X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=61d296405e87a62bc2cfa2e071ba8f4353fdd756;hp=de8fc5fa6c7c4e6c0c9e4ec3ce3571c076cf9375;hb=9efbe94f74c2ea61522fcdb3e3d012fde6034fcd;hpb=26319cf2ffc188af7fc324143881d45fd7c322c8 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index de8fc5fa..61d29640 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -44,18 +44,18 @@ We'll be using trees where the nodes are integers, e.g., Our first task will be to replace each leaf with its double: - let rec treemap (newleaf : 'a -> 'b) (t : 'a tree) : 'b tree = + let rec tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree = match t with - | Leaf x -> Leaf (newleaf x) - | Node (l, r) -> Node (treemap newleaf l, - treemap newleaf r);; + | Leaf i -> Leaf (leaf_modifier i) + | Node (l, r) -> Node (tree_map leaf_modifier l, + tree_map leaf_modifier r);; -`treemap` takes a function that transforms old leaves into new leaves, +`tree_map` takes a function that transforms old leaves into new leaves, and maps that function over all the leaves in the tree, leaving the structure of the tree unchanged. For instance: let double i = i + i;; - treemap double t1;; + tree_map double t1;; - : int tree = Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) @@ -70,28 +70,29 @@ structure of the tree unchanged. For instance: | | 14 22 -We could have built the doubling operation right into the `treemap` -code. However, because what to do to each leaf is a parameter, we can +We could have built the doubling operation right into the `tree_map` +code. However, because we've left what to do to each leaf as a parameter, we can decide to do something else to the leaves without needing to rewrite -`treemap`. For instance, we can easily square each leaf instead by +`tree_map`. For instance, we can easily square each leaf instead by supplying the appropriate `int -> int` operation in place of `double`: - let square x = x * x;; - treemap square t1;; + let square i = i * i;; + tree_map square t1;; - : int tree =ppp Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -Note that what `treemap` does is take some global, contextual +Note that what `tree_map` does is take some unchanging contextual information---what to do to each leaf---and supplies that information -to each subpart of the computation. In other words, `treemap` has the +to each subpart of the computation. In other words, `tree_map` has the behavior of a reader monad. Let's make that explicit. -In general, we're on a journey of making our treemap function more and +In general, we're on a journey of making our `tree_map` function more and more flexible. So the next step---combining the tree transformer with -a reader monad---is to have the treemap function return a (monadized) +a reader monad---is to have the `tree_map` function return a (monadized) tree that is ready to accept any `int -> int` function and produce the updated tree. +\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11)))) \f . _____|____ @@ -107,82 +108,111 @@ updated tree. That is, we want to transform the ordinary tree `t1` (of type `int tree`) into a reader object of type `(int -> int) -> int tree`: something that, when you apply it to an `int -> int` function `f` returns an `int -tree` in which each leaf `x` has been replaced with `f x`. +tree` in which each leaf `i` has been replaced with `f i`. With previous readers, we always knew which kind of environment to expect: either an assignment function (the original calculator simulation), a world (the intensionality monad), an integer (the -Jacobson-inspired link monad), etc. In this situation, it will be -enough for now to expect that our reader will expect a function of -type `int -> int`. +Jacobson-inspired link monad), etc. In the present case, we expect that our "environment" will be some function of type `int -> int`. "Looking up" some `int` in the environment will return us the `int` that comes out the other side of that function. type 'a reader = (int -> int) -> 'a;; (* mnemonic: e for environment *) let reader_unit (a : 'a) : 'a reader = fun _ -> a;; let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> f (u e) e;; -It's easy to figure out how to turn an `int` into an `int reader`: +It would be a simple matter to turn an *integer* into an `int reader`: - let int2int_reader : 'a -> 'b reader = fun (a : 'a) -> fun (op : 'a -> 'b) -> op a;; - int2int_reader 2 (fun i -> i + i);; + let int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;; + int_readerize 2 (fun i -> i + i);; - : int = 4 -But what do we do when the integers are scattered over the leaves of a -tree? A binary tree is not the kind of thing that we can apply a +But how do we do the analagous transformation when our `int`s are scattered over the leaves of a tree? How do we turn an `int tree` into a reader? +A tree is not the kind of thing that we can apply a function of type `int -> int` to. - let rec treemonadizer (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = +But we can do this: + + let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = match t with - | Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x')) - | Node (l, r) -> reader_bind (treemonadizer f l) (fun x -> - reader_bind (treemonadizer f r) (fun y -> + | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i')) + | Node (l, r) -> reader_bind (tree_monadize f l) (fun x -> + reader_bind (tree_monadize f r) (fun y -> reader_unit (Node (x, y))));; This function says: give me a function `f` that knows how to turn -something of type `'a` into an `'b reader`, and I'll show you how to -turn an `'a tree` into an `'a tree reader`. In more fanciful terms, -the `treemonadizer` function builds plumbing that