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Zygomorphism

Functional Programming

Tutorial Video

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This video demonstrates the "Zygomorphism" functional Rust example. Difficulty level: Expert. Key concepts covered: Functional Programming. A zygomorphism (zygo) is a mutually recursive fold where two computations run simultaneously, one depending on the other. Key difference from OCaml: 1. **HKT requirement**: These morphisms ideally require higher

Tutorial

The Problem

A zygomorphism (zygo) is a mutually recursive fold where two computations run simultaneously, one depending on the other. The classic example: compute both the value and its depth simultaneously in a single tree traversal. Zygomorphisms generalize paramorphisms and are more efficient than running two separate traversals.

🎯 Learning Outcomes

• The categorical definition and the mathematical laws that must hold

• How to implement this pattern in Rust despite the lack of higher-kinded types

• The relationship to more familiar functional idioms (fold, unfold, map)

• Key concepts: zygo, mutually recursive fold, paired accumulation

• Where this pattern appears in production systems and when simpler alternatives suffice

Code Example

#![allow(clippy::all)]
//! # Zygomorphism
//! Two folds running in parallel.

pub fn zygo<A, B, C>(
    xs: &[A],
    nil_b: B,
    cons_b: impl Fn(&A, &B) -> B,
    nil_c: C,
    cons_c: impl Fn(&A, &B, C) -> C,
) -> C
where
    B: Clone,
{
    let mut bs: Vec<B> = vec![nil_b.clone()];
    for x in xs.iter().rev() {
        bs.push(cons_b(x, bs.last().unwrap()));
    }
    bs.reverse();
    let mut c = nil_c;
    for (i, x) in xs.iter().enumerate().rev() {
        c = cons_c(x, &bs[i + 1], c);
    }
    c
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_zygo() {
        let sum = zygo(&[1, 2, 3], 0, |x, b| x + b, 0, |_, _, c| c + 1);
        assert_eq!(sum, 3);
    }
}

(* Zygomorphism in OCaml *)
(* Compute average and variance in one pass *)

let zygo f g xs =
  (* f is the "helper" algebra, g uses f's intermediate results *)
  List.fold_left (fun (acc_f, acc_g) x ->
    let new_f = f acc_f x in
    let new_g = g acc_g acc_f x in
    (new_f, new_g)
  ) (f [] (List.hd xs), g [] [] (List.hd xs))
  (List.tl xs)

(* Mean and sum of squares in one pass *)
let mean_and_ssq xs =
  let n = float_of_int (List.length xs) in
  let (sum, ssq) = List.fold_left (fun (s,q) x -> (s+.x, q+.x*.x)) (0.,0.) xs in
  (sum /. n, ssq /. n -. (sum/.n) *. (sum/.n))

let () =
  let xs = [2.;4.;4.;4.;5.;5.;7.;9.] in
  let (mean, variance) = mean_and_ssq xs in
  Printf.printf "mean=%.2f variance=%.2f stddev=%.2f\n" mean variance (sqrt variance)

Key Differences

HKT requirement: These morphisms ideally require higher-kinded types for full generality; Rust uses GATs or associated types as approximations.

Performance: Rust's implementations are more verbose but compile to efficient machine code; OCaml's implementations are more concise with similar runtime performance.

Practical adoption: In Haskell, recursive schemes from recursion-schemes are widely used; in Rust and OCaml, direct recursion is more common in practice.

Theoretical value: Understanding these patterns deepens intuition for all recursive programming, even when direct recursion is used in production code.

OCaml Approach

OCaml's pattern matching and recursive types make morphism implementations natural. The Fix type and F-algebra/coalgebra patterns translate directly, though without Haskell's typeclass machinery:

(* OCaml recursive schemes use:
   - Recursive variant types for F-algebras
   - Higher-order functions for the morphism
   - GADTs for type-safe fixed points in advanced cases *)

Full Source

#![allow(clippy::all)]
//! # Zygomorphism
//! Two folds running in parallel.

pub fn zygo<A, B, C>(
    xs: &[A],
    nil_b: B,
    cons_b: impl Fn(&A, &B) -> B,
    nil_c: C,
    cons_c: impl Fn(&A, &B, C) -> C,
) -> C
where
    B: Clone,
{
    let mut bs: Vec<B> = vec![nil_b.clone()];
    for x in xs.iter().rev() {
        bs.push(cons_b(x, bs.last().unwrap()));
    }
    bs.reverse();
    let mut c = nil_c;
    for (i, x) in xs.iter().enumerate().rev() {
        c = cons_c(x, &bs[i + 1], c);
    }
    c
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_zygo() {
        let sum = zygo(&[1, 2, 3], 0, |x, b| x + b, 0, |_, _, c| c + 1);
        assert_eq!(sum, 3);
    }
}

(* Zygomorphism in OCaml *)
(* Compute average and variance in one pass *)

let zygo f g xs =
  (* f is the "helper" algebra, g uses f's intermediate results *)
  List.fold_left (fun (acc_f, acc_g) x ->
    let new_f = f acc_f x in
    let new_g = g acc_g acc_f x in
    (new_f, new_g)
  ) (f [] (List.hd xs), g [] [] (List.hd xs))
  (List.tl xs)

(* Mean and sum of squares in one pass *)
let mean_and_ssq xs =
  let n = float_of_int (List.length xs) in
  let (sum, ssq) = List.fold_left (fun (s,q) x -> (s+.x, q+.x*.x)) (0.,0.) xs in
  (sum /. n, ssq /. n -. (sum/.n) *. (sum/.n))

let () =
  let xs = [2.;4.;4.;4.;5.;5.;7.;9.] in
  let (mean, variance) = mean_and_ssq xs in
  Printf.printf "mean=%.2f variance=%.2f stddev=%.2f\n" mean variance (sqrt variance)

✓ Tests Rust test suite

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_zygo() {
        let sum = zygo(&[1, 2, 3], 0, |x, b| x + b, 0, |_, _, c| c + 1);
        assert_eq!(sum, 3);
    }
}

Deep Comparison

Zygomorphism

Two folds with one depending on the other

Exercises

Laws verification: Write tests that verify the categorical laws for this morphism on a specific data type.

New data type: Apply the morphism to a different recursive data type (e.g., apply catamorphism to a rose tree instead of a binary tree).

Comparison: Implement the same computation using direct recursion and the morphism — measure whether the morphism version composes more cleanly.

Open Source Repos

functional-rust

View the source for this example on GitHub — OCaml and Rust side by side in the repo.

Rust