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Generalised folds and unfolds for R

Preface

This is an experiment in porting the Haskell recursion-schemes package to R.

The library provides generalised folding and unfolding functions, which can be applied to a large number of data structures. The library provides just enough of the Haskell language infrastructure to make it possible to use the following higher-order functions:

• The catamorphism and its categorical dual the anamorphism
• The paramorphism and its categorical dual the apomorphism
• The histomorphism and its categorical dual the futumorphism

Refolds like the hylomorphism are not implemented, but using building-blocks and the examples of the code provided, it should be feasible to implement these.

Tutorial articles (vignettes) and package license

The first package vignette is a tutorial on making use of the various morphisms with the familiar list type. This is primarily a demonstration of each function rather than an explanation of how to implement a "base" functor for a given type.

The second package vignette explores implementation of fmap for the data.tree package's Node type, implementation of a "base" functor for this type. Because of that package's unpredictable behaviour, the vignette then explores implementation of a basic Tree type. Finally, it explores implementation of a monad instance for the basic Tree type, and testing it against the monad laws.

The third package vignette explores using fixed-points instead of project()/embed(), and demonstrates this using Peano numbers. When I was building/testing the library, I used fixed-points instead of the project()/embed() functions, but I later replaced that code with those functions, and also adapted the "base" functor of a list to demonstrate the package, so this library is to an extent adapted from the Haskell recursion-schemes package. It isn't clear to me what the implications are in terms of software licensing when adapting code/ideas (other libraries which implement recursion-schemes such as SCALA's matryoshka don't share the same license as recursion-schemes), so I've applied the same license (BSD 2-Clause) with a note of attribution. My own code is licensed under BSD 2-Clause.

Recursion schemes

My initial explorations of catamorphisms and anamorphisms was largely inspired by the following, excellent, series of blog-posts: https://blog.ploeh.dk/2017/10/04/from-design-patterns-to-category-theory/ (Seemann's book Code that Fits in Your Head also happens to be very good.)

Other resources which are fairly accessible include the following:

• https://jtobin.io/practical-recursion-schemes (this is a series)
• https://blog.sumtypeofway.com/posts/introduction-to-recursion-schemes.html (this is a series)
• https://fho.f12n.de/posts/2014-05-07-dont-fear-the-cat.html
• https://arxiv.org/pdf/2202.13633.pdf

I think that others are better able to explain the theory behind the various recursion schemes in a more accurate and insightful manner than I can, so the function reference tends to be fairly sparse, although every function is documented, and the tutorial vignettes describe usage. Some of the terminology can also be confusing, and Haskell examples provided by others tend to be more succint and clearer than my corresponding implementation in R.

Additional features implemented

Futu and histo depend on the free monad and the cofree comonad, and para and apo depend on the either monad and tuples. As such, the following language features are also implemented:

• S3 methods for functors (fmap())
• S3 methods for monads (bind a.k.a. >>=, discard a.k.a. >>, and join())
• S3 methods for comonads (extract(), extend(), and duplicate())
• Infix pipes which work like the Haskell infix operators, surrounded by percentage signs similar to magrittr pipes

Thus enabling the following:

• The free monad and the cofree comonad
• An fmap and bind implementation for a list
• "Base" functor for a list, which represents the list functor at a given point in recursion
• The Either monad, as well as Tuple, with a fmap implementation for both and a bind implementation for Either. Not all tuples are monads.

Note that Haskell has another typeclass, which bears mentioning, Applicative. In fact, all monads are applicative functors. However, some Applicative instances are pretty unintuitive, and there is invariably more than one possible Applicative instance, so I've not implemented Applicative for any of the types in this package.

Miscellaneous remarks

Pattern-matching—not supported in base R, and the various libraries do not work like in Haskell—would have made many of my R functions much more succint.

Type signatures would have been helpful especially when implementing building-blocks like fmap and free/cofree. Indeed, R is not strongly-typed—it will freely coerce values—but the S3 class system does at least enable writing functions which are specific to a given functor/monad, e.g. the bind implementation for Either only applies to Either values.