[This section used to be near the end of the lecture notes for week 6]
+We begin by reasoning about what should happen when someone tries to
+divide by zero. This will lead us to a general programming technique
+called a *monad*, which we'll see in many guises in the weeks to come.
+
Integer division presupposes that its second argument
(the divisor) is not zero, upon pain of presupposition failure.
Here's what my OCaml interpreter says:
The definition of `div'` shows exactly what extra needs to be said in
order to trigger the no-division-by-zero presupposition.
-For linguists: this is a complete theory of a particularly simply form
-of presupposition projection (every predicate is a hole).
-
-
-
-
-Monads in General
------------------
-
-Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [[Towards Monads]].
-In those notes, we saw a way to separate thinking about error
-conditions (such as trying to divide by zero) from thinking about
-normal arithmetic computations. We did this by making use of the
-`option` type: in each place where we had something of type `int`, we
-put instead something of type `int option`, which is a sum type
-consisting either of one choice with an `int` payload, or else a `None`
-choice which we interpret as signaling that something has gone wrong.
-
-The goal was to make normal computing as convenient as possible: when
-we're adding or multiplying, we don't have to worry about generating
-any new errors, so we do want to think about the difference between
-`int`s and `int option`s. We tried to accomplish this by defining a
-`bind` operator, which enabled us to peel away the `option` husk to get
-at the delicious integer inside. There was also a homework problem
-which made this even more convenient by mapping any binary operation
-on plain integers into a lifted operation that understands how to deal
-with `int option`s in a sensible way.
-
[Linguitics note: Dividing by zero is supposed to feel like a kind of
presupposition failure. If we wanted to adapt this approach to
building a simple account of presupposition projection, we would have
not surprisingly, these refinements will require some more
sophisticated techniques than the super-simple option monad.]
+
+Monads in General
+-----------------
+
+We've just seen a way to separate thinking about error conditions
+(such as trying to divide by zero) from thinking about normal
+arithmetic computations. We did this by making use of the `option`
+type: in each place where we had something of type `int`, we put
+instead something of type `int option`, which is a sum type consisting
+either of one choice with an `int` payload, or else a `None` choice
+which we interpret as signaling that something has gone wrong.
+
+The goal was to make normal computing as convenient as possible: when
+we're adding or multiplying, we don't have to worry about generating
+any new errors, so we would rather not think about the difference
+between `int`s and `int option`s. We tried to accomplish this by
+defining a `bind` operator, which enabled us to peel away the `option`
+husk to get at the delicious integer inside. There was also a
+homework problem which made this even more convenient by defining a
+`lift` operator that mapped any binary operation on plain integers
+into a lifted operation that understands how to deal with `int
+option`s in a sensible way.
+
So what exactly is a monad? We can consider a monad to be a system
that provides at least the following three elements:
discussing earlier (whose value is written `()`). It's also only
very loosely connected to the "return" keyword in many other
programming languages like C. But these are the names that the literature
- uses.
+ uses. [The rationale for "unit" comes from the monad laws
+ (see below), where the unit function serves as an identity,
+ just like the unit number (i.e., 1) serves as the identity
+ object for multiplication. The rationale for "return" comes
+ from a misguided desire to resonate with C programmers and
+ other imperative types.]
The unit/return operation is a way of lifting an ordinary object into
the monadic box you've defined, in the simplest way possible. You can think
be defined so as to make sure that the result of `f x` was also
a singing box. If `f` also wanted to insert a song, you'd have to decide
whether both songs would be carried through, or only one of them.
+ (Are you beginning to realize how wierd and wonderful monads
+ can be?)
There is no single `bind` function that dictates how this must go.
For each new monadic type, this has to be worked out in an
`unit`/return function, and the `bind` function.
-A note on notation: Haskell uses the infix operator `>>=` to stand
-for `bind`. Chris really hates that symbol. Following Wadler, he prefers to
-use an infix five-pointed star ⋆, or on a keyboard, `*`. Jim on the other hand
-thinks `>>=` is what the literature uses and students won't be able to
-avoid it. Moreover, although ⋆ is OK (though not a convention that's been picked up), overloading the multiplication symbol invites its own confusion
-and Jim feels very uneasy about that. If not `>>=` then we should use
-some other unfamiliar infix symbol (but `>>=` already is such...)
+A note on notation: Haskell uses the infix operator `>>=` to stand for
+`bind`: wherever you see `u >>= f`, that means `bind u f`.
+Wadler uses ⋆, but that hasn't been widely adopted (unfortunately).
-In any case, the course leaders will work this out somehow. In the meantime,
-as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
-if you ever see this notation:
+Also, if you ever see this notation:
do
x <- u
y <- v
f x y
-is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x
--> bind v (fun y -> f x y))`. Those who did last week's homework may recognize
-this last expression.
+is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u
+(fun x -> bind v (fun y -> f x y))`. Those who did last week's
+homework may recognize this last expression. You can think of the
+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, 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>