Implementing Scott Encoding in TypeScript

Scott encoding, also known as tagless final encoding, is a powerful technique for representing data structures in a way that makes it easier to define operations over them without explicit recursion or pattern matching on the structure itself. This challenge will guide you through implementing a basic Scott encoding for a simple algebraic data type (ADT) in TypeScript.

Problem Description

Your task is to create a TypeScript implementation of Scott encoding for a Tree data structure. A Tree can either be a Leaf containing a value or a Node containing a left subtree and a right subtree.

You need to define the Tree type and a foldTree function that acts as the Scott encoder. This foldTree function will take a Tree and a set of "interpreters" (functions corresponding to each constructor of the Tree) and produce a result.

Key Requirements:

1. Define the Tree type: This type should represent a binary tree with values at the leaves.
2. Implement foldTree: This function will be the core of your Scott encoding. It should accept:
   • The Tree to be encoded.
   • An object containing two functions:
     • leaf: A function to handle Leaf constructors. It should take the value of the leaf and return some result A.
     • node: A function to handle Node constructors. It should take the results of folding the left and right subtrees (both of type A) and return a new result of type A.
3. Type Safety: Ensure all types are correctly defined and enforced by TypeScript.

Expected Behavior:

The foldTree function should abstract away the recursive structure of the Tree. Instead of recursively calling foldTree on subtrees within the foldTree implementation itself, the leaf and node functions provided by the caller will be responsible for defining how to process the structure.

Edge Cases:

• Empty trees are not possible with this definition.
• Consider trees with only one node.

Examples

Example 1: Calculating the sum of all values in a Tree.

// Define Tree type
interface Leaf<T> {
  type: 'leaf';
  value: T;
}

interface Node<T> {
  type: 'node';
  left: Tree<T>;
  right: Tree<T>;
}

type Tree<T> = Leaf<T> | Node<T>;

// Helper functions to create trees (for demonstration)
function leaf<T>(value: T): Leaf<T> {
  return { type: 'leaf', value };
}

function node<T>(left: Tree<T>, right: Tree<T>): Node<T> {
  return { type: 'node', left, right };
}

// The foldTree function (your implementation)
// ... implement foldTree here ...

// Interpreter for sum
const sumInterpreter = {
  leaf: <T>(value: T) => value, // If it's a leaf, the sum is just the value
  node: <T>(leftSum: number, rightSum: number) => leftSum + rightSum, // If it's a node, sum the results from children
};

// Example Tree
const myTree: Tree<number> = node(
  leaf(1),
  node(
    leaf(2),
    leaf(3)
  )
);

// Using foldTree to calculate sum
const totalSum = foldTree(myTree, sumInterpreter);
console.log(totalSum); // Expected Output: 6

Example 2: Flattening a Tree into an array of its leaf values.

// Using the same Tree type and helper functions as Example 1

// Interpreter for flattening
const flattenInterpreter = {
  leaf: <T>(value: T) => [value], // If it's a leaf, return an array with that value
  node: <T>(leftArray: T[], rightArray: T[]) => [...leftArray, ...rightArray], // If it's a node, concatenate the arrays from children
};

// Example Tree
const anotherTree: Tree<string> = node(
  leaf("a"),
  node(
    leaf("b"),
    leaf("c")
  )
);

// Using foldTree to flatten
const flattenedArray = foldTree(anotherTree, flattenInterpreter);
console.log(flattenedArray); // Expected Output: ["a", "b", "c"]

Example 3: Counting the number of leaves in a Tree.

// Using the same Tree type and helper functions as Example 1

// Interpreter for counting leaves
const countLeavesInterpreter = {
  leaf: <T>(value: T) => 1, // A leaf counts as 1
  node: <T>(leftCount: number, rightCount: number) => leftCount + rightCount, // A node's count is the sum of its children's counts
};

// Example Tree
const complexTree: Tree<number> = node(
  node(leaf(1), leaf(2)),
  node(
    leaf(3),
    node(leaf(4), leaf(5))
  )
);

// Using foldTree to count leaves
const leafCount = foldTree(complexTree, countLeavesInterpreter);
console.log(leafCount); // Expected Output: 5

Constraints

• The Tree type must be defined using discriminated unions in TypeScript.
• The foldTree function must have a generic type parameter A representing the return type of the interpreters.
• The leaf interpreter function should have the signature (value: T) => A.
• The node interpreter function should have the signature (leftResult: A, rightResult: A) => A.
• Performance is not a primary concern for this challenge, but the implementation should be idiomatic TypeScript.

Notes

• Think about how foldTree can be implemented without explicit recursion within foldTree itself. The recursion is implicitly handled by the way the leaf and node interpreters are structured to call foldTree on their subtrees.
• This approach allows you to define new operations on the Tree without modifying the Tree type definition or the foldTree function itself. You just provide a new set of interpreters.
• The concept is similar to how functional programming languages handle algebraic data types, allowing for elegant and extensible code.