+[[!toc]]
-So far, we've defined monads as single-layered things. Though in the Groenendijk, Stokhof, and Veltman 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 wraps a Reader monad box around it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad:
+##Multi-layered monadic boxes##
+
+So far, we've defined monads as single-layered boxes. Though in the Groenendijk, Stokhof, and Veltman 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 wraps Readerish monad packaging around 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 *)
e) (u e)`. We did that so as to factor out the parts where any Reader monad is
being supplied as an argument to another function. That will help make some patterns that are coming up more salient.
-That was the plain Reader monad. Now if we want instead to wrap some other monad M inside Reader packaging. How could we do it?
+That was the plain Reader monad. Now if we want instead to wrap some other monad M inside Readerish packaging. How could we do it?
-Well, one way to proceed would be to just let values of the other monad M be the `'a` in your `'a reader`. Then you could apply `reader_bind` to get at the wrapped `'a M`, and then apply `M.bind` to get at *its* wrapped `'a`. This sometimes works. It's what we did in the hints to GSV assignment, where we said we "combined State and Set in an ad hoc way."
+Well, one way to proceed would be to just let values of the other monad M be the `'a` in your `'a reader`. Then you could apply `reader_bind` to get at the wrapped `'a M`, and then apply `M.bind` to get at *its* wrapped `'a`. This sometimes works. It's what we did in the hints to GSV assignment, where as we said, we "combined State and Set in an ad hoc way."
-But there are two problems: (1) It's cumbersome having to work with *both* `reader_bind` and `M.bind`. It's be nice to figure out some systematic way to connect the plumbing of the different monadic layers, though, so that we could have a *single* `bind` that took our `'a M_inside_Reader`, and sequenced it with a single `'a -> 'b M_inside_Reader` function. Similarly for `unit`. This is what the ReaderT monad transformer will let us do.
+But there are two problems: (1) It's cumbersome having to work with *both* `reader_bind` and `M.bind`. It'd be nice to figure out some systematic way to connect the plumbing of the different monadic layers, so that we could have a *single* `bind` that took our `'a M_inside_Reader`, and sequenced it with a single `'a -> 'b M_inside_Reader` function. Similarly for `unit`. This is what the ReaderT monad transformer will let us do.
-(2) For some combinations of monads, the best way to implement an X monadic wrapper around an inner M monad won't be equivalent to either an `('a m) x` or an `('a x) m`. It will be a tighether intermingling of the two. So some natural activities will remain out of reach until we equip ourselves to go beyond `('a m) x`s and so on.
+(2) For some combinations of monads, the best way to implement a Tish monadic wrapper around an inner M monad won't be equivalent to either an `('a m) t` or an `('a t) m`. It will be a tighter intermingling of the two. So some natural activities will remain out of reach until we equip ourselves to go beyond `('a m) t`s and so on.
What we want in general are monadic transformers. For example, a ReaderT transformer will be parameterized not just on the type of its innermost contents `'a`, but also on the monadic box `M` that wraps `'a`. It will make use of monad `M`'s existing operations `M.unit` and `M.bind`, together with the Reader patterns for unit and bind, to define a new monad ReaderT(M) that fuses the behavior of Reader and M.
-Here's how we do it:
+To be clear: ReaderT isn't itself a monad. It's a kind of Readerish packaging (wrapping paper) that converts one monadic box M into another monadic box ReaderT(M).
+
+Here's how it's implemented:
(* monadic operations for the ReaderT monadic transformer *)
let bind (u : 'a readerT(M)) (f : 'a -> 'b readerT(M)) : 'b readerT(M) =
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`
+Notice the key differences: where before `unit` was implemented by a function that just returned `a`, now we
+instead return `M.unit a`. Where before `bind` just supplied value `u e`
of type `'a reader` as an argument to a function, now we instead
`M.bind` the corresponding value to the function. Notice also the differences
in the types.
let unit (a : 'a) : 'a = a;;
let bind (u : 'a) (f : 'a -> 'b) : 'b = f u;;
-and you used the ReaderT transformer to wrap the Identity monad inside Reader packaging. What do you suppose you would get?
