Towards Monads: Safe division
[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:
# 12/0;;
Exception: Division_by_zero.
So we want to explicitly allow for the possibility that
division will return something other than a number.
We'll use OCaml's option
type, which works like this:
# type 'a option = None | Some of 'a;;
# None;;
- : 'a option = None
# Some 3;;
- : int option = Some 3
So if a division is normal, we return some number, but if the divisor is
zero, we return None
. As a mnemonic aid, we'll append a '
to the end of our new divide function.
let div' (x:int) (y:int) = match y with 0 -> None | _ -> Some (x / y);; (* val div' : int -> int -> int option = fun # div' 12 2;; - : int option = Some 6 # div' 12 0;; - : int option = None # div' (div' 12 2) 3;; Characters 4-14: div' (div' 12 2) 3;; ^^^^^^^^^^ Error: This expression has type int option but an expression was expected of type int *)
This starts off well: dividing 12 by 2, no problem; dividing 12 by 0, just the behavior we were hoping for. But we want to be able to use the output of the safe-division function as input for further division operations. So we have to jack up the types of the inputs:
let div' (u:int option) (v:int option) = match u with None -> None | Some x -> (match v with Some 0 -> None | Some y -> Some (x / y));; (* val div' : int option -> int option -> int option = # div' (Some 12) (Some 2);; - : int option = Some 6 # div' (Some 12) (Some 0);; - : int option = None # div' (div' (Some 12) (Some 0)) (Some 3);; - : int option = None *)
Beautiful, just what we need: now we can try to divide by anything we want, without fear that we're going to trigger any system errors.
I prefer to line up the match
alternatives by using OCaml's
built-in tuple type:
let div' (u:int option) (v:int option) = match (u, v) with (None, _) -> None | (_, None) -> None | (_, Some 0) -> None | (Some x, Some y) -> Some (x / y);;
So far so good. But what if we want to combine division with other arithmetic operations? We need to make those other operations aware of the possibility that one of their arguments has triggered a presupposition failure:
let add' (u:int option) (v:int option) = match (u, v) with (None, _) -> None | (_, None) -> None | (Some x, Some y) -> Some (x + y);; (* val add' : int option -> int option -> int option = # add' (Some 12) (Some 4);; - : int option = Some 16 # add' (div' (Some 12) (Some 0)) (Some 4);; - : int option = None *)
This works, but is somewhat disappointing: the add'
operation
doesn't trigger any presupposition of its own, so it is a shame that
it needs to be adjusted because someone else might make trouble.
But we can automate the adjustment. The standard way in OCaml,
Haskell, etc., is to define a bind
operator (the name bind
is not
well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a '
after the name "bind" as well.
let bind' (u: int option) (f: int -> (int option)) = match u with None -> None | Some x -> f x;; let add' (u: int option) (v: int option) = bind' u (fun x -> bind' v (fun y -> Some (x + y)));; let div' (u: int option) (v: int option) = bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));; (* # div' (div' (Some 12) (Some 2)) (Some 3);; - : int option = Some 2 # div' (div' (Some 12) (Some 0)) (Some 3);; - : int option = None # add' (div' (Some 12) (Some 0)) (Some 3);; - : int option = None *)
Compare the new definitions of add'
and div'
closely: the definition
for add'
shows what it looks like to equip an ordinary operation to
survive in dangerous presupposition-filled world. Note that the new
definition of add'
does not need to test whether its arguments are
None objects or real numbers---those details are hidden inside of the
bind'
function.
The definition of div'
shows exactly what extra needs to be said in
order to trigger the no-division-by-zero presupposition.
[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
to do several things. First, we would have to make use of the
polymorphism of the option
type. In the arithmetic example, we only
made use of int option
s, but when we're composing natural language
expression meanings, we'll need to use types like N option
, Det option
,
VP option
, and so on. But that works automatically, because we can use
any type for the 'a
in 'a option
. Ultimately, we'd want to have a
theory of accommodation, and a theory of the situations in which
material within the sentence can satisfy presuppositions for other
material that otherwise would trigger a presupposition violation; but,
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:
A complex type that's built around some more basic type. Usually the complex type will be polymorphic, and so can apply to different basic types. In our division example, the polymorphism of the
'a option
type provides a way of building an option out of any other type of object. People often use a container metaphor: ifu
has typeint option
, thenu
is a box that (may) contain an integer.type 'a option = None | Some of 'a;;
A way to turn an ordinary value into a monadic value. In OCaml, we did this for any integer
x
by mapping it to the optionSome x
. In the general case, this operation is known asunit
orreturn.
Both of those names are terrible. This operation is only very loosely connected to theunit
type we were 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. [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 of the singleton function as an example: it takes an ordinary object and returns a set containing that object. In the example we've been considering:
let unit x = Some x;; val unit : 'a -> 'a option = <fun>
So
unit
is a way to put something inside of a monadic box. It's crucial to the usefulness of monads that there will be monadic boxes that aren't the result of that operation. In the Option/Maybe monad, for instance, there's also the empty boxNone
. In another (whimsical) example, you might have, in addition to boxes merely containing integers, special boxes that contain integers and also sing a song when they're opened.The unit/return operation will always be the simplest, conceptually 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
. 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:let bind u f = match u with None -> None | Some x -> f x;; val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
Note the type:
bind
takes two arguments: first, a monadic box (in this case, an'a option
); and second, a function from ordinary objects to monadic boxes.bind
then returns a monadic value: in this case, a'b option
(you can start with, e.g.,int option
s and end withbool option
s).Intuitively, the interpretation of what
bind
does is this: the first argument is a monadic valueu
, which evaluates to a box that (maybe) contains some ordinary value, call itx
. Then the second argument usesx
to compute a new monadic value. Conceptually, then, we havelet bind u f = (let x = unbox u in f x);;
The guts of the definition of the
bind
operation amount to specifying how to unbox the monadic valueu
. In thebind
operator for the Option monad, we unboxed the monadic value by matching it with the patternSome x
---wheneveru
happened to be a box containing an integerx
, this allowed us to get our hands on thatx
and feed it tof
.If the monadic box didn't contain any ordinary value, we instead pass through the empty box unaltered.
In a more complicated case, like our whimsical "singing box" example from before, if the monadic value happened to be a singing box containing an integer
x
, then thebind
operation would probably be defined so as to make sure that the result off x
was also a singing box. Iff
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 useful way.
So the "Option/Maybe monad" consists of the polymorphic option
type, the
unit
/return function, and the bind
function.
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).
Also, if you ever see this notation:
do
x <- u
f x
That's a Haskell shorthand for u >>= (\x -> f x)
, that is, bind u f
.
Similarly:
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. 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. (See our page on Translating between OCaml Scheme and Haskell.) It won't work in
OCaml. In fact, the <-
symbol already means something different in OCaml,
having to do with mutable record fields. We'll be discussing mutation someday
soon.)
As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:
# type 'a list
The unit
/return operation is:
# let unit x = [x];;
val unit : 'a -> 'a list = <fun>
That is, the simplest way to lift an 'a
into an 'a list
is just to make a
singleton list of that 'a
. Finally, the bind
operation is:
# let bind u f = List.concat (List.map f u);;
val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
What's going on here? Well, consider List.map f u
first. This goes through all
the members of the list u
. There may be just a single member, if u = unit x
for some x
. Or on the other hand, there may be no members, or many members. In
any case, we go through them in turn and feed them to f
. Anything that gets fed
to f
will be an 'a
. f
takes those values, and for each one, returns a 'b list
.
For example, it might return a list of all that value's divisors. Then we'll
have a bunch of 'b list
s. The surrounding List.concat ( )
converts that bunch
of 'b list
s into a single 'b list
:
# List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
- : int list = [1; 1; 2; 1; 3; 1; 2; 4]
So now we've seen two monads: the Option/Maybe monad, and the List monad. For any
monadic system, there has to be a specification of the complex monad type,
which will be parameterized on some simpler type 'a
, and the unit
/return
operation, and the bind
operation. These will be different for different
monadic systems.
Many monadic systems will also define special-purpose operations that only make sense for that system.
Although the unit
and bind
operation are defined differently for different
monadic systems, there are some general rules they always have to follow.
The Monad Laws
Just like good robots, monads must obey three laws designed to prevent 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 -> '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 forf
:# 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>
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 writeu >>= f
or equivalently( >>= ) u f
instead of>>= u 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 toint 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;; - : int option = Some 3 # unit 2 >>= divide 6;; - : int option = Some 3 # divide 6 0;; - : int option = None # 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)
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
.Wadler and others try to make this look nicer by phrasing it like this, where U, V, and W are schematic for any expressions with the relevant monadic type:
(U >>= fun x -> V) >>= fun y -> W == U >>= fun x -> (V >>= fun y -> W)
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;; - : int option = Some 3 # Some 3 >>= (fun x -> unit x >>= unit);; - : int option = Some 3 # Some 3 >>= divide 6 >>= divide 2;; - : int option = Some 1 # Some 3 >>= (fun x -> divide 6 x >>= divide 2);; - : int option = Some 1 # Some 3 >>= divide 2 >>= divide 6;; - : int option = None # Some 3 >>= (fun x -> divide 2 x >>= divide 6);; - : int option = None
Of course, associativity must hold for arbitrary functions of type
'a -> 'b m
, wherem
is the monad type. It's easy to convince yourself that thebind
operation for the Option monad obeys associativity by dividing the inputs into cases: ifu
matchesNone
, both computations will result inNone
; ifu
matchesSome x
, andf x
evalutes toNone
, then both computations will again result inNone
; and if the value off x
matchesSome y
, then both computations will evaluate tog y
.Right identity: unit is a right identity for bind. That is,
u >>= unit == u
for all monad objectsu
. For instance,# Some 3 >>= unit;; - : int option = Some 3 # None >>= unit;; - : 'a option = None
More details about monads
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, 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.
See also:
- Haskell wikibook on Monad Laws.
- Yet Another Haskell Tutorial on Monad Laws
- Haskell wikibook on Understanding Monads
- Haskell wikibook on Advanced Monads
- Haskell wikibook on do-notation
- Yet Another Haskell Tutorial on do-notation
Here are some papers that introduced monads into functional programming:
Eugenio Moggi, Notions of Computation and Monads: Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible.
Philip Wadler. The essence of functional programming: invited talk, 19'th Symposium on Principles of Programming Languages, ACM Press, Albuquerque, January 1992.
Philip Wadler. Monads for Functional Programming: in M. Broy, editor, Marktoberdorf Summer School on Program Design Calculi, Springer Verlag, NATO ASI Series F: Computer and systems sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer, editors, Advanced Functional Programming, Springer Verlag, LNCS 925, 1995. Some errata fixed August 2001.
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 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 may sometimes see:
u >> v
That just means:
u >>= fun _ -> v
that is:
bind u (fun _ -> v)
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:
# let even x = (x mod 2 = 0);;
val g : int -> bool = <fun>
even
has the type int -> bool
. Now what if we want to convert it into an operation on the Option/Maybe monad?
# let lift g = fun u -> bind u (fun x -> Some (g x));;
val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
lift even
will now be a function from int option
s to bool option
s. We can
also define a lift operation for binary functions:
# let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));;
val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = <fun>
lift2 (+)
will now be a function from int option
s and int option
s to int option
s. This should look familiar to those who did the homework.
The lift
operation (just lift
, not lift2
) is sometimes also called the map
operation. (In Haskell, they say fmap
or <$>
.) And indeed when we're working with the List monad, lift f
is exactly List.map f
!
Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used Some (g x)
and so on; in the general case we'd use unit (g x)
, using the specific unit
operation for the monad we're working with.
In general, any lift/map operation can be relied on to satisfy these laws:
* lift id = id
* lift (compose f g) = compose (lift f) (lift g)
where id
is fun x -> x
and compose f g
is fun x -> f (g x)
. If you think about the special case of the map operation on lists, this should make sense. List.map id lst
should give you back lst
again. And you'd expect these
two computations to give the same result:
List.map (fun x -> f (g x)) lst
List.map f (List.map g lst)
Another general monad operation is called ap
in Haskell---short for "apply." (They also use <*>
, but who can remember that?) This works like this:
ap [f] [x; y] = [f x; f y]
ap (Some f) (Some x) = Some (f x)
and so on. Here are the laws that any ap
operation can be relied on to satisfy:
ap (unit id) u = u
ap (ap (ap (unit compose) u) v) w = ap u (ap v w)
ap (unit f) (unit x) = unit (f x)
ap u (unit x) = ap (unit (fun f -> f x)) u
Another general monad operation is called join
. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the bind
operation for the List monad, there was a step where
we went from:
[[1]; [1;2]; [1;3]; [1;2;4]]
to:
[1; 1; 2; 1; 3; 1; 2; 4]
That is the join
operation.
All of these operations can be defined in terms of bind
and unit
; or alternatively, some of them can be taken as primitive and bind
can be defined in terms of them. Here are various interdefinitions:
lift f u = u >>= compose unit f
lift f u = ap (unit f) u
lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y)))
lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v
ap u v = u >>= (fun f -> lift f v)
ap u v = lift2 id u v
join m2 = m2 >>= id
u >>= f = join (lift f u)
u >> v = u >>= (fun _ -> v)
u >> v = lift2 (fun _ -> id) u v
Monad outlook
We're going to be using monads for a number of different things in the 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.
But first, we'll look at several linguistic applications for monads, based on what's called the Reader monad.