608 Expert

Catamorphism

Functional Programming

Tutorial Video

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This video demonstrates the "Catamorphism" functional Rust example. Difficulty level: Expert. Key concepts covered: Functional Programming. A catamorphism (fold/cata) is a generalized fold over a recursive data structure. Key difference from OCaml: 1. **HKT requirement**: These morphisms ideally require higher

Tutorial

The Problem

A catamorphism (fold/cata) is a generalized fold over a recursive data structure. Any fold can be expressed as a catamorphism. The F-algebra (algebra of the base functor) specifies how to collapse one level; the catamorphism applies it recursively bottom-up. This generalizes List.fold_right, tree fold, and any recursive descent. The term comes from Meijer et al. 1991 Bananas, Lenses, Envelopes and Barbed Wire.

🎯 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: cata, F-algebra, fold over Fix, bottom-up recursion

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

Code Example

#![allow(clippy::all)]
//! # Catamorphism (Fold)
//! Generalized fold over recursive structures.

pub fn cata_list<A, B>(list: &[A], nil: B, cons: impl Fn(&A, B) -> B) -> B {
    list.iter().rev().fold(nil, |acc, x| cons(x, acc))
}

pub fn sum(xs: &[i32]) -> i32 {
    cata_list(xs, 0, |x, acc| x + acc)
}
pub fn product(xs: &[i32]) -> i32 {
    cata_list(xs, 1, |x, acc| x * acc)
}
pub fn length<A>(xs: &[A]) -> usize {
    cata_list(xs, 0, |_, acc| acc + 1)
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_sum() {
        assert_eq!(sum(&[1, 2, 3]), 6);
    }
    #[test]
    fn test_product() {
        assert_eq!(product(&[1, 2, 3, 4]), 24);
    }
    #[test]
    fn test_length() {
        assert_eq!(length(&[1, 2, 3]), 3);
    }
}

(* Catamorphisms in OCaml *)

(* Nat catamorphism *)
let cata_nat zero succ n =
  let rec go = function
    | 0 -> zero
    | n -> succ (go (n-1))
  in go n

let to_int n = cata_nat 0 (fun x -> x+1) n
let double  n = cata_nat 0 (fun x -> x+2) n

(* List catamorphism = fold_right *)
let cata_list nil cons xs = List.fold_right cons xs nil

let sum   = cata_list 0 (+)
let prod  = cata_list 1 ( * )
let rev   = cata_list [] (fun x acc -> acc @ [x])
let len   = cata_list 0 (fun _ acc -> acc+1)

let () =
  Printf.printf "sum [1..5]  = %d\n" (sum [1;2;3;4;5]);
  Printf.printf "prod [1..5] = %d\n" (prod [1;2;3;4;5]);
  Printf.printf "len [1..5]  = %d\n" (len [1;2;3;4;5])

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)]
//! # Catamorphism (Fold)
//! Generalized fold over recursive structures.

pub fn cata_list<A, B>(list: &[A], nil: B, cons: impl Fn(&A, B) -> B) -> B {
    list.iter().rev().fold(nil, |acc, x| cons(x, acc))
}

pub fn sum(xs: &[i32]) -> i32 {
    cata_list(xs, 0, |x, acc| x + acc)
}
pub fn product(xs: &[i32]) -> i32 {
    cata_list(xs, 1, |x, acc| x * acc)
}
pub fn length<A>(xs: &[A]) -> usize {
    cata_list(xs, 0, |_, acc| acc + 1)
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_sum() {
        assert_eq!(sum(&[1, 2, 3]), 6);
    }
    #[test]
    fn test_product() {
        assert_eq!(product(&[1, 2, 3, 4]), 24);
    }
    #[test]
    fn test_length() {
        assert_eq!(length(&[1, 2, 3]), 3);
    }
}

(* Catamorphisms in OCaml *)

(* Nat catamorphism *)
let cata_nat zero succ n =
  let rec go = function
    | 0 -> zero
    | n -> succ (go (n-1))
  in go n

let to_int n = cata_nat 0 (fun x -> x+1) n
let double  n = cata_nat 0 (fun x -> x+2) n

(* List catamorphism = fold_right *)
let cata_list nil cons xs = List.fold_right cons xs nil

let sum   = cata_list 0 (+)
let prod  = cata_list 1 ( * )
let rev   = cata_list [] (fun x acc -> acc @ [x])
let len   = cata_list 0 (fun _ acc -> acc+1)

let () =
  Printf.printf "sum [1..5]  = %d\n" (sum [1;2;3;4;5]);
  Printf.printf "prod [1..5] = %d\n" (prod [1;2;3;4;5]);
  Printf.printf "len [1..5]  = %d\n" (len [1;2;3;4;5])

✓ Tests Rust test suite

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_sum() {
        assert_eq!(sum(&[1, 2, 3]), 6);
    }
    #[test]
    fn test_product() {
        assert_eq!(product(&[1, 2, 3, 4]), 24);
    }
    #[test]
    fn test_length() {
        assert_eq!(length(&[1, 2, 3]), 3);
    }
}

Deep Comparison

Catamorphism

Fold: tear down structure bottom-up

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