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Monads
------

Start by (re)reading the discussion of monads in the lecture notes for
week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
In those notes, we saw a way to separate thining 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 monad: 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 just an integer, or else some special value which
we could interpret as signaling that something had 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
ints and int options.  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 bindary operation
on plain integers into a lifted operation that understands how to deal
with int options 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
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 options, but when we're composing natural language
expression meanings, we'll need to use types like `N int`, `Det Int`,
`VP int`, 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.]

So what examctly is a monad?  As usual, we're not going to be pedantic
about it, but for our purposes, we can consider a monad to be a system
that provides at least the following three elements:

* A way to build a complex type from some basic type.  In the 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 `x` has type `int option`, then `x` 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 n by mapping an arbitrary integer `n` to
  the option `Some n`.  To be official, we can define a function
  called unit:

    let unit x = Some x;;
    val unit : 'a -> 'a option = <fun>

    So `unit` is a way to put something inside of a box.

* A bind operation (note the type):

     let bind m f = match m with None -> None | Some n -> f n;;
     val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>

     `bind` takes two arguments (a monadic object and a function from
     ordinary objects to monadic objects), and returns a monadic
     object.

     Intuitively, the interpretation of what `bind` does is like this:
     the first argument computes a monadic object m, which will
     evaluate to a box containing some ordinary value, call it `x`.
     Then the second argument uses `x` to compute a new monadic
     value.  Conceptually, then, we have

    let bind m f = (let x = unwrap m in f x);;

    The guts of the definition of the `bind` operation amount to
    specifying how to unwrap the monadic object `m`.  In the bind
    opertor for the option monad, we unwraped the option monad by
    matching the monadic object `m` with `Some n`--whenever `m`
    happend to be a box containing an integer `n`, this allowed us to
    get our hands on that `n` and feed it to `f`.

So the "Option monad" consists of the polymorphic option type, the
unit function, and the bind function.  

A note on notation: some people use the infix operator `>==` to stand
for `bind`.  I really hate that symbol.  Following Wadler, I prefer to
infix five-pointed star, or on a keyboard, `*`.


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 -> 'a M`, where `'a M` is a monadic
     object, we have `(unit x) * f == f x`.  For instance, `unit` is a
     function of type `'a -> 'a option`, so we have

<pre>
# let ( * ) m f = match m with None -> None | Some n -> f n;;
val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
# let unit x = Some x;;
val unit : 'a -> 'a option = <fun>
# unit 2 * unit;;
- : int option = Some 2
</pre>

        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
        `m * f` or `( * ) m f` instead of `* m f`.

*    Associativity: bind obeys a kind of associativity, like this:

    (m * f) * g == m * (fun x -> f x * g)

    If you don't understand why the lambda form is necessary, you need
    to look again at the type of bind.  This is important.  

    For an illustration of associativity in the option monad:

<pre>
Some 3 * unit * unit;; 
- : int option = Some 3
Some 3 * (fun x -> unit x * unit);;
- : int option = Some 3
</pre>

        Of course, associativity must hold for arbitrary functions of
        type `'a -> M 'a`, 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 `m`
        matches `None`, both computations will result in `None`; if
        `m` matches `Some n`, and `f n` evalutes to `None`, then both
        computations will again result in `None`; and if the value of
        `f n` matches `Some r`, then both computations will evaluate
        to `g r`.

*    Right identity: unit is a right identity for bind.  That is, 
     `m * unit == m` for all monad objects `m`.  For instance,

<pre>
# Some 3 * unit;;
- : int option = Some 3
</pre>

Now, 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 * n == n == n * 1` for all
`n`, and `(a * b) * c == a * (b * c)` for all `a`, `b`, and `c`.  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 if we generalize bind so that both arguments are
of type `'a -> M 'a`, then we get plain identity laws and
associativity laws, and the monad laws are exactly like the monoid
laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>).


Monad outlook
-------------

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,
which will enable us to model mutation: variables whose values appear
to change as the computation progresses.  Later, we will study the
Continuation monad.

The intensionality monad
------------------------

In the meantime, we'll see a linguistic application for monads:
intensional function application.  In Shan (2001) [Monads for natural
language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that
making expressions sensitive to the world of evaluation is
conceptually the same thing as making use of a *reader monad* (which
we'll see again soon).  This technique was beautifully re-invented
by Ben-Avi and Winter (2007) in their paper [A modular
approach to
intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
though without explicitly using monads.

All of the code in the discussion below can be found here: [[intensionality-monad.ml]].
To run it, download the file, start Ocaml, and say `# #use
"intensionality-monad.ml";;`. 

Here's the idea: since people can have different attitudes towards
different propositions that happen to have the same truth value, we
can't have sentences denoting simple truth values.  Then if John
believed that the earth was round, it would force him to believe
Fermat's last theorem holds, since both propositions are equally true.
The traditional solution is to allow sentences to denote a function
from worlds to truth values, what Montague called an intension.  
So if `s` is the type of possible worlds, we have the following
situation:


<pre>
Extensional types                 Intensional types       Examples
-------------------------------------------------------------------

S         s->t                    s->t                    John left
DP        s->e                    s->e                    John
VP        s->e->t                 s->(s->e)->t            left
Vt        s->e->e->t              s->(s->e)->(s->e)->t    saw
Vs        s->t->e->t              s->(s->t)->(s->e)->t    thought
</pre>

This system is modeled on the way Montague arranged his grammar.
(There are significant simplifications: for instance, determiner
phrases are thought of as corresponding to individuals rather than to
generalized quantifiers.)  If you're curious about the initial `s`'s
in the extensional types, they're there because the behavior of these
expressions depends on which world they're evaluated at.  If you are
in a situation in which you can hold the evaluation world constant,
you can further simplify the extensional types.  (Usually, the
dependence of the extension of an expression on the evaluation world
is hidden in a superscript, or built into the lexical interpretation
function.)

