X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=0d9e33df3425822ec12e1b4b8aab11fa2594475c;hp=315cb68a9e58726a8d8c6cb65aeca027794d82be;hb=HEAD;hpb=bf8d964cc93f6b0b44a432bca8c94b1374c05e1f diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn deleted file mode 100644 index 315cb68a..00000000 --- a/manipulating_trees_with_monads.mdwn +++ /dev/null @@ -1,528 +0,0 @@ -[[!toc]] - -Manipulating trees with monads ------------------------------- - -This topic develops an idea based on a detailed suggestion of Ken -Shan's. We'll build a series of functions that operate on trees, -doing various things, including replacing leaves, counting nodes, and -converting a tree to a list of leaves. The end result will be an -application for continuations. - -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 -of intensionality to an extensional grammar, but we have not yet seen -the utility of replacing one monad with other. - -First, we'll be needing a lot of trees for the remainder of the -course. Here again is a type constructor for leaf-labeled, binary trees: - - type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree) - -[How would you adjust the type constructor to allow for labels on the -internal nodes?] - -We'll be using trees where the nodes are integers, e.g., - - - let t1 = Node (Node (Leaf 2, Leaf 3), - Node (Leaf 5, Node (Leaf 7, - Leaf 11))) - . - ___|___ - | | - . . - _|_ _|__ - | | | | - 2 3 5 . - _|__ - | | - 7 11 - -Our first task will be to replace each leaf with its double: - - let rec tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree = - match t with - | Leaf i -> Leaf (leaf_modifier i) - | Node (l, r) -> Node (tree_map leaf_modifier l, - tree_map leaf_modifier r);; - -`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;; - tree_map double t1;; - - : int tree = - Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) - - . - ___|____ - | | - . . - _|__ __|__ - | | | | - 4 6 10 . - __|___ - | | - 14 22 - -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 -`tree_map`. For instance, we can easily square each leaf instead by -supplying the appropriate `int -> int` operation in place of `double`: - - 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 `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. - -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) -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 . - _____|____ - | | - . . - __|___ __|___ - | | | | - f 2 f 3 f 5 . - __|___ - | | - f 7 f 11 - -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 `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, 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 would be a simple matter to turn an *integer* into an `int reader`: - - 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 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. - -But we can do this: - - 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'))));; - -This function says: give me a function `f` that knows how to turn -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;; - - : 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, `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: - - # 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 -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 leaves in -the tree. - - type 'a state = int -> 'a * int;; - 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 `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 tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = - match t with - | 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 leaves in the tree: - - # 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))), 5) - - . - ___|___ - | | - . . - _|__ _|__ - | | | | - 2 3 5 . - _|__ - | | - 7 11 - -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 - - # tree_monadize (fun i -> [ [i; square i] ]) t1;; - - : int list tree list = - [Node - (Node (Leaf [2; 4], Leaf [3; 9]), - Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))] - -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. 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 a = fun k -> k a;; - let continuation_bind u f = fun k -> u (fun a -> f a k);; - - let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = - match t with - | 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. 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. - -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. 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 -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 t -> t);; - - : 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 `tree_monadize`: - - (* Simulating the tree reader: distributing a operation over the leaves *) - # 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 -> 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 -> 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 - - 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 ---------------------- - -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 (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);; - -For once, let's check the Monad laws. The left identity law is easy: - - 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 `f a`: - -\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5))) - - . . - __|__ __|__ - | | | | - a1 . f a1 . - _|__ __|__ - | | | | - . a5 . f a5 - bind _|__ f = __|__ - | | | | - . a4 . f a4 - __|__ __|___ - | | | | - a2 a3 f a2 f a3 - -Given this equivalence, the right identity law - - Right identity: bind u unit = u - -falls out once we realize that - - bind (Leaf a) unit = unit a = Leaf a - -As for the associative law, - - 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))))) - - . - ____|____ - . . | | - bind __|__ f = __|_ = . . - | | | | __|__ __|__ - a1 a2 f a1 f a2 | | | | - a1 a1 a1 a1 - -Now when we bind this tree to `g`, we get - - . - _____|______ - | | - . . - __|__ __|__ - | | | | - 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 -built the exact same final tree. - -So binary trees are a monad. - -Haskell combines this monad with the Option monad to provide a monad -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'))));; - -