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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.

<pre>
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
*)
</pre>

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:

<pre>
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 = <fun>
# 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
*)
</pre>

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:

<pre>
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);;
</pre>

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:

<pre>
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 = <fun>
# add' (Some 12) (Some 4);;
- : int option = Some 16
# add' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>

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.

<pre>
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
*)
</pre>

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: if `u` has type `int option`,
        then `u` 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 option `Some x`.  In the general case, this operation is
        known as `unit` or `return.` Both of those names are terrible. This
        operation is only very loosely connected to the `unit` 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 box `None`. 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 with `bool option`s).

        Intuitively, the interpretation of what `bind` does is this:
        the first argument is a monadic value `u`, which 
        evaluates to a box that (maybe) contains some ordinary value, call it `x`.
        Then the second argument uses `x` to compute a new monadic
        value.  Conceptually, then, we have

                let 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 value `u`.  In the `bind`
        operator for the option monad, we unboxed the monadic value by
        matching it with the pattern `Some x`---whenever `u`
        happened to be a box containing an integer `x`, this allowed us to
        get our hands on that `x` and feed it to `f`.

        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 the `bind` operation would probably
        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
        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 &#8902;, 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. 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 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>

        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 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 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;;
                - : 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`.

        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`, 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
`u` matches `Some x`, and `f x` evalutes to `None`, then both
computations will again result in `None`; and if the value of
`f x` matches `Some y`, then both computations will evaluate
to `g y`.

*       **Right identity: unit is a right identity for bind.**  That is, 
        `u >>= unit == u` for all monad objects `u`.  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](/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:

*       [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): 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](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
<!--    This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
        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.-->

*       [Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
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.
<!--    The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.-->


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.

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*.

##[[Reader monad for Variable Binding]]##

##[[Reader monad for Intensionality]]##