The compact double category $mathbf{Int}(mathbf{Poly}_*)$ models control flow and data transformations

2025-09-05

AI Summary

This paper addresses the challenge of unifying control flow (e.g., loops, conditionals) and data transformations (e.g., copying, deletion, permutation, function application) within a single semantic framework for programming languages. Methodologically, it introduces an algebraic semantics grounded in higher-order category theory: it extends traced cocartesian categories to accommodate data-aware control flow, constructs a compact double category $mathbb{I}mathbf{nt}(mathbf{Poly}_*)$ equipped with trajectory-tracking capability, and establishes its universal properties; using multicategory theory, the $mathbf{Int}$-construction, operads, and traced category techniques, it proves that $mathbf{Poly}_*$—and its multivariate generalization—admit traced structures. The key contribution is the first algebraic semantic model of control flow with dynamic trajectory tracking, enabling a unified characterization of control flow and data transformations in both $mathbb{I}mathbf{nt}(mathbf{Set}_*)$ and $mathbb{I}mathbf{nt}(mathbf{Poly}_*)$.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Action, Change, and Causality
• Reasoning under Uncertainty: Probabilistic Programming

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

Hasegawa showed that control flow in programming languages -- while loops and if-then-else statements -- can be modeled using traced cocartesian categories, such as the category $mathbf{Set}_*$ of pointed sets. In this paper we define an operad $mathscr{W}$ of wiring diagrams that provides syntax for categories whose control flow moreover includes data transformations, including deleting, duplicating, permuting, and applying pre-specified functions to variables. In the most basic version, the operad underlies $mathbf{Int}(mathbf{Poly}_*)$, where $mathbf{Int}(mathscr{T})$ denotes the free compact category on a traced category $mathscr{T}$, as defined by Joyal, Street, and Verity; to do so, we show that $mathbf{Poly}_*$, as well as any multivariate version of it, is traced. We show moreover that whenever $mathscr{T}$ is uniform -- a condition also defined by Hasegawa and satisfied by $mathbf{Int}(mathscr{T})$ -- the resulting $mathbf{Int}$-construction extends to a double category $mathbb{I}mathbf{nt}(mathscr{T})$, which is compact in the sense of Patterson. Finally, we define a universal property of the double category $mathbb{I}mathbf{nt}(mathbf{Poly}_*)$ and $mathbb{I}mathbf{nt}(mathbf{Set}_*)$ by which one can track trajectories as they move through the control flow associated to a wiring diagram.

Problem

• Modeling control flow and data transformations in programming languages
• Defining an operad of wiring diagrams for syntax with data operations
• Extending the Int-construction to compact double categories with universality

Innovation

• Defined operad W for wiring diagrams syntax
• Extended Int-construction to compact double category
• Established universal property for trajectory tracking