Functional Pearl: Implementing trace in JavaScript

Question

Ross Paterson: Arrows and Computation introduces the trace function (on page 11):

trace :: ((a, c) -> (b, c)) -> a -> b
trace f a = let (b, c) = f (a, c) in b

The trace function is useful for modularizing the magic feedback step in circular programs. For example, consider Richard Bird's famous repmin function which finds the minimum leaf value of a tree and creates an identical tree with every leaf value replaced by the minimum leaf value, both in a single pass by making clever use of lazy evaluation and local recursion (as provided by letrec):

data Tree = Leaf Int | Node Tree Tree deriving Show

repmin :: Tree -> Tree
repmin = trace repmin'

repmin' :: (Tree, Int) -> (Tree, Int)
-- put the minimum value m into the leaf and return the old value n as the minimum
repmin' (Leaf n, m) = (Leaf m, n)
-- copy the minimum value m into both the left and right subtrees and
-- set the minimum value m to the minimum of both the left and right subtrees
repmin' (Node l r, m) = let (l', lmin) = repmin' l m in
                        let (r', rmin) = repmin' r m in
                        (Node l' r', lmin `min` rmin)

Anyway, I was wondering how to implement the trace function in JavaScript such that we can implement repmin as follows:

function Leaf(value) {
    this.value = value;
}

function Node(left, right) {
    this.left  = left;
    this.right = right;
}

var repmin = trace(function repmin(tree, min) {
    switch (tree.constructor) {
    case Leaf:
        return [new Leaf(min), tree.value];
    case Node:
        var [left,  lmin] = repmin(tree.left,  min);
        var [right, rmin] = repmin(tree.right, min);
        return [new Node(left, right), Math.min(lmin, rmin)];
    }
});

In order to implement trace we need local recursion as provided by letrec so that we can write something like:

function trace(f) {
    return function (a) {
        var [b, c] = f(a, c);
        return b;
    };
}

I originally thought of making c a promise. However, that changes the semantics of trace. So, can you think of a way to implement trace in JavaScript without changing its semantics?

You can't really do that, but maybe you can fake it with proxy objects. — melpomene, Oct 30 '16 at 15:31

score 5 · Accepted Answer · answered Oct 30 '16 at 16:08

Actually, you only need lazy evaluation because assignments in JavaScript are like letrec. Lazy evaluation is generally implemented using thunks. Hence, you can implement trace as follows:

function trace(f) {
    return function (a) {
        var [b, c] = f(a, () => c);
        return b;
    };
}

Using this definition of trace, the repmin function can remain the same:

var repmin = trace(function repmin(tree, min) {
    switch (tree.constructor) {
    case Leaf:
        return [new Leaf(min), tree.value];
    case Node:
        var [left,  lmin] = repmin(tree.left,  min);
        var [right, rmin] = repmin(tree.right, min);
        return [new Node(left, right), Math.min(lmin, rmin)];
    }
});

However, you'd want to make your data constructors possibly lazy using getters:

function Leaf(value) {
    Object.defineProperty(this, "value", descOf(value));
}

function Node(left, right) {
    Object.defineProperty(this, "left",  descOf(left));
    Object.defineProperty(this, "right", descOf(right));
}

function descOf(value) {
    var desc = { enumerable: true };
    var prop = typeof value === "function" &&
        value.length === 0 ? "get" : "value";
    desc[prop] = value;
    return desc;
}

Putting it all together:

var tree = new Node(new Node(new Leaf(1), new Leaf(2)),
                    new Node(new Leaf(3), new Leaf(4)));

console.log("Input: ", show(tree));

console.log("Output:", show(repmin(tree)));

function show(tree) {
    switch (tree.constructor) {
    case Leaf: return "Leaf(" + tree.value + ")";
    case Node: return "Node(" + show(tree.left) + ", " + show(tree.right) + ")";
    }
}

<script>
function trace(f) {
    return function (a) {
        var [b, c] = f(a, () => c);
        return b;
    };
}

var repmin = trace(function repmin(tree, min) {
    switch (tree.constructor) {
    case Leaf:
        return [new Leaf(min), tree.value];
    case Node:
        var [left,  lmin] = repmin(tree.left,  min);
        var [right, rmin] = repmin(tree.right, min);
        return [new Node(left, right), Math.min(lmin, rmin)];
    }
});

function Leaf(value) {
    Object.defineProperty(this, "value", descOf(value));
}

function Node(left, right) {
    Object.defineProperty(this, "left",  descOf(left));
    Object.defineProperty(this, "right", descOf(right));
}

function descOf(value) {
    var desc = { enumerable: true };
    var prop = typeof value === "function" &&
        value.length === 0 ? "get" : "value";
    desc[prop] = value;
    return desc;
}
</script>

The only problem is that you won't be able to write functions like:

var sqr = trace((x, y) => [y * y, x]);

This is because the * operator is not lazy. Hence, you would have to define a lazy mul function:

var sqr = trace((x, y) => [mul(y, y), x]);

console.log(evaluate(sqr(10)));

function mul(a, b) {
    return function () {
        return evaluate(a) * evaluate(b);
    };
}

function evaluate(value) {
    return typeof value === "function" &&
        value.length === 0 ? value() : value;
}

function trace(f) {
    return function (a) {
        var [b, c] = f(a, () => c);
        return b;
    };
}

Hope that helps.

Functional Pearl: Implementing trace in JavaScript

1 Answers1