From 008afaa826fadb5feab482b89d798903004a7b63 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Thu, 2 Dec 2010 10:36:53 -0500 Subject: [PATCH] tree_monadize is TreeT.bind Signed-off-by: Jim Pryor --- manipulating_trees_with_monads.mdwn | 137 +++++++++++++++++++++++++++++++++++- 1 file changed, 135 insertions(+), 2 deletions(-) diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index e3ed6f3a..16097633 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -314,11 +314,11 @@ 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) = 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: @@ -393,3 +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 application, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way.) But 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)` 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 circuitious... + * 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 that doesn't 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 return `M.unit a`. Where before we just supplied a 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 -> 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.unit (f a) + | 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'))));; + + -- 2.11.0