X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=2ec15d6a6c9a8ee37ec39304a5f4d1ee75befffc;hp=445722bebc24e1334fb6fddc506108e61894ea44;hb=9fe62083953213cce34fc4458e36666902c5ee4b;hpb=b1d420acee7b904af41aabe6db71e872baf251f5 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 445722be..2ec15d6a 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -3,11 +3,13 @@ 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. +This topic develops an idea based on a suggestion of Ken Shan's. +We'll build a series of functions that operate on trees, doing various +things, including updating leaves with a Reader monad, counting nodes +with a State monad, replacing leaves with a List monad, and converting +a tree into a list of leaves with a Continuation monad. It will turn +out that the continuation monad can simulate the behavior of each of +the other monads. From an engineering standpoint, we'll build a tree transformer that deals in monads. We can modify the behavior of the system by swapping @@ -20,7 +22,7 @@ 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) + 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?] @@ -30,7 +32,7 @@ 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))) + Leaf 11))) . ___|___ | | @@ -71,14 +73,15 @@ structure of the tree unchanged. For instance: 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`: +code. However, because we've made what to do to each leaf 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 + - : int tree = 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 @@ -106,14 +109,25 @@ updated tree. 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. +tree`) into a reader monadic 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`. + +[Application note: this kind of reader object could provide a model +for Kaplan's characters. It turns an ordinary tree into one that +expects contextual information (here, the `λ f`) that can be +used to compute the content of indexicals embedded arbitrarily deeply +in the tree.] + +With our previous applications of the Reader monad, we always knew +which kind of environment to expect: either an assignment function, as +in the original calculator simulation; a world, as in the +intensionality monad; an individual, as in 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;; @@ -218,14 +232,52 @@ Then we can count the number of leaves in the tree: ___|___ | | . . - _|__ _|__ + _|__ _|__ , 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. +Note that the value returned is a pair consisting of a tree and an +integer, 5, which represents the count of the leaves in the tree. + +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. + +We can use the state monad to replace leaves with a number +corresponding to that leave's ordinal position. When we do so, we +reveal the order in which the monadic tree forces evaluation: + + # tree_monadize (fun a -> fun s -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5) + +The key thing to notice is that instead of copying `a` into the +monadic box, we throw away the `a` and put a copy of the state in +instead. + +Reversing the order requires reversing the order of the state_bind +operations. It's not obvious that this will type correctly, so think +it through: + + let rec tree_monadize_rev (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 r) (fun r' -> + state_bind (tree_monadize f l) (fun l' -> + state_unit (Node (l', r'))));; + + # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5) + +We will need below to depend on controlling the order in which nodes +are visited when we use the continuation monad to solve the +same-fringe problem. One more revealing example before getting down to business: replacing `state` everywhere in `tree_monadize` with `list` gives us @@ -240,11 +292,11 @@ 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. - - - +[Why is the argument to `tree_monadize` `int -> int list list` instead +of `int -> int list`? Well, as usual, the List monad bind operation +will erase the outer list box, so if we want to replace the leaves +with lists, we have to nest the replacement lists inside a disposable +box.] Now for the main point. What if we wanted to convert a tree to a list of leaves? @@ -306,12 +358,37 @@ generalizing the type of the Continuation monad to 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). +Using continuations to solve the same fringe problem +---------------------------------------------------- + +We've seen two solutions to the same fringe problem so far. +The simplest is to map each tree to a list of its leaves, then compare +the lists. But if the fringes differ in an early position, we've +wasted our time visiting the rest of the tree. + +The second solution was to use tree zippers and mutable state to +simulate coroutines. We would unzip the first tree until we found the +next leaf, then store the zipper structure in the mutable variable +while we turned our attention to the other tree. Because we stop as +soon as we find the first mismatched leaf, this solution does not have +the flaw just mentioned of the solution that maps both trees to a list +of leaves before beginning comparison. + +Since zippers are just continuations reified, we expect that the +solution in terms of zippers can be reworked using continuations, and +this is indeed the case. To make this work in the most convenient +way, we need to use the fully general type for continuations just mentioned. + +tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);; + + 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. Just as mere lists are in fact a monad, +so are trees. Here is the type constructor, unit, and bind: type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; let tree_unit (a: 'a) : 'a tree = Leaf a;; @@ -417,112 +494,5 @@ What's this have to do with the `tree_monadize` functions we defined earlier? ... 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 -> Leaf b) (f a) - | 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'))));; - +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. We discuss that further here: [[Monad Transformers]].