrams beyond biproduct categories
• Apply tape diagrams to Kleisli categories
• Represent symmetric monoidal monads graphically

Innovation

• Extends tape diagrams to Kleisli categories
• Handles symmetric monoidal monads
• Uses algebraic theories for presentation