X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=315cb68a9e58726a8d8c6cb65aeca027794d82be;hp=81dc451068072c3839e6f9e4bdf2f531011fef51;hb=bf8d964cc93f6b0b44a432bca8c94b1374c05e1f;hpb=0feebbaaa58403d836d7ea6166cf709dd3faf1a8 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 81dc4510..315cb68a 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -13,7 +13,7 @@ From an engineering standpoint, we'll build a tree transformer that deals in monads. We can modify the behavior of the system by swapping one monad for another. We've already seen how adding a monad can add a layer of funtionality without disturbing the underlying system, for -instance, in the way that the reader monad allowed us to add a layer +instance, in the way that the Reader monad allowed us to add a layer of intensionality to an extensional grammar, but we have not yet seen the utility of replacing one monad with other. @@ -81,14 +81,14 @@ supplying the appropriate `int -> int` operation in place of `double`: - : int tree =ppp Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -Note that what `tree_map` 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, `tree_map` has the -behavior of a reader monad. Let's make that explicit. +behavior of a Reader monad. Let's make that explicit. 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 `tree_map` 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. @@ -113,9 +113,7 @@ 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 the present case, it will be -enough to expect that our "environment" will be some 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;; @@ -135,16 +133,44 @@ But we can do this: let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = match t with - | 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))));; + | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b)) + | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' -> + reader_bind (tree_monadize f r) (fun r' -> + reader_unit (Node (l', r'))));; 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 `'b tree reader`. 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 +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 + ------------ + +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;; @@ -161,17 +187,15 @@ result: - : 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 s -> (a, s);; - let state_bind_and_count u f = fun s -> let (a, s') = u s in f a (s' + 1);; + let state_bind u f = fun s -> let (a, s') = u s in f a s';; Gratifyingly, we can use the `tree_monadize` function without any modification whatsoever, except for replacing the (parametric) type @@ -179,16 +203,16 @@ modification whatsoever, except for replacing the (parametric) type let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = match t with - | Leaf i -> state_bind_and_count (f i) (fun i' -> state_unit (Leaf i')) - | Node (l, r) -> state_bind_and_count (tree_monadize f l) (fun x -> - state_bind_and_count (tree_monadize f r) (fun y -> - state_unit (Node (x, y))));; + | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b)) + | Node (l, r) -> state_bind (tree_monadize f l) (fun l' -> + state_bind (tree_monadize f r) (fun r' -> + state_unit (Node (l', r'))));; -Then we can count the number of nodes in the tree: +Then we can count the number of leaves in the tree: - # tree_monadize 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) . ___|___ @@ -201,17 +225,7 @@ 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 `tree_monadize` with `list` gives us @@ -224,10 +238,11 @@ 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. + @@ -240,29 +255,28 @@ of leaves? let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = match t with - | 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))));; + | Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b)) + | Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' -> + continuation_bind (tree_monadize f r) (fun r' -> + continuation_unit (Node (l', r'))));; -We use the continuation monad described above, and insert the -`continuation` type in the appropriate place in the `tree_monadize` code. -We then compute: +We use the Continuation monad described above, and insert the +`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 `'b`-wrapping Continuation monads, it will give us back a way to turn `int tree`s into corresponding `'b tree`-wrapping Continuation monads. - # tree_monadize (fun a k -> a :: (k a)) t1 (fun t -> []);; - - : int list = [2; 3; 5; 7; 11] +So for example, we compute: - + # 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? Think first about what the operation `fun a -> fun k -> a :: k a` does when you apply it to a plain `int`, and the continuation `fun _ -> []`. Then given what we've said about `tree_monadize`, what should we expect `tree_monadize (fun a -> fun k -> a :: k a` to do? -The continuation monad is amazingly flexible; we can use it to +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 `continuation_unit` as our first argument to `tree_monadize`, and then apply the result to the identity function: - # tree_monadize continuation_unit t1 (fun i -> i);; + # tree_monadize continuation_unit t1 (fun t -> t);; - : int tree = Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) @@ -270,38 +284,41 @@ That is, nothing happens. But we can begin to substitute more interesting functions for the first argument of `tree_monadize`: (* Simulating the tree reader: distributing a operation over the leaves *) - # tree_monadize (fun a k -> k (square a)) t1 (fun i -> i);; + # tree_monadize (fun a -> fun k -> k (square a)) t1 (fun t -> t);; - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) (* Simulating the int list tree list *) - # tree_monadize (fun a k -> k [a; square a]) t1 (fun i -> i);; + # tree_monadize (fun a -> fun k -> k [a; square a]) t1 (fun t -> t);; - : int list tree = Node (Node (Leaf [2; 4], Leaf [3; 9]), Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) (* Counting leaves *) - # tree_monadize (fun a k -> 1 + k a) t1 (fun i -> 0);; + # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);; - : int = 5 We could simulate the tree state example too, but it would require -generalizing the type of the continuation monad to +generalizing the type of the Continuation monad to type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;; -The