GATlab: Modeling and Programming with Generalized Algebraic Theories (2404.04837v3)

Abstract: Categories and categorical structures are increasingly recognized as useful abstractions for modeling in science and engineering. To uniformly implement category-theoretic mathematical models in software, we introduce GATlab, a domain-specific language for algebraic specification embedded in a technical programming language. GATlab is based on generalized algebraic theories (GATs), a logical system extending algebraic theories with dependent types so as to encompass category theory. Using GATlab, the programmer can specify generalized algebraic theories and their models, including both free models, based on symbolic expressions, and computational models, defined by arbitrary code in the host language. Moreover, the programmer can define maps between theories and use them to declaratively migrate models of one theory to models of another. In short, GATlab aims to provide a unified environment for both computer algebra and software interface design with generalized algebraic theories. In this paper, we describe the design, implementation, and applications of GATlab.

• The paper presents GATlab, a Julia-embedded language that unifies category-theoretic modeling with dependent type programming for Generalized Algebraic Theories.
• GATlab provides an algebraic specification language for defining theories and implementing models in module-style or trait-style, supporting theory morphisms for model migration.
• It enables users to create both symbolic and computational models, facilitates complex computations like string diagram evaluation, and offers a structured foundation for scientific and engineering tools.

Overview of "GATlab: Modeling and Programming with Generalized Algebraic Theories"

This paper presents GATlab, an advanced domain-specific language embedded within Julia, designed to facilitate the use of Generalized Algebraic Theories (GATs) for modeling and programming. GATs extend traditional algebraic theories by incorporating dependent types, enabling a more robust representation of categorical structures. GATlab allows programmers to specify GATs and utilize them to create both symbolic models and computational models defined by arbitrary host language code.

The authors outline the design, implementation, and applications of GATlab, emphasizing its role as a unified environment for both category-theoretic models and software interface design. Unlike existing approaches that separately address categorical structures and dependent type programming languages, GATlab integrates these methodologies into a coherent framework, offering a powerful tool for theoretical and practical advancements in scientific and engineering modeling.

Key Technical Contributions

• Advanced Type System Integration: GATlab augments existing technical computing languages with an advanced type system, enabling uniform computation over categorical structures without retrofitting dependent type theory onto loosely typed languages.
• User-Defined Theories and Models: The platform provides an algebraic specification language with a standard library encompassing over 90 reusable theories. Users can implement models of these theories in two distinct styles: module-style (akin to ML modules) and trait-style (similar to Haskell's typeclasses).
• Expressive Framework for Model Migration: Through theory morphisms, GATlab supports the declarative and algebraic migration of models from one theory to another, enhancing composability and reusability in model design.
• Support for Symbolic and Computational Models: GATlab facilitates the creation of free or symbolic models, supporting advanced computations such as scheduling and evaluating string diagrams, while also allowing integration with computational semantics.

Strong Numerical Results and Bold Claims

The paper does not present empirical numerical results but makes several bold claims regarding the implications of GATlab's design choices. One such claim is that the indexed approach to dependent types reduces redundant storage significantly compared to the fibered approach, leading to more efficient data representation and manipulation. Additionally, the paper asserts that GATlab's unique handling of dependent types and morphisms in computational semantics provides a new structural and symbolic foundation applicable to a broad spectrum of scientific and engineering tools.

Implications and Future Developments

GATlab's implementation offers significant theoretical and practical implications. Theoretically, it advances the algebraic specifications needed to handle complex categorical structures within a programming language. Practically, it simplifies the software development process in scientific computing by providing robust tools for modeling and simulating categorical structures.

Future developments may focus on enhancing the symbolic computation capabilities of GATlab, integrating more sophisticated types of rewriting systems such as e-graphs and Groebner bases in collaboration with existing systems like Metatheory.jl. Additionally, exploring randomized testing for theory validation and model verification could yield new insights and improvements in software reliability. There's also potential exploration of frameworks that naturally represent 2-categories, like cartesian double theories, offering further depth in categorical computer algebra systems.

Overall, GATlab signifies a valuable contribution to both applied category theory and programming language design, providing a rich framework for further exploration in the modeling of complex mathematical structures.