<pre>
let div' (u:int option) (v:int option) =
- match v with
+ match u with
None -> None
- | Some 0 -> None
- | Some y -> (match u with
- None -> None
- | Some x -> Some (x / y));;
+ | Some x -> (match v with
+ Some 0 -> None
+ | Some y -> Some (x / y));;
(*
val div' : int option -> int option -> int option = <fun>
most straightforward way to lift an ordinary value into a monadic value
of the monadic type in question.
-* Thirdly, an operation that's often called `bind`. This is another
+* Thirdly, an operation that's often called `bind`. As we said before, this is another
unfortunate name: this operation is only very loosely connected to
what linguists usually mean by "binding." In our option/maybe monad, the
bind operation is:
them from hurting the people that use them or themselves.
* **Left identity: unit is a left identity for the bind operation.**
- That is, for all `f:'a -> 'a m`, where `'a m` is a monadic
+ That is, for all `f:'a -> 'b m`, where `'b m` is a monadic
type, we have `(unit x) >>= f == f x`. For instance, `unit` is itself
a function of type `'a -> 'a m`, so we can use it for `f`:
function to be defined (in this case, the name of the function
is `>>=`, pronounced "bind") is an infix operator, so we write
`u >>= f` or equivalently `( >>= ) u f` instead of `>>= u
- f`. Now, for a less trivial instance of a function from `int`s
- to `int option`s:
+ f`.
# unit 2;;
- : int option = Some 2
# unit 2 >>= unit;;
- : int option = Some 2
+ Now, for a less trivial instance of a function from `int`s to `int option`s:
+
# let divide x y = if 0 = y then None else Some (x/y);;
val divide : int -> int -> int option = <fun>
# divide 6 2;;
(u >>= f) >>= g == u >>= (fun x -> f x >>= g)
- If you don't understand why the lambda form is necessary (the "fun
- x" part), you need to look again at the type of `bind`.
+ If you don't understand why the lambda form is necessary (the
+ "fun x -> ..." part), you need to look again at the type of `bind`.
Some examples of associativity in the option monad (bear in
mind that in the Ocaml implementation of integer division, 2/3
- : int option = None
Of course, associativity must hold for *arbitrary* functions of
-type `'a -> 'a m`, where `m` is the monad type. It's easy to
+type `'a -> 'b m`, where `m` is the monad type. It's easy to
convince yourself that the `bind` operation for the option monad
obeys associativity by dividing the inputs into cases: if `u`
matches `None`, both computations will result in `None`; if
Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.-->
-* [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from <https://www.cs.indiana.edu/cgi-pub/c311/doku.php?id=home> but the link above is to a local copy.
-
-There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes us laugh.
+There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you.
In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category
Theory](/advanced_topics/monads_in_category_theory) notes do so, for example.
-------------
We're going to be using monads for a number of different things in the
-weeks to come. The first main application will be the State monad,
+weeks to come. One major application will be the State monad,
which will enable us to model mutation: variables whose values appear
to change as the computation progresses. Later, we will study the
Continuation monad.
-In the meantime, we'll look at several linguistic applications for monads, based
-on what's called the *reader monad*.
+But first, we'll look at several linguistic applications for monads, based
+on what's called the *Reader monad*.
##[[Reader monad]]##