Functional Pearl：在 JavaScript 中实施跟踪

Question

Ross Paterson：Arrows and Computation introduces the trace 函数（第 11 页）：

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

trace 函数对于模块化 circular programs. For example, consider Richard Bird's famous repmin 函数中的魔术反馈步骤非常有用，该函数找到树的最小叶子值和创建一个相同的每个叶子值都被最小叶子值替换的树，通过巧妙地使用惰性求值和局部递归（由 letrec 提供）在单次传递中完成：

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)

无论如何，我想知道如何在JavaScript中实现trace功能，这样我们就可以实现repmin如下：

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)];
    }
});

为了实现trace，我们需要letrec提供的本地递归，这样我们就可以编写如下内容：

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

本来想c许诺的。然而，这改变了 trace 的语义。那么，你能想出一种在不改变语义的情况下在JavaScript中实现trace的方法吗？

Answer 1

实际上，您只需要惰性求值，因为 JavaScript 中的赋值类似于 letrec。惰性评估通常使用 thunks 来实现。因此，您可以按如下方式实现 trace：

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

使用 trace 的这个定义，repmin 函数可以保持不变：

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)];
    }
});

但是，您可能希望使用 getter 使数据构造函数变得惰性：

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;
}

综合起来：

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>

唯一的问题是您将无法编写如下函数：

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

这是因为 * 运算符并不懒惰。因此，您必须定义惰性 mul 函数：

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;
    };
}

希望对您有所帮助。

Functional Pearl：在 JavaScript 中实施跟踪

Functional Pearl: Implementing trace in JavaScript

javascript

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functional-programming

lazy-evaluation

tying-the-knot