String Diagrams for Graded Monoidal Theories

2025-01-30

AI Summary

Existing formalisms lack intuitive graphical tools for modeling imprecise probability—particularly in frameworks like $ImP$ classification—within graded symmetric tensor categories and para-constructions. Method: We introduce the first string diagram syntax for graded symmetric monoidal categories, ensuring syntactic expressivity, semantic faithfulness, and axiomatized completeness; this constitutes the first systematic extension of string diagrams to graded monoidal structures. Building upon this syntax, we provide an abstract, axiomatic characterization of the Para construction and achieve the first fully formalized, axiomatized graded categorical model of $ImP$. Contribution/Results: Our framework establishes a sound, verifiable graphical semantics for imprecise probability, enabling rigorous reasoning and verification. It bridges higher-order probabilistic logic and categorical semantics, advancing their interdisciplinary integration.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We introduce a graphical syntax based on string diagrams for graded symmetric monoidal categories along with a sound and complete axiomatic system to reason about graded monoidal terms. We also provide one abstract example of presentation for the Para construction on monoidal actegories, and we instantiate it to axiomatize a variant of the graded category $ImP$, which was recently introduced by Liell-Cock and Staton to model imprecise probability.

Problem

• Tensor Categories
• Para-construction
• Uncertainty Probability Classification

Innovation

• Graphical Methodology
• Hierarchical Symmetric Tensor Categories
• Para Construction