+and you used the ReaderT transformer to wrap the Identity monad inside Readerish packaging. What do you suppose you would get?
The relations between the State monad and the StateT monadic transformer are parallel:
fun s -> M.bind (u s) (fun (a, s') -> f a s');;
-Do you see the pattern? Where before we'd return an `'a * store` value, now we instead use `M.unit` to return an `('a * store) M` value. Where before we'd supply a `'a state` value `(u s)` as an argument to a function, now we instead `M.bind` it to that function.
+Do you see the pattern? Where before `unit` was implemented by a function that returned an `'a * store` value, now we instead use `M.unit` to return an `('a * store) M` value. Where before `bind` supplied an `'a state` value `(u s)` as an argument to a function, now we instead `M.bind` it to that function.
-Once again, what do you think you'd get if you wrapped a StateT monadic box around an Identity monad?
+Once again, what do you think you'd get if you wrapped StateT monadic packaging around an Identity monad?
-Notice that an `'a stateT(M)` is not an `'a M state`. The pattern by which the types are transformed from an Blah monad to a BlahT monad transformer is:
- 't0 ---> 't0 M
- 't1 -> 't0 ---> 't1 -> 't0 M
- ('t1 -> 't0) -> 't0 ---> ('t1 -> 't0 M) -> 't0 M
+We spell out all the common monads, their common dedicated operations (such as `lookup` for the Reader monad), and monad transformer cousins of all of these, in an OCaml [[monad library]]. Read the linked page for details about how to use the library, and some design choices we made. Our [[State Monad Tutorial]] gives some more examples of using the library.
When a T monadic layer encloses an inner M monad, the T's interface is the most exposed one. To use operations defined in the inner M monad, you'll need to use T's `elevate` function on them. This brings the M-based operation into the T-around-M monadic bundle. Haskell calls this `lift` (MORE...)
-We said that when T is around M, you can rely on T's interface to be most exposed. That is intuitive. What you cannot also assume is that the implementing type has the T-like type structure surrounding an M-like structure. Often it will be reverse: a ListT(Maybe) is implemented by a `'a list option`, not by an `'a option list`. Until you've tried to write the code to a monadic transformer library yourself, this will probably remain counter-intuitive. But you don't need to concern yourself with it in practise. Think of what you have as a ListT(Maybe); don't worry about whether the underlying implementation is as an `'a list option` or an `'a option list` or as something more complicated. (As we noted, an `'a stateT(M)` is not an `('a M) state` nor an `('a state) M`.)
+We said that when T encloses M, you can rely on T's interface to be most exposed. That is intuitive. What you cannot also assume is that the implementing type has a Tish structure surrounding an Mish structure. Often it will be reverse: a ListT(Maybe) is implemented by a `'a list option`, not by an `'a option list`. Until you've tried to write the code to a monadic transformer library yourself, this will probably remain counter-intuitive. But you don't need to concern yourself with it in practise. Think of what you have as a ListT(Maybe); don't worry about whether the underlying implementation is as an `'a list option` or an `'a option list` or as something more complicated.
-Apart from whose interface is outermost, the behavior of a StateT(Maybe) and a MaybeT(State) will partly coincide. But in certain crucial respects they will diverge, and you need to think carefully about which behavior you want and what the appropriate layering is for your needs. (MORE...)
+Notice from the code for StateT, above, that an `'a stateT(M)` is not an `('a M) state`; neither is it an `('a state) M`. The pattern by which we transform the types from a Blah monad to a BlahT monad transformer is:
+ 't0 ---> 't0 M
+ 't1 -> 't0 ---> 't1 -> 't0 M
+ ('t1 -> 't0) -> 't0 ---> ('t1 -> 't0 M) -> 't0 M
-Ken Shan's paper [Monads for natural language semantics](http://arxiv.org/abs/cs/0205026v1) (2001) discusses how to systematically move from some plain monads to the corresponding monad transformers. But as he notes, his algorithm isn't the only one possible, and it only applies to monads whose type has a certain form. (Reader and State have thet form, List for example doesn't.)