connects all of the -leaves of a tree into one connected monadic network; it threads the -`'b reader` monad through the leaves. +something of type `'a` into an `'b reader`---this is a function of the same type that you could bind an `'a reader` to---and I'll show you how to +turn an `'a tree` into an `'b tree reader`. That is, if you show me how to do this: + + ------------ + 1 ---> | 1 | + ------------ + +then I'll give you back the ability to do this: + + ____________ + . | . | + __|___ ---> | __|___ | + | | | | | | + 1 2 | 1 2 | + ------------ + +And how will that boxed tree behave? Whatever actions you perform on it will be transmitted down to corresponding operations on its leaves. For instance, our `int reader` expects an `int -> int` environment. If supplying environment `e` to our `int reader` doubles the contained `int`: + + ------------ + 1 ---> | 1 | applied to e ~~> 2 + ------------ - # treemonadizer int2int_reader t1 (fun i -> i + i);; +Then we can expect that supplying it to our `int tree reader` will double all the leaves: + + ____________ + . | . | . + __|___ ---> | __|___ | applied to e ~~> __|___ + | | | | | | | | + 1 2 | 1 2 | 2 4 + ------------ + +In more fanciful terms, the `tree_monadize` function builds plumbing that connects all of the leaves of a tree into one connected monadic network; it threads the +`'b reader` monad through the original tree's leaves. + + # tree_monadize int_readerize t1 double;; - : int tree = Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) Here, our environment is the doubling function (`fun i -> i + i`). If -we apply the very same `int tree reader` (namely, `treemonadizer -int2int_reader t1`) to a different `int -> int` function---say, the +we apply the very same `int tree reader` (namely, `tree_monadize +int_readerize t1`) to a different `int -> int` function---say, the squaring function, `fun i -> i * i`---we get an entirely different result: - # treemonadizer int2int_reader t1 (fun i -> i * i);; + # tree_monadize int_readerize t1 square;; - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -Now that we have a tree transformer that accepts a reader monad as a +Now that we have a tree transformer that accepts a *reader* monad as a parameter, we can see what it would take to swap in a different monad. -For instance, we can use a state monad to count the number of nodes in + +For instance, we can use a state monad to count the number of leaves in the tree. type 'a state = int -> 'a * int;; - let state_unit a = fun i -> (a, i);; - let state_bind u f = fun i -> let (a, i') = u i in f a (i' + 1);; + let state_unit a = fun s -> (a, s);; + let state_bind u f = fun s -> let (a, s') = u s in f a s';; -Gratifyingly, we can use the `treemonadizer` function without any +Gratifyingly, we can use the `tree_monadize` function without any modification whatsoever, except for replacing the (parametric) type `'b reader` with `'b state`, and substituting in the appropriate unit and bind: - let rec treemonadizer (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = + let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = match t with - | Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x')) - | Node (l, r) -> state_bind (treemonadizer f l) (fun x -> - state_bind (treemonadizer f r) (fun y -> + | Leaf i -> state_bind (f i) (fun i' -> state_unit (Leaf i')) + | Node (l, r) -> state_bind (tree_monadize f l) (fun x -> + state_bind (tree_monadize f r) (fun y -> state_unit (Node (x, y))));; -Then we can count the number of nodes in the tree: +Then we can count the number of leaves in the tree: - # treemonadizer state_unit t1 0;; + # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;; - : int tree * int = - (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13) + (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5) . ___|___ @@ -195,22 +225,12 @@ Then we can count the number of nodes in the tree: | | 7 11 -Notice that we've counted each internal node twice---it's a good -exercise to adjust the code to count each node once. - - - +Why does this work? Because the operation `fun a -> fun s -> (a, s+1)` takes an `int` and wraps it in an `int state` monadic box that increments the state. When we give that same operations to our `tree_monadize` function, it then wraps an `int tree` in a box, one that does the same state-incrementing for each of its leaves. One more revealing example before getting down to business: replacing -`state` everywhere in `treemonadizer` with `list` gives us +`state` everywhere in `tree_monadize` with `list` gives us - # treemonadizer (fun x -> [ [x; square x] ]) t1;; + # tree_monadize (fun i -> [ [i; square i] ]) t1;; - : int list tree list = [Node (Node (Leaf [2; 4], Leaf [3; 9]), @@ -218,64 +238,74 @@ One more revealing example before getting down to business: replacing Unlike the previous cases, instead of turning a tree into a function from some input to a result, this transformer replaces each `int` with -a list of `int`'s. +a list of `int`'s. We might also have done this with a Reader Monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List Monad an operation like `fun -> [ i; [2*i; 3*i] ]`. Use small trees for your experiment. + + + + Now for the main point. What if we wanted to convert a tree to a list of leaves? type ('a, 'r) continuation = ('a -> 'r) -> 'r;; - let continuation_unit x c = c x;; - let continuation_bind u f c = u (fun a -> f a c);; + let continuation_unit a = fun k -> k a;; + let continuation_bind u f = fun k -> u (fun a -> f a k);; - let rec treemonadizer (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = + let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = match t with - | Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x')) - | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x -> - continuation_bind (treemonadizer f r) (fun y -> + | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i')) + | Node (l, r) -> continuation_bind (tree_monadize f l) (fun x -> + continuation_bind (tree_monadize f r) (fun y -> continuation_unit (Node (x, y))));; We use the continuation monad described above, and insert the -`continuation` type in the appropriate place in the `treemonadizer` code. -We then compute: +`continuation` type in the appropriate place in the `tree_monadize` code. Then if we give the `tree_monadize` function an operation that converts `int`s into continuations expecting `'b` arguments, it will give us back a way to turn `int tree`s into continuations that expect `'b tree` arguments. The effect of giving the continuation such an argument will be to distribute across the `'b tree`'s leaves effects that parallel the effects that the `'b`-expecting continuations would have on their `'b`s. + +So for example, we compute: - # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);; + # tree_monadize (fun a -> fun k -> a :: (k a)) t1 (fun t -> []);; - : int list = [2; 3; 5; 7; 11] -We have found a way of collapsing a tree into a list of its leaves. +We have found a way of collapsing a tree into a list of its leaves. Can you trace how this is working? The continuation monad is amazingly flexible; we can use it to simulate some of the computations performed above. To see how, first -note that an interestingly uninteresting thing happens if we use the -continuation unit as our first argument to `treemonadizer`, and then +note that an interestingly uninteresting thing happens if we use +`continuation_unit` as our first argument to `tree_monadize`, and then apply the result to the identity function: - # treemonadizer continuation_unit t1 (fun x -> x);; + # tree_monadize continuation_unit t1 (fun i -> i);; - : int tree = Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) That is, nothing happens. But we can begin to substitute more -interesting functions for the first argument of `treemonadizer`: +interesting functions for the first argument of `tree_monadize`: (* Simulating the tree reader: distributing a operation over the leaves *) - # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);; + # tree_monadize (fun a -> fun k -> k (square a)) t1 (fun i -> i);; - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) (* Simulating the int list tree list *) - # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);; + # tree_monadize (fun a -> fun k -> k [a; square a]) t1 (fun i -> i);; - : int list tree = Node (Node (Leaf [2; 4], Leaf [3; 9]), Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) (* Counting leaves *) - # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; + # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun i -> 0);; - : int = 5 We could simulate the tree state example too, but it would require generalizing the type of the continuation monad to - type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;; + type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;; + +If you want to see how to parameterize the definition of the `tree_monadize` function, so that you don't have to keep rewriting it for each new monad, see [this code](/code/tree_monadize.ml). + The binary tree monad --------------------- @@ -284,37 +314,37 @@ Of course, by now you may have realized that we have discovered a new monad, the binary tree monad: type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; - let tree_unit (x: 'a) = Leaf x;; + let tree_unit (a: 'a) = Leaf a;; let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = match u with - | Leaf x -> f x + | Leaf a -> f a | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));; For once, let's check the Monad laws. The left identity law is easy: - Left identity: bind (unit a) f = bind (Leaf a) f = fa + Left identity: bind (unit a) f = bind (Leaf a) f = f a To check the other two laws, we need to make the following observation: it is easy to prove based on `tree_bind` by a simple induction on the structure of the first argument that the tree resulting from `bind u f` is a tree with the same strucure as `u`, -except that each leaf `a` has been replaced with `fa`: +except that each leaf `a` has been replaced with `f a`: -\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) +\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5))) . . __|__ __|__ | | | | - a1 . fa1 . + a1 . f a1 . _|__ __|__ | | | | - . a5 . fa5 + . a5 . f a5 bind _|__ f = __|__ | | | | - . a4 . fa4 + . a4 . f a4 __|__ __|___ | | | | - a2 a3 fa2 fa3 + a2 a3 f a2 f a3 Given this equivalence, the right identity law @@ -326,31 +356,31 @@ falls out once we realize that As for the associative law, - Associativity: bind (bind u f) g = bind u (\a. bind (fa) g) + Associativity: bind (bind u f) g = bind u (\a. bind (f a) g) we'll give an example that will show how an inductive proof would proceed. Let `f a = Node (Leaf a, Leaf a)`. Then -\tree (. (. (. (. (a1)(a2))))) -\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) )) +\tree (. (. (. (. (a1) (a2))))) +\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))))) . ____|____ . . | | bind __|__ f = __|_ = . . | | | | __|__ __|__ - a1 a2 fa1 fa2 | | | | + a1 a2 f a1 f a2 | | | | a1 a1 a1 a1 Now when we bind this tree to `g`, we get - . - ____|____ - | | - . . - __|__ __|__ - | | | | - ga1 ga1 ga1 ga1 + . + _____|______ + | | + . . + __|__ __|__ + | | | | + g a1 g a1 g a1 g a1 At this point, it should be easy to convince yourself that using the recipe on the right hand side of the associative law will