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Programming language: Haskell
License: GNU Affero General Public License v3.0 or later
Tags: Recursion     Yaya    
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Yet another … yet another recursion scheme library for Haskell.


Recursion schemes allow you to separate any recursion from your business logic, writing step-wise operations that can be applied in a way that guarantees termination (or, dually, progress).

How is this possible? You can’t have totality and Turing-completeness, can you? Oh, but you can – there is a particular type, Partial a (encoded with a fixed-point) that handles potential non-termination, akin to the way that Maybe a handles exceptional cases. It can be folded into IO in your main function, so that the runtime can execute a Turing-complete program that was modeled totally.


  • [yaya](core/README.md) – the safe operations and data types for working with recursion
  • [yaya-hedgehog](hedgehog/README.md) – utilities for testing your Yaya-using code with Hedgehog
  • [yaya-unsafe](unsafe/README.md) – unsafe instances and operations

other libraries


Yaya is a sister library to Turtles – the same approach, but implemented in Scala. Here are some differences to be aware of:

  • the Zoo modules in Turtles are both larger and their use is more encouraged, because Scala’s inference makes it harder to use gcata etc. directly;
  • the Unsafe and Native modules have different contents, because different structures are strict or lazy between the two languages. E.g., in Scala, scala.collection.immutable.List is strict, so the Recursive instance is in Native, while the Corecursive instance is in Unsafe, but Haskell’s Data.List is lazy, so the Corecursive instance is in Native while the Recursive instance is in Unsafe.

Differences from recursion-schemes

  • uses multi-parameter type classes rather than a type family for the Base functor, because the latter frequently requires constraints in the form of Recursive t, Base t ~ f, so we prefer Recursive t f.
  • total operations, with anything potentially partial relegated to the yaya-unsafe package, which has consequences:
    • Recursive (Mu f) f and Corecursive (Nu f) f are the only “safe” instances of recursive data types
  • a separate Steppable class with project and embed, increasing the number of operations that can be defined without needing unsafe instances.
  • relatively few operations are defined, with the intent being that g{cata|ana} be used directly instead. The more commonly known operations are provided in the Zoo modules, with the intent that they provide a mapping from names you may have heard to how Yaya expects you to accomplish them.
  • pattern functors tend to be named independently of their fixed-points. E.g., we use Maybe directly instead of some NatF, XNor a b instead of ListF, and a `AndMaybe` b instead of NonEmptyF.

NB: There are a number of instances (e.g., Corecursive [a] (XNor a)) that are actually safe, but they rely on Haskell’s own recursion. We could potentially add a module/package in between the safe and unsafe ones, containing Corecursive instances for types that are lazy in their recursive parameters and Recursive instances for ones that are strict.

Non-philosophical differences:

  • no TH yet – this might be a philosophical difference, actually. On the one hand, it useful to be able to migrate existing code by quickly creating a pattern functor, but on the other, this stuff is generally unsafe. So, it should at least be relegated to yaya-unsafe.
  • Yaya has productive metamorphisms (stream, etc.). The naïve composition of cata and ana has no benefits, but it has been difficult to generalize Gibbons’ streaming transformers to data structures other than lists.

Differences from compdata

I’m not as familiar with compdata, so I’ll have to look at it more before fleshing this out.

Non-philosophical differences

  • no type-indexed recursion schemes – ideally Yaya would achieve this via PolyKinds.