The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
from worlds to extensions: intransitive verb phrases like "left" now
take intensional concepts as arguments (type s->e) rather than plain
individuals (type e), and attitude verbs like "think" now take
propositions (type s->t) rather than truth values (type t).

The intenstional types are more complicated than the intensional
types.  Wouldn't it be nice to keep the complicated types to just
those attitude verbs that need to worry about intensions, and keep the
rest of the grammar as extensional as possible?  This desire is
parallel to our earlier desire to limit the concern about division by
zero to the division function, and let the other functions ignore
division-by-zero problems as much as possible.

So here's what we do:

In Ocaml, we'll use integers to model possible worlds:

    type s = int;;
    type e = char;;
    type t = bool;;

Characters (characters in the computational sense, i.e., letters like
`'a'` and `'b'`, not Kaplanian characters) will model individuals, and
Ocaml booleans will serve for truth values.

<pre>
type 'a intension = s -> 'a;;
let unit x (w:s) = x;;

let ann = unit 'a';;
let bill = unit 'b';;
let cam = unit 'c';;
</pre>

In our monad, the intension of an extensional type `'a` is `s -> 'a`,
a function from worlds to extensions.  Our unit will be the constant
function (an instance of the K combinator) that returns the same
individual at each world.

Then `ann = unit 'a'` is a rigid designator: a constant function from
worlds to individuals that returns `'a'` no matter which world is used
as an argument.

Let's test compliance with the left identity law:

<pre>
# let bind m f (w:s) = f (m w) w;;
val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = <fun>
# bind (unit 'a') unit 1;;
- : char = 'a'
</pre>

We'll assume that this and the other laws always hold.

We now build up some extensional meanings:

    let left w x = match (w,x) with (2,'c') -> false | _ -> true;;

This function says that everyone always left, except for Cam in world
2 (i.e., `left 2 'c' == false`).

Then the way to evaluate an extensional sentence is to determine the
extension of the verb phrase, and then apply that extension to the
extension of the subject:

<pre>
let extapp fn arg w = fn w (arg w);;

extapp left ann 1;;
# - : bool = true

extapp left cam 2;;
# - : bool = false
</pre>

`extapp` stands for "extensional function application".
So Ann left in world 1, but Cam didn't leave in world 2.

A transitive predicate:

    let saw w x y = (w < 2) && (y < x);;
    extapp (extapp saw bill) ann 1;; (* true *)
    extapp (extapp saw bill) ann 2;; (* false *)

In world 1, Ann saw Bill and Cam, and Bill saw Cam.  No one saw anyone
in world two.

Good.  Now for intensions:

    let intapp fn arg w = fn w arg;;

The only difference between intensional application and extensional
application is that we don't feed the evaluation world to the argument.
(See Montague's rules of (intensional) functional application, T4 -- T10.)
In other words, instead of taking an extension as an argument,
Montague's predicates take a full-blown intension.  

But for so-called extensional predicates like "left" and "saw", 
the extra power is not used.  We'd like to define intensional versions
of these predicates that depend only on their extensional essence.
Just as we used bind to define a version of addition that interacted
with the option monad, we now use bind to intensionalize an
extensional verb:

<pre>
let lift pred w arg = bind arg (fun x w -> pred w x) w;;

intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
</pre>

Because `bind` unwraps the intensionality of the argument, when the
lifted "left" receives an individual concept (e.g., `unit 'a'`) as
argument, it's the extension of the individual concept (i.e., `'a'`)
that gets fed to the basic extensional version of "left".  (For those
of you who know Montague's PTQ, this use of bind captures Montague's
third meaning postulate.)

Likewise for extensional transitive predicates like "saw":

<pre>
let lift2 pred w arg1 arg2 = 
  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
</pre>

Crucially, an intensional predicate does not use `bind` to consume its
arguments.  Attitude verbs like "thought" are intensional with respect
to their sentential complement, but extensional with respect to their
subject (as Montague noticed, almost all verbs in English are
extensional with respect to their subject; a possible exception is "appear"):

<pre>
let think (w:s) (p:s->t) (x:e) = 
  match (x, p 2) with ('a', false) -> false | _ -> p w;;
</pre>

Ann disbelieves any proposition that is false in world 2.  Apparently,
she firmly believes we're in world 2.  Everyone else believes a
proposition iff that proposition is true in the world of evaluation.

<pre>
intapp (lift (intapp think
                     (intapp (lift left)
                             (unit 'b'))))
       (unit 'a') 
1;; (* true *)
</pre>

So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).

The `lift` is there because "think Bill left" is extensional wrt its
subject.  The important bit is that "think" takes the intension of
"Bill left" as its first argument.

<pre>
intapp (lift (intapp think
                     (intapp (lift left)
                             (unit 'c'))))
       (unit 'a') 
1;; (* false *)
</pre>

But even in world 1, Ann doesn't believe that Cam left (even though he
did: `intapp (lift left) cam 1 == true`).  Ann's thoughts are hung up
on what is happening in world 2, where Cam doesn't leave.

*Small project*: add intersective ("red") and non-intersective
 adjectives ("good") to the fragment.  The intersective adjectives
 will be extensional with respect to the nominal they combine with
 (using bind), and the non-intersective adjectives will take
 intensional arguments.

Finally, note that within an intensional grammar, extensional funtion
application is essentially just bind:

<pre>
# let swap f x y = f y x;;
# bind cam (swap left) 2;;
- : bool = false
</pre>