binary tree monad +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 --------------------- Of course, by now you may have realized that we have discovered a new -monad, the binary tree monad: +monad, the Binary Tree monad: type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; - let tree_unit (a: 'a) = Leaf a;; + let tree_unit (a: 'a) : 'a tree = Leaf a;; let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = match u with | Leaf a -> f a - | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));; + | 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: @@ -376,3 +393,136 @@ called a [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) that is intended to represent non-deterministic computations as a tree. + +What's this have to do with tree\_mondadize? +-------------------------------------------- + +So we've defined a Tree monad: + + type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; + let tree_unit (a: 'a) : 'a tree = Leaf a;; + let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = + match u with + | Leaf a -> f a + | Node (l, r) -> Node (tree_bind l f, tree_bind r f);; + +What's this have to do with the `tree_monadize` functions we defined earlier? + + let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = + match t with + | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b)) + | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' -> + reader_bind (tree_monadize f r) (fun r' -> + reader_unit (Node (l', r'))));; + +... and so on for different monads? + +The answer is that each of those `tree_monadize` functions is adding a Tree monad *layer* to a pre-existing Reader (and so on) monad. So far, we've defined monads as single-layered things. Though in the Groenendijk, Stokhoff, and Veltmann homework, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way. In practice, one often wants to combine the abilities of several monads. Corresponding to each monad like Reader, there's a corresponding ReaderT **monad transformer**. That takes an existing monad M and adds a Reader monad layer to it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad: + + (* monadic operations for the Reader monad *) + + type 'a reader = + env -> 'a;; + let unit (a : 'a) : 'a reader = + fun e -> a;; + let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = + fun e -> (fun v -> f v e) (u e);; + +We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`, in order to factor out the parts where any Reader monad is being supplied as an argument to another function. Then if we want instead to add a Reader layer to some arbitrary other monad M, with its own M.unit and M.bind, here's how we do it: + + (* monadic operations for the ReaderT monadic transformer *) + + (* We're not giving valid OCaml code, but rather something + * that's conceptually easier to digest. + * How you really need to write this in OCaml is more circuitous... + * see http://lambda.jimpryor.net/code/tree_monadize.ml for some details. *) + + type ('a, M) readerT = + env -> 'a M;; + (* this is just an 'a M reader; but don't rely on that pattern to generalize *) + + let unit (a : 'a) : ('a, M) readerT = + fun e -> M.unit a;; + + let bind (u : ('a, M) readerT) (f : 'a -> ('b, M) readerT) : ('b, M) readerT = + fun e -> M.bind (u e) (fun v -> f v e);; + +Notice the key differences: where before we just returned `a`, now we instead return `M.unit a`. Where before we just supplied value `u e` of type `'a reader` as an argument to a function, now we instead `M.bind` the `'a reader` to that function. Notice also the differences in the types. + +What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad: + + type 'a identity = 'a;; + let unit (a : 'a) : 'a = a;; + let bind (u : 'a) (f : 'a -> 'b) : 'b = f u;; + +and you used the ReaderT transformer to add a Reader monad layer to the Identity monad. What do you suppose you would get? + +The relations between the State monad and the StateT monadic transformer are parallel: + + (* monadic operations for the State monad *) + + type 'a state = + store -> ('a * store);; + + let unit (a : 'a) : 'a state = + fun s -> (a, s);; + + let bind (u : 'a state) (f : 'a -> 'b state) : 'b state = + fun s -> (fun (a, s') -> f a s') (u s);; + +We've used `(fun (a, s') -> f a s') (u s)` instead of the more familiar `let (a, s') = u s in f a s'` in order to factor out the part where a value of type `'a state` is supplied as an argument to a function. Now StateT will be: + + (* monadic operations for the StateT monadic transformer *) + + type ('a, M) stateT = + store -> ('a * store) M;; + (* notice this is not an 'a M state *) + + let unit (a : 'a) : ('a, M) stateT = + fun s -> M.unit (a, s);; + + let bind (u : ('a, M) stateT) (f : 'a -> ('b, M) stateT) : ('b, M) stateT = + fun s -> M.bind (u s) (fun (a, s') -> f a s');; + +Do you see the pattern? Where ordinarily we'd return an `'a` value, now we instead return an `'a M` value. Where ordinarily we'd supply a `'a state` value as an argument to a function, now we instead `M.bind` it to that function. + +Okay, now let's do the same thing for our Tree monad. + + (* monadic operations for the Tree monad *) + + type 'a tree = + Leaf of 'a | Node of ('a tree) * ('a tree);; + + let unit (a: 'a) : 'a tree = + Leaf a;; + + let rec bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = + match u with + | Leaf a -> (fun b -> b) (f a) (* see below *) + | Node (l, r) -> (fun l' r' -> Node (l', r')) (bind l f) (bind r f);; + + (* monadic operations for the TreeT monadic transformer *) + + type ('a, M) treeT = + 'a tree M;; + + let unit (a: 'a) : ('a, M) tree = + M.unit (Leaf a);; + + let rec bind (u : ('a, M) tree) (f : 'a -> ('b, M) tree) : ('b, M) tree = + match u with + | Leaf a -> M.bind (f a) (fun b -> M.unit (Leaf b)) + | Node (l, r) -> M.bind (bind l f) (fun l' -> + M.bind (bind r f) (fun r' -> + M.unit (Node (l', r'));; + +Compare this definition of `bind` for the TreeT monadic transformer to our earlier definition of `tree_monadize`, specialized for the Reader monad: + + let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = + match t with + | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b)) + | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' -> + reader_bind (tree_monadize f r) (fun r' -> + reader_unit (Node (l', r'))));; + +