+Ken Shan's paper [Monads for natural language semantics](http://arxiv.org/abs/cs/0205026v1) (2001) discusses how to systematically move from some plain monads to the corresponding monad transformers. But as he notes, his algorithm isn't the only one possible, and it only applies to monads whose type has a certain form. (Reader and State have that form; List for example doesn't.)
As best we know, figuring out how a monad transformer should be defined is still something of an art, not something that can be done mechanically. However, you can think that all of the art goes into deciding what ReaderT and so on should be; having figured that out, plain Reader would follow as the simple case where ReaderT is parameterized on the Identity monad.
-We spell out all the common monads, their common dedicated operations (such as `lookup` for the Reader monad), and monad transformer cousins of all of these, in an OCaml [[monad library]]. Read the linked page for details about how to use the library, and some design choices we made. Our [[State Monad Tutorial]] gives some more examples of using the library.
+Apart from whose interface is outermost, the behavior of a StateT(Maybe) and a MaybeT(State) will partly coincide. But in certain crucial respects they will diverge, and you need to think carefully about which behavior you want and what the appropriate layering is for your needs. (MORE...)
+
+Tree Monads
+===========
-Our library includes a `Tree_monad`, for binary, leaf-labeled trees. There are other kinds of trees you might want to monadize, but we took the name `Tree_monad` for this one. Like the Haskell [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) monad, our `Tree_monad` also incorporates an option-like layer. (See the comments in our library code about `plus` and `zero` for some discussion of why.)
+Our [[monad library]] includes a `Tree_monad`, for binary, leaf-labeled trees. There are other kinds of trees you might want to monadize, but we took the name `Tree_monad` for this one. Like the Haskell [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) monad, our `Tree_monad` also incorporates an Optionish layer. (See the comments in our library code about `plus` and `zero` for discussion of why.)
-So how does our `Tree_monad` behave? Simplified, its implementation would look something like this:
+So how does our `Tree_monad` behave? Simplified, its implementation looks something like this:
(* monadic operations for the Tree monad *)
| |
4 8
-Except, as we mentioned, our implementation of the Tree monad incorporates an option-like layer too. So `f' 2` should be not `Leaf 2` but `Some (Leaf 2)`. What if `f'` also mapped `1` to `None` and `4` to `Some (Node (Leaf 2, Leaf 4))`. Then binding the tree `Node (Leaf 1, Node (Leaf 2, Leaf 4))` to `f'` would delete thr branch corresponding to the original `Leaf 1`, and would splice in the results for `f' 2` and `f' 4`, yielding:
+Except, as we mentioned, our implementation of the Tree monad incorporates an Optionish layer too. So `f' 2` should be not `Leaf 2` but `Some (Leaf 2)`. What if `f'` also mapped `1` to `None` and `4` to `Some (Node (Leaf 2, Leaf 4))`. Then binding the tree `Node (Leaf 1, Node (Leaf 2, Leaf 4))` to `f'` would delete thr branch corresponding to the original `Leaf 1`, and would splice in the results for `f' 2` and `f' 4`, yielding:
.
_|__ >>= f' ~~>
| |
2 4
-As always, the functions you bind an `'a tree` to need not map `'a`s to `'a tree`s; they can map them to `'b tree`s instead. For instance, we could map `Node (Leaf 1, Node (Leaf 2, Leaf 4))` instead to `Node (Leaf "two", Node (Leaf "two", Leaf "four"))`.
+As always, the functions you bind an `'a tree` to need not map `'a`s to `'a tree`s; they can map them to `'b tree`s instead. For instance, we could transform `Node (Leaf 1, Node (Leaf 2, Leaf 4))` instead into `Node (Leaf "two", Node (Leaf "two", Leaf "four"))`.
-As we mention in the notes to the [[monad library]], the library encapsulates the implementation of its monadic types. So to work with it you have to use the primitives it provides. You can't say:
+As we [mention in the notes](/monad_library), our monad library encapsulates the implementation of its monadic types. So to work with it you have to use the primitives it provides. You can't say:
# Tree_monad.(orig_tree >>= fun a -> match a with
| 4 -> Some (Node (Leaf 2, Leaf 4))