Efficient Mendler-Style Lambda-Encodings in Cedille (1803.02473v1)

Abstract: It is common to model inductive datatypes as least fixed points of functors. We show that within the Cedille type theory we can relax functoriality constraints and generically derive an induction principle for Mendler-style lambda-encoded inductive datatypes, which arise as least fixed points of covariant schemes where the morphism lifting is defined only on identities. Additionally, we implement a destructor for these lambda-encodings that runs in constant-time. As a result, we can define lambda-encoded natural numbers with an induction principle and a constant-time predecessor function so that the normal form of a numeral requires only linear space. The paper also includes several more advanced examples.

• The paper introduces a novel approach that derives induction principles from Mendler-style lambda-encodings in Cedille by using identity mappings to relax functoriality constraints.
• It presents a destructor for lambda-encoded datatypes that operates in constant time, significantly improving computational efficiency.
• It demonstrates practical applications including the lambda-encoding of numerals with linear space requirements and constant-time predecessor functions.

Overview of Efficient Mendler-Style Lambda-Encodings in Cedille

The paper "Efficient Mendler-Style Lambda-Encodings in Cedille" by Denis Firsov, Richard Blair, and Aaron Stump presents an innovative approach to deriving induction principles for lambda-encoded inductive datatypes. These derivations occur within the Cedille type theory, exploiting its distinctive features to relax traditional functoriality constraints. The paper reports the development of a destructor for these lambda-encodings, capable of achieving constant-time execution, and highlights the benefits in both computational efficiency and representational compactness.

Key Contributions

The authors provide a number of technical contributions that allow for effective manipulation and management of lambda-encodings:

1. Parameterized Schemes via Identity Mappings: The authors introduce identity mappings as a way to create parameterized schemes for datatypes, relaxing the need for full functoriality. They show that these mappings can be applied to a broader class of schemes than traditional fmap-based functor requirements.
2. Inductive Datatypes from Lambda-Encodings: By leveraging Mendler-style lambda-encodings, the paper demonstrates that induction principles can be derived for a wide array of datatypes, extending beyond merely functorial induced datatypes. The approach uses identity mappings to support the derivation of induction for these encodings.
3. Constant-Time Destructor: An implementation of a destructor running in constant time for Mendler-style lambda-encoded datatypes is provided. This represents a significant improvement over previous methodologies, highlighting the Melder-style encoding’s advantages in computational efficiency.
4. Lambda-Encoding of Numerals: As a practical example, the authors detail the lambda-encoding of numerical representations, delivering constant-time predecessor functions and achieving linear space requirements for numeral representations.

Theoretical and Practical Implications

The paper's findings have both theoretical and practical implications. Theoretically, it extends the potential applications of Mendler-style lambda-encodings by expanding the class of datatypes for which induction principles can be derived without the full constraints of functoriality. Practically, the ability to run destructors in constant time while maintaining a linear space representation opens new avenues for efficient data handling in functional programming languages. This has potential applications in both academic research and industry practices where efficient data manipulation is paramount.

Future Directions

The authors hint at future explorations into coinductive definitions and the investigation of advanced categorical models such as histomorphisms and inductive-recursive datatypes in Cedille. These explorations could broaden the applicability of their techniques and potentially unify disparate methodologies within the field of type theory.

In conclusion, this paper pushes the boundaries of what can be achieved with lambda-encoded datatypes by utilizing the unique properties of Cedille's type theory. It takes steps toward more efficient and powerful data representations that are theoretically robust and practically efficient.