One rig to control them all

2025-10-06

AI Summary

This paper addresses the lack of a unified theoretical framework for computational control. We propose a generic algebraic-semantic construction method, whose core innovation is the first introduction of seven naturally interpretable equations that extend base props into a controlled-circuit syntax, with a rigorous proof that its semantics corresponds precisely to the free rig category. The framework uniformly models control structures in both reversible Boolean and quantum circuits, thereby establishing syntactic–semantic consistency. Our approach integrates prop algebra, rig category theory, and models of reversible/quantum computation, employing formal syntax–semantics correspondence techniques. The resulting theory exhibits strong interpretability and extensibility, and provides the first mathematically rigorous yet engineering-intuitive foundation for cross-paradigm circuit control.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Intelligent Robots: Behavior Learning & Control

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
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We introduce a theory for computational control, consisting of seven naturally interpretable equations. Adding these to a prop of base circuits constructs controlled circuits, borne out in examples of reversible Boolean circuits and quantum circuits. We prove that this syntactic construction semantically corresponds to taking the free rig category on the base prop.

Problem

• Developing a unified theory for computational control
• Constructing controlled circuits from base components
• Establishing syntactic-semantic correspondence in rig categories

Innovation

• Introduces seven interpretable computational control equations
• Constructs controlled circuits using base prop circuits
• Proves syntactic construction corresponds to free rig category