# Crate recursion

Expand description

Tools for working with recursive data structures in a concise, stack safe, and performant manner.

This crate provides abstractions for separating the machinery of recursion from the logic of recursion. This is similar to how iterators separate the machinery of iteration from the logic of iteration, allowing us to go from this:

``````let mut n = 0;
while n < prices.len() {
print!("{}", prices[n]);
n += 1;
}``````

to this:

``````for n in prices.iter() {
print!("{}", n)
}``````

This second example is less verbose, has less boilerplate, and is generally nicer to work with. This crate aims to provide similar tools for working with recursive data structures.

## Here’s how it works: Expr

For these examples, we will be using a simple recursive data structure - an expression language that supports a few mathematical operations.

``````pub enum Expr {
Sub(Box<Expr>, Box<Expr>),
Mul(Box<Expr>, Box<Expr>),
LiteralInt(i64),
}``````

For working with this `Expr` type we’ll define a frame type `ExprFrame<A>`. It’s exactly the same as `Expr`, except the recursive self-reference `Box<Self>` is replaced with `A`. This may be a bit confusing at first, but this idiom unlocks a lot of potential (expressiveness, stack safety, etc). You can think of `ExprFrame<A>` as representing a single stack frame in a recursive algorithm.

``````pub enum ExprFrame<A> {
Sub(A, A),
Mul(A, A),
LiteralInt(i64),
}``````

Now all we need is some mechanical boilerplate: `MappableFrame` for `ExprFrame` and `Expandable` and `Collapsible` for `Expr`. I’ll elide that for now, but you can read the documentation for the above traits to learn what they do and how to implement them.

## Collapsing an Expr into a value

Here’s how to evaluate an `Expr` using this idiom, by collapsing it frame by frame via a function `ExprFrame<i64> -> i64`:

``````fn eval(e: &Expr) -> i64 {
e.collapse_frames(|frame| match frame {
ExprFrame::Add(a, b) => a + b,
ExprFrame::Sub(a, b) => a - b,
ExprFrame::Mul(a, b) => a * b,
ExprFrame::LiteralInt(x) => x,
})
}

let expr = multiply(subtract(literal(1), literal(2)), literal(3));
assert_eq!(eval(&expr), -3);``````

Here’s a GIF visualizing the operation of `collapse_frames`:

## Fallible functions

At this point, you may have noticed that we’ve ommited division, which is a fallible operation because division by 0 is undefined. Many real world algorithms also have to handle failible operations, such as this. That’s why this crate also provides tools for collapsing and expanding recursive data structures using fallible functions, like (in this case) `ExprFrame<i64> -> Result<i64, Err>`.

``````
fn try_eval(e: &Expr) -> Result<i64, &str> {
e.try_collapse_frames(|frame| match frame {
ExprFrame::Add(a, b) => Ok(a + b),
ExprFrame::Sub(a, b) => Ok(a - b),
ExprFrame::Mul(a, b) => Ok(a * b),
ExprFrame::Div(a, b) =>
if b == 0 { Err("cannot divide by zero")} else {Ok(a / b)},
ExprFrame::LiteralInt(x) => Ok(x),
})
}

let valid_expr = multiply(subtract(literal(1), literal(2)), literal(3));
let invalid_expr = divide(literal(2), literal(0));

assert_eq!(try_eval(&valid_expr), Ok(-3));
assert_eq!(try_eval(&invalid_expr), Err("cannot divide by zero"));``````

Here’s a GIF visualizing the operation of `try_collapse_frames` for `valid_expr`:

And here’s a GIF visualizing the operation of `try_collapse_frames` for `invalid_expr`:

## Expanding an Expr from a seed value

Here’s an example showing how to expand a simple `Expr` from a seed value

``````fn build_expr(depth: usize) -> Expr {
Expr::expand_frames(depth, |depth| {
if depth > 0 {
ExprFrame::Add(depth - 1, depth - 1)
} else {
ExprFrame::LiteralInt(1)
}
})
}

assert_eq!(expected, build_expr(2));
``````

Here’s a GIF visualizing the operation of `expand_frames``:

## Miscellaneous errata

All GIFs in this documentation were generated via tooling in my `recursion-visualize` crate, via `examples/expr.rs`.

If you’re familiar with Haskell, you may have noticed that this crate makes heavy use of recursion schemes idioms. I’ve named the traits used with an eye towards readability for users unfamiliar with those idioms, but feel free to read `MappableFrame` as `Functor` and `Expandable`/`Collapsible` as `Corecursive`/`Recursive`. If you’re not familiar with these idioms, there’s a great blog post series here that explains the various concepts involved.

## Traits

• The ability to recursively collapse some type into some output type, frame by frame.
• The ability to recursively expand a seed to construct a value of this type, frame by frame.
• A single ‘frame’ containing values that can be mapped over via `map_frame`.