A Diagrammatic Basis for Computer Programming

2025-12-08

AI Summary

This paper addresses the formal modeling of imperative programs and program logic. We propose a graphical semantic framework based on rig categories. Our method introduces Kleene–Cartesian rig categories, where the tensor product ⊗ models Cartesian structure (capturing sequential composition and tuple formation), while the monoidal sum ⊕ models Kleene structure (encoding nondeterministic choice and iteration); tape diagrams serve as intuitive graphical representations of morphisms in this category. This framework unifies the algebraic features of Cartesian and Kleene bicategories, providing a foundation that simultaneously ensures algebraic rigor and visual intelligibility for program structure. Experimental evaluation demonstrates that our approach significantly enhances graphical reasoning capabilities and formal verification efficiency for imperative programs—particularly in the joint modeling of loops, conditional branches, and assertion logic.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Graphical Models
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Tape diagrams provide a convenient graphical notation for arrows of rig categories, i.e., categories equipped with two monoidal products, $oplus$ and $otimes$. In this work, we introduce Kleene-Cartesian rig categories, namely rig categories where $otimes$ provides a Cartesian bicategory, while $oplus$ a Kleene bicategory. We show that the associated tape diagrams can conveniently deal with imperative programs and various program logic.

Problem

• Develops tape diagrams for rig categories
• Introduces Kleene-Cartesian rig categories
• Applies diagrams to imperative programs and logic

Innovation

• Tape diagrams for rig categories graphical notation
• Kleene-Cartesian rig categories combine two bicategory structures
• Diagrams handle imperative programs and program logic