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
+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 monad: in each place where we had something of type `int`, we
+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
+`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 bindary operation
+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 options in a sensible way.
+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
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
+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
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
+So what exactly 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.
+* 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;;`
+ 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:
+* A way to turn an ordinary value into a monadic value. In OCaml, we
+ did this for any integer `n` by mapping it to
+ the option `Some n`. To be official, we can define a function
+ called unit:
- `let unit x = Some x;;`
+ let unit x = Some x;;
- `val unit : 'a -> 'a option = <fun>`
+ val unit : 'a -> 'a option = <fun>
- So `unit` is a way to put something inside of a box.
+ So `unit` is a way to put something inside of a box.
-* A bind operation (note the type):
+* A bind operation (note the type):
- `let bind m f = match m with None -> None | Some n -> f n;;`
+ let bind m f = match m with None -> None | Some n -> f n;;
- `val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>`
+ 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.
+ `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
+ 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);;`
+ 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`.
+ 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.
+So the "option monad" consists of the polymorphic option type, the
+unit function, and the bind function. With the option monad, we can
+think of the "safe division" operation
-A note on notation: some people use the infix operator `>==` to stand
+<pre>
+# let divide num den = if den = 0 then None else Some (num/den);;
+val divide : int -> int -> int option = <fun>
+</pre>
+
+as basically a function from two integers to an integer, except with
+this little bit of option frill, or option plumbing, on the side.
+
+A note on notation: Haskell uses 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, `*`.
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
+* **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;;
+- : int option = Some 2
# unit 2 * unit;;
- : int option = Some 2
+
+# 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
</pre>
-The parentheses is the magic for telling Ocaml that the
+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:
+* **Associativity: bind obeys a kind of associativity**. Like this:
- `(m * f) * g == m * (fun x -> f x * g)`
+ (m * f) * g == m * (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.
+ 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.
- For an illustration of associativity in the option monad:
+ Some examples of associativity in the option monad:
<pre>
-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;;
+- : 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
</pre>
Of course, associativity must hold for arbitrary functions of
-type `'a -> M 'a`, where `M` is the monad type. It's easy to
+type `'a -> 'a 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 `m`
matches `None`, both computations will result in `None`; if
`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,
+* **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
+# None * unit;;
+- : 'a option = None
</pre>
Now, if you studied algebra, you'll remember that a *monoid* is an
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
+of type `'a -> 'a m`, 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>, near the bottom).
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
+To run it, download the file, start OCaml, and say
- # #use "intensionality-monad.ml";;
+ # #use "intensionality-monad.ml";;
Note the extra `#` attached to the directive `use`.
So here's what we do:
-In Ocaml, we'll use integers to model possible worlds:
+In OCaml, we'll use integers to model possible worlds:
- type s = int;;
- type e = char;;
- type t = bool;;
+ 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.
+OCaml booleans will serve for truth values.
<pre>
type 'a intension = s -> 'a;;
We now build up some extensional meanings:
- let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+ 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`).
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 *)
+ 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;;
+ 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.