notation like this: take the singing box `u` and evaluate it (which
includes listening to the song). Take the int contained in the
singing box (the end result of evaluting `u`) and bind the variable
-`x` to that int. So `x <- u` means "Sing me up an int, and I'll call
-it `x`".
+`x` to that int. So `x <- u` means "Sing me up an int, which I'll call
+`x`".
(Note that the above "do" notation comes from Haskell. We're mentioning it here
because you're likely to see it when reading about monads. It won't work in
* **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
- type, we have `(unit x) * f == f x`. For instance, `unit` is itself
+ 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`:
# let unit x = Some x;;
val unit : 'a -> 'a option = <fun>
- # let ( * ) u f = match u with None -> None | Some x -> f x;;
- val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+ # let ( >>= ) u f = match u with None -> None | Some x -> f x;;
+ val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
The parentheses is the magic for telling OCaml that the
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 `( * ) u f` instead of `* u f`. Now:
+ 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:
# unit 2;;
- : int option = Some 2
- # unit 2 * unit;;
+ # unit 2 >>= unit;;
- : int option = Some 2
# let divide x y = if 0 = y then None else Some (x/y);;
val divide : int -> int -> int option = <fun>
# divide 6 2;;
- : int option = Some 3
- # unit 2 * divide 6;;
+ # unit 2 >>= divide 6;;
- : int option = Some 3
# divide 6 0;;
- : int option = None
- # unit 0 * divide 6;;
+ # unit 0 >>= divide 6;;
- : int option = None
* **Associativity: bind obeys a kind of associativity**. Like this:
- (u * f) * g == u * (fun x -> f x * g)
+ (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`.
- Some examples of associativity in the option monad:
+ Some examples of associativity in the option monad (bear in
+ mind that in the Ocaml implementation of integer division, 2/3
+ evaluates to zero, throwing away the remainder):
- # Some 3 * unit * unit;;
+ # Some 3 >>= unit >>= unit;;
- : int option = Some 3
- # Some 3 * (fun x -> unit x * unit);;
+ # Some 3 >>= (fun x -> unit x >>= unit);;
- : int option = Some 3
- # Some 3 * divide 6 * divide 2;;
+ # Some 3 >>= divide 6 >>= divide 2;;
- : int option = Some 1
- # Some 3 * (fun x -> divide 6 x * divide 2);;
+ # Some 3 >>= (fun x -> divide 6 x >>= divide 2);;
- : int option = Some 1
- # Some 3 * divide 2 * divide 6;;
+ # Some 3 >>= divide 2 >>= divide 6;;
- : int option = None
- # Some 3 * (fun x -> divide 2 x * divide 6);;
+ # Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
- : int option = None
Of course, associativity must hold for *arbitrary* functions of
to `g y`.
* **Right identity: unit is a right identity for bind.** That is,
- `u * unit == u` for all monad objects `u`. For instance,
+ `u >>= unit == u` for all monad objects `u`. For instance,
- # Some 3 * unit;;
+ # Some 3 >>= unit;;
- : int option = Some 3
- # None * unit;;
+ # None >>= unit;;
- : 'a option = None
If you studied algebra, you'll remember that a *monoid* is an
associative operation with a left and right identity. For instance,
the natural numbers along with multiplication form a monoid with 1
-serving as the left and right identity. That is, temporarily using
-`*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
+serving as the left and right identity. That is, `1 * u == u == u * 1` for all
`u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`. As
presented here, a monad is not exactly a monoid, because (unlike the
arguments of a monoid operation) the two arguments of the bind are of
different types. But it's possible to make the connection between
monads and monoids much closer. This is discussed in [Monads in Category
-Theory](/advanced_notes/monads_in_category_theory).
+Theory](/advanced_topics/monads_in_category_theory).
See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
Here are some papers that introduced monads into functional programming:
* [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 me laugh.
+There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes us laugh.
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_notes/monads_in_category_theory) notes do so, for example.
+Theory](/advanced_topics/monads_in_category_theory) notes do so, for example.
Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.
You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
-The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example:
+The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example:
# let even x = (x mod 2 = 0);;
val g : int -> bool = <fun>