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Anamorphism

Functional Programming

Tutorial Video

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This video demonstrates the "Anamorphism" functional Rust example. Difficulty level: Expert. Key concepts covered: Functional Programming. An anamorphism (unfold/ana) is the categorical dual of a catamorphism — it builds a recursive structure top-down from a seed. Key difference from OCaml: 1. **HKT requirement**: These morphisms ideally require higher

Tutorial

The Problem

An anamorphism (unfold/ana) is the categorical dual of a catamorphism — it builds a recursive structure top-down from a seed. While catamorphisms consume structures, anamorphisms produce them. The F-coalgebra specifies how to expand one level from a seed value. Examples: generating a list from a range, unfolding a tree from a depth-first generator, producing a stream from a state.

🎯 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: ana, F-coalgebra, unfold, top-down structure generation

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

Code Example

#![allow(clippy::all)]
//! # Anamorphism (Unfold)
//! Generalized unfold to build recursive structures.

pub fn ana<S, A>(seed: S, next: impl Fn(S) -> Option<(A, S)>) -> Vec<A> {
    let mut result = Vec::new();
    let mut s = seed;
    while let Some((a, s2)) = next(s) {
        result.push(a);
        s = s2;
    }
    result
}

pub fn range(start: i32, end: i32) -> Vec<i32> {
    ana(start, |n| if n < end { Some((n, n + 1)) } else { None })
}

pub fn iterate<A: Clone>(init: A, f: impl Fn(&A) -> A, n: usize) -> Vec<A> {
    ana((init, n), |(a, count)| {
        if count > 0 {
            Some((a.clone(), (f(&a), count - 1)))
        } else {
            None
        }
    })
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_range() {
        assert_eq!(range(0, 5), vec![0, 1, 2, 3, 4]);
    }
    #[test]
    fn test_iterate() {
        assert_eq!(iterate(1, |x| x * 2, 4), vec![1, 2, 4, 8]);
    }
}

(* Anamorphism in OCaml *)

(* unfold: generate a list from a seed *)
let unfold f seed =
  let rec go acc s =
    match f s with
    | None        -> List.rev acc
    | Some (x, s') -> go (x :: acc) s'
  in go [] seed

let range lo hi    = unfold (fun i -> if i >= hi then None else Some (i, i+1)) lo
let fibs max_n     = unfold (fun (a,b) -> if a > max_n then None else Some (a, (b, a+b))) (0,1)
let to_digits n    = unfold (fun n -> if n=0 then None else Some (n mod 10, n/10)) n

let () =
  Printf.printf "range 1..5: %s\n" (String.concat "," (List.map string_of_int (range 1 6)));
  Printf.printf "fibs<=100: %s\n" (String.concat "," (List.map string_of_int (fibs 100)));
  Printf.printf "digits 1234: %s\n" (String.concat "," (List.map string_of_int (to_digits 1234)))

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)]
//! # Anamorphism (Unfold)
//! Generalized unfold to build recursive structures.

pub fn ana<S, A>(seed: S, next: impl Fn(S) -> Option<(A, S)>) -> Vec<A> {
    let mut result = Vec::new();
    let mut s = seed;
    while let Some((a, s2)) = next(s) {
        result.push(a);
        s = s2;
    }
    result
}

pub fn range(start: i32, end: i32) -> Vec<i32> {
    ana(start, |n| if n < end { Some((n, n + 1)) } else { None })
}

pub fn iterate<A: Clone>(init: A, f: impl Fn(&A) -> A, n: usize) -> Vec<A> {
    ana((init, n), |(a, count)| {
        if count > 0 {
            Some((a.clone(), (f(&a), count - 1)))
        } else {
            None
        }
    })
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_range() {
        assert_eq!(range(0, 5), vec![0, 1, 2, 3, 4]);
    }
    #[test]
    fn test_iterate() {
        assert_eq!(iterate(1, |x| x * 2, 4), vec![1, 2, 4, 8]);
    }
}

(* Anamorphism in OCaml *)

(* unfold: generate a list from a seed *)
let unfold f seed =
  let rec go acc s =
    match f s with
    | None        -> List.rev acc
    | Some (x, s') -> go (x :: acc) s'
  in go [] seed

let range lo hi    = unfold (fun i -> if i >= hi then None else Some (i, i+1)) lo
let fibs max_n     = unfold (fun (a,b) -> if a > max_n then None else Some (a, (b, a+b))) (0,1)
let to_digits n    = unfold (fun n -> if n=0 then None else Some (n mod 10, n/10)) n

let () =
  Printf.printf "range 1..5: %s\n" (String.concat "," (List.map string_of_int (range 1 6)));
  Printf.printf "fibs<=100: %s\n" (String.concat "," (List.map string_of_int (fibs 100)));
  Printf.printf "digits 1234: %s\n" (String.concat "," (List.map string_of_int (to_digits 1234)))

✓ Tests Rust test suite

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_range() {
        assert_eq!(range(0, 5), vec![0, 1, 2, 3, 4]);
    }
    #[test]
    fn test_iterate() {
        assert_eq!(iterate(1, |x| x * 2, 4), vec![1, 2, 4, 8]);
    }
}

Deep Comparison

Anamorphism

Unfold: build structure from seed

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