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Deriving Dependently-Typed OOP from First Principles - Extended Version with Additional Appendices

Core Concepts
The authors explore the field of dependently-typed object-oriented programming by deriving it from first principles using the principle of duality.
The content delves into the challenges and solutions in formalizing a system for dependently-typed object-oriented programming. It discusses core concepts, design constraints, and formalization details. The expression problem is addressed through data type declarations and pattern matching definitions. Challenges like judgmental equality preservation under defunctionalization are discussed. Solutions include user-annotated labels for comatches and avoiding eta-equality to maintain consistency during transformations.
The expression problem describes how most types can easily be extended with new ways to produce the type or new ways to consume the type, but not both. Dependently-typed programming almost exclusively follows the functional programming model rather than the object-oriented model. Defunctionalization eliminates higher-order functions by replacing lambda abstractions with constructors of a data type. Refunctionalization re-introduces higher-order functions by replacing occurrences of constructors with lambda abstractions.
"The expression problem is one way to elucidate the difference between functional or data-oriented programs and object-oriented programs." "Dependently-typed programming almost exclusively follows the functional programming model." "Defunctionalization is a whole-program transformation that eliminates higher-order functions." "Refunctionalization is its partial inverse, reintroducing higher-order functions."

Deeper Inquiries

How does preserving judgmental equality impact program transformations?

Preserving judgmental equality is crucial for maintaining the consistency and correctness of program transformations, such as defunctionalization and refunctionalization. When programs are transformed through these processes, different lambda abstractions or comatches may be mapped to distinct constructors in the resulting data or codata types. If judgmental equality is not preserved, it can lead to inconsistencies where terms that were originally considered equal are no longer treated as such after transformation. For example, if two lambda abstractions ๐œ†๐‘ฅ.๐‘ฅ and ๐œ†๐‘ฆ.๐‘ฆ are initially judged as equal due to their syntactic form, but after transformation they get mapped to different constructors in a data type, then this breaks the original judgmental equality between them. This inconsistency can result in typechecking errors and incorrect behavior of the transformed programs. Therefore, by ensuring that judgmental equality is maintained throughout program transformations, we uphold the integrity of the codebase and prevent unexpected issues from arising during compilation and execution.

How do user-annotated labels for comatches impact program development?

Supporting user-annotated labels for comatches provides programmers with a way to give meaningful names to these constructs within their codebase. These labels serve as identifiers that help developers distinguish between different comatches and provide context about their purpose or functionality. The implications of using user-annotated labels for comatches include: Clarity: By assigning descriptive names to comatches, developers can enhance readability and understandability of the codebase. It makes it easier for team members to grasp the intent behind each comatch without needing detailed comments or documentation. Maintainability: Named comatches facilitate maintenance tasks such as debugging, refactoring, and extending existing functionality. Developers can quickly locate specific parts of code based on their labeled identifiers. Consistency: User-defined labels promote consistency in naming conventions across projects or modules within a project. Consistent naming practices contribute to better organization and structure in software development efforts. 4..Tooling Support: IDEs (Integrated Development Environments) can leverage these annotations for providing intelligent suggestions during coding sessions like auto-completion features based on named entities. Overall,user-annotated labels enhance developer productivity by improving code comprehension,reusability,and maintainability.

How does avoiding eta-equality affect normalization during typechecking?

Avoiding eta-equality means that certain reductions involving functions' extensionality principle (eta-conversion) are not performed during normalization process while typechecking.In functional programming languages,the eta-rule states that two functions f,g: A -> B are considered equal if forall x:A,f(x)=g(x).However,in some contexts,this rule might lead divergence problems,hence its avoidance has been suggested Implications: 1.Reduced Complexity:By excluding eta-reductions,typechecking algorithms become simpler since they do not need consider all possible function extensions. 2.Type Soundness: Avoiding eta-equality helps maintain strong typing guarantees,since it prevents potential inconsistencies introduced by excessive reduction rules. 3.Program Efficiency: While eta-reduction may sometimes optimize performance,it also introduces complexities which could outweigh any benefits.Avoidance ensures predictable behavior without sacrificing efficiency 4.Consistency:Normalization without relying on eta-equality promotes consistent evaluation strategies across different contexts,making it easier predict how expressions will be reduced under various conditions In conclusion,Avoiding Eta-Equivalence simplifies normalization procedures,demonstrates more predictable behavior,and enhances overall robustness